Notation by work
Paper XI: Packets, Arithmetic Realization, and Conditional Comparison
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671 symbol spellings in 1417 mathematical passages across 23 selected source files. Subscripts and superscripts have their own entries alongside their base symbols.
The same symbol can have several meanings here. A passage classified as a definition may define another part of its expression. Read the linked context before assigning a meaning.
All 671 entries are available below.
+Definition passage · 88 passages
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Definition passage: Schrödinger operators · Read passage · Paper_XI_Traditional_Final.tex, line 816
[Schrödinger operators] On H= C^X, define (ρ(x,ξ,z)f)(u) = ψ(z+ξ(u)+frac12ξ(x))f(u+x).
Exact mathematical passage
\bigl(\rho(x,\xi,z)f\bigr)(u) = \psi\left(z+\xi(u)+\frac12\xi(x)\right)f(u+x).
=Definition passage · 451 passages
Search aliases: =
Definition passage: Monodromy packet chain · Read passage · Paper_XI_Traditional_Final.tex, line 251
[Monodromy packet chain] Let V_0,…,V_m-1 be copies of the space of τ. Put V=⊕_j=0^m-1V_j. A monodromy packet chain is a nilpotent operator N with N(V_j)⊆ V_j-1 for j>0, with N(V_0)=0, together with a Frobenius operator whose conjugation scales N by q^-1.
Exact mathematical passage
V=\bigoplus_{j=0}^{m-1}V_j.
ADefinition passage · 78 passages
Search aliases: A
Definition passage: One break Clifford datum · Read passage · Paper_XI_Traditional_Final.tex, line 1016
[One break Clifford datum] A one break Clifford datum consists of finite groups and representations Ntriangleleft G, A=G/N, Π:G→GL( X), ρ:N→GL(W), with the following properties.
Exact mathematical passage
N\triangleleft G, \qquad A=G/N, \qquad \Pi:G\to\GL(\calX), \qquad \rho:N\to\GL(W),
aDefinition passage · 114 passages
Search aliases: a
Definition passage: Clifford cocycle class · Read passage · Paper_XI_Traditional_Final.tex, line 1064
[Clifford cocycle class] Choose scalars c(a,b) such that T_aT_b=c(a,b)T_a+b. The cohomology class [c_ρ]∈ H^2(A, C^×) is the Clifford cocycle class of the datum.
Exact mathematical passage
c(a,b)
A_{v_1v_0}Definition passage · 1 passage
Search aliases: A_v_1v_0
Definition passage: Path transport · Read passage · Paper_XI_Traditional_Final.tex, line 1625
[Path transport] For an oriented path γ=(v_0→ v_1→⋯→ v_m), define A_γ=A_v_m v_m-1⋯ A_v_1v_0, B_γ=B_v_m v_m-1⋯ B_v_1v_0. For a loop based at r, these are the based holonomies.
Exact mathematical passage
A_\gamma=A_{v_m v_{m-1}}\cdots A_{v_1v_0}, \qquad B_\gamma=B_{v_m v_{m-1}}\cdots B_{v_1v_0}.
A_{v_m v_{m-1}}Definition passage · 1 passage
Search aliases: A_v_m v_m-1
Definition passage: Path transport · Read passage · Paper_XI_Traditional_Final.tex, line 1625
[Path transport] For an oriented path γ=(v_0→ v_1→⋯→ v_m), define A_γ=A_v_m v_m-1⋯ A_v_1v_0, B_γ=B_v_m v_m-1⋯ B_v_1v_0. For a loop based at r, these are the based holonomies.
Exact mathematical passage
A_\gamma=A_{v_m v_{m-1}}\cdots A_{v_1v_0}, \qquad B_\gamma=B_{v_m v_{m-1}}\cdots B_{v_1v_0}.
A_\gammaDefinition passage · 3 passages
Search aliases: A_γ gamma
Definition passage: Path transport · Read passage · Paper_XI_Traditional_Final.tex, line 1625
[Path transport] For an oriented path γ=(v_0→ v_1→⋯→ v_m), define A_γ=A_v_m v_m-1⋯ A_v_1v_0, B_γ=B_v_m v_m-1⋯ B_v_1v_0. For a loop based at r, these are the based holonomies.
Exact mathematical passage
A_\gamma=A_{v_m v_{m-1}}\cdots A_{v_1v_0}, \qquad B_\gamma=B_{v_m v_{m-1}}\cdots B_{v_1v_0}.
\absDefinition passage · 24 passages
Search aliases: abs
Definition passage: Normalized Gauss phase · Read passage · Paper_XI_Traditional_Final.tex, line 750
[Normalized Gauss phase] For nontrivial χ, define γ(χ,ψ) = q^-1/2G(χ,ψ). Then absγ(χ,ψ)=1.
Exact mathematical passage
\abs{\gamma(\chi,\psi)}=1
\ArfDefinition passage · 7 passages
Search aliases: Arf operatorname
Definition passage: Arf invariant · Read passage · Paper_XI_Traditional_Final.tex, line 1493
[Arf invariant] Choose a symplectic basis (e_i,f_i)_i=1^r. Define Arf(q) = ∑_i=1^rq(e_i)q(f_i) ∈ F_2.
Exact mathematical passage
\Arf(q) = \sum_{i=1}^{r}q(e_i)q(f_i) \in\FF_2.
BDefinition passage · 36 passages
Search aliases: B
Definition passage: Path transport · Read passage · Paper_XI_Traditional_Final.tex, line 1625
[Path transport] For an oriented path γ=(v_0→ v_1→⋯→ v_m), define A_γ=A_v_m v_m-1⋯ A_v_1v_0, B_γ=B_v_m v_m-1⋯ B_v_1v_0. For a loop based at r, these are the based holonomies.
Exact mathematical passage
A_\gamma=A_{v_m v_{m-1}}\cdots A_{v_1v_0}, \qquad B_\gamma=B_{v_m v_{m-1}}\cdots B_{v_1v_0}.
bDefinition passage · 56 passages
Search aliases: b
Definition passage: Clifford cocycle class · Read passage · Paper_XI_Traditional_Final.tex, line 1064
[Clifford cocycle class] Choose scalars c(a,b) such that T_aT_b=c(a,b)T_a+b. The cohomology class [c_ρ]∈ H^2(A, C^×) is the Clifford cocycle class of the datum.
Exact mathematical passage
c(a,b)
B_{v_1v_0}Definition passage · 1 passage
Search aliases: B_v_1v_0
Definition passage: Path transport · Read passage · Paper_XI_Traditional_Final.tex, line 1625
[Path transport] For an oriented path γ=(v_0→ v_1→⋯→ v_m), define A_γ=A_v_m v_m-1⋯ A_v_1v_0, B_γ=B_v_m v_m-1⋯ B_v_1v_0. For a loop based at r, these are the based holonomies.
Exact mathematical passage
A_\gamma=A_{v_m v_{m-1}}\cdots A_{v_1v_0}, \qquad B_\gamma=B_{v_m v_{m-1}}\cdots B_{v_1v_0}.
B_{v_m v_{m-1}}Definition passage · 1 passage
Search aliases: B_v_m v_m-1
Definition passage: Path transport · Read passage · Paper_XI_Traditional_Final.tex, line 1625
[Path transport] For an oriented path γ=(v_0→ v_1→⋯→ v_m), define A_γ=A_v_m v_m-1⋯ A_v_1v_0, B_γ=B_v_m v_m-1⋯ B_v_1v_0. For a loop based at r, these are the based holonomies.
Exact mathematical passage
A_\gamma=A_{v_m v_{m-1}}\cdots A_{v_1v_0}, \qquad B_\gamma=B_{v_m v_{m-1}}\cdots B_{v_1v_0}.
B_\gammaDefinition passage · 3 passages
Search aliases: B_γ gamma
Definition passage: Path transport · Read passage · Paper_XI_Traditional_Final.tex, line 1625
[Path transport] For an oriented path γ=(v_0→ v_1→⋯→ v_m), define A_γ=A_v_m v_m-1⋯ A_v_1v_0, B_γ=B_v_m v_m-1⋯ B_v_1v_0. For a loop based at r, these are the based holonomies.
Exact mathematical passage
A_\gamma=A_{v_m v_{m-1}}\cdots A_{v_1v_0}, \qquad B_\gamma=B_{v_m v_{m-1}}\cdots B_{v_1v_0}.
b_\rhoDefinition passage · 5 passages
Search aliases: b_ρ rho
Definition passage: One break symplectic envelope · Read passage · Paper_XI_Traditional_Final.tex, line 1094
[One break symplectic envelope] The one break symplectic envelope is V_ρ=A/rad b_ρ.
Exact mathematical passage
\boxed{ V_\rho=A/\rad b_\rho. }
CDefinition passage · 39 passages
Search aliases: C
Definition passage: Bridge datum · Read passage · Paper_XI_Traditional_Final.tex, line 1206
item a one break Galois envelope (V_ρ,b_ρ) item a type side quotient (V_θ,h_θ) item an equivariant symplectic isometry ι:V_ρ⟶ V_θ for a finite cyclic stabilizer C=⟨ c⟩ item the same nontrivial central character on the two Heisenberg groups item when p=2, compatible quadratic refinements and an isomorphism of the resulting Heisenberg central extensions lifting ι.
Exact mathematical passage
C=\langle c\rangle
cDefinition passage · 57 passages
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Definition passage: Clifford cocycle class · Read passage · Paper_XI_Traditional_Final.tex, line 1064
[Clifford cocycle class] Choose scalars c(a,b) such that T_aT_b=c(a,b)T_a+b. The cohomology class [c_ρ]∈ H^2(A, C^×) is the Clifford cocycle class of the datum.
Exact mathematical passage
c(a,b)
c_\rhoDefinition passage · 2 passages
Search aliases: c_ρ rho
Definition passage: Clifford cocycle class · Read passage · Paper_XI_Traditional_Final.tex, line 1069
[Clifford cocycle class] Choose scalars c(a,b) such that T_aT_b=c(a,b)T_a+b. The cohomology class [c_ρ]∈ H^2(A, C^×) is the Clifford cocycle class of the datum.
Exact mathematical passage
[c_\rho]\in H^2(A,\CC^{\times})
\calHDefinition passage · 4 passages
Search aliases: calH H mathcal
Definition passage: Schrödinger operators · Read passage · Paper_XI_Traditional_Final.tex, line 815
[Schrödinger operators] On H= C^X, define (ρ(x,ξ,z)f)(u) = ψ(z+ξ(u)+frac12ξ(x))f(u+x).
Exact mathematical passage
\calH=\CC^X
\calXDefinition passage · 4 passages
Search aliases: calX X mathcal
Definition passage: One break Clifford datum · Read passage · Paper_XI_Traditional_Final.tex, line 1016
[One break Clifford datum] A one break Clifford datum consists of finite groups and representations Ntriangleleft G, A=G/N, Π:G→GL( X), ρ:N→GL(W), with the following properties.
Exact mathematical passage
N\triangleleft G, \qquad A=G/N, \qquad \Pi:G\to\GL(\calX), \qquad \rho:N\to\GL(W),
\CCDefinition passage · 22 passages
Search aliases: CC C mathbb
Definition passage: Schrödinger operators · Read passage · Paper_XI_Traditional_Final.tex, line 815
[Schrödinger operators] On H= C^X, define (ρ(x,ξ,z)f)(u) = ψ(z+ξ(u)+frac12ξ(x))f(u+x).
Exact mathematical passage
\calH=\CC^X
\CC^XDefinition passage · 2 passages
Search aliases: CC^X C^X CC mathbb
Definition passage: Schrödinger operators · Read passage · Paper_XI_Traditional_Final.tex, line 815
[Schrödinger operators] On H= C^X, define (ρ(x,ξ,z)f)(u) = ψ(z+ξ(u)+frac12ξ(x))f(u+x).
Exact mathematical passage
\calH=\CC^X
\CC^{\times}Definition passage · 8 passages
Search aliases: CC^× C^× CC times mathbb times
Definition passage: Clifford cocycle class · Read passage · Paper_XI_Traditional_Final.tex, line 1069
[Clifford cocycle class] Choose scalars c(a,b) such that T_aT_b=c(a,b)T_a+b. The cohomology class [c_ρ]∈ H^2(A, C^×) is the Clifford cocycle class of the datum.
Exact mathematical passage
[c_\rho]\in H^2(A,\CC^{\times})
\detDefinition passage · 19 passages
Search aliases: det
Definition passage: Odd finite Weil phase · Read passage · Paper_XI_Traditional_Final.tex, line 1312
[Odd finite Weil phase] For fixed point free g, define Φ_p(g,V) = (frac-1p)^r (fracdet(g-I)p).
Exact mathematical passage
\Phi_p(g,V) = \left(\frac{-1}{p}\right)^r \left(\frac{\det(g-I)}{p}\right).
eDefinition passage · 34 passages
Search aliases: e
Definition passage: Arf invariant · Read passage · Paper_XI_Traditional_Final.tex, line 1492
[Arf invariant] Choose a symplectic basis (e_i,f_i)_i=1^r. Define Arf(q) = ∑_i=1^rq(e_i)q(f_i) ∈ F_2.
Exact mathematical passage
(e_i,f_i)_{i=1}^{r}
e_iDefinition passage · 2 passages
Search aliases: e_i
Definition passage: Arf invariant · Read passage · Paper_XI_Traditional_Final.tex, line 1492
[Arf invariant] Choose a symplectic basis (e_i,f_i)_i=1^r. Define Arf(q) = ∑_i=1^rq(e_i)q(f_i) ∈ F_2.
Exact mathematical passage
(e_i,f_i)_{i=1}^{r}
fDefinition passage · 39 passages
Search aliases: f
Definition passage: Schrödinger operators · Read passage · Paper_XI_Traditional_Final.tex, line 816
[Schrödinger operators] On H= C^X, define (ρ(x,ξ,z)f)(u) = ψ(z+ξ(u)+frac12ξ(x))f(u+x).
Exact mathematical passage
\bigl(\rho(x,\xi,z)f\bigr)(u) = \psi\left(z+\xi(u)+\frac12\xi(x)\right)f(u+x).
f_iDefinition passage · 2 passages
Search aliases: f_i
Definition passage: Arf invariant · Read passage · Paper_XI_Traditional_Final.tex, line 1492
[Arf invariant] Choose a symplectic basis (e_i,f_i)_i=1^r. Define Arf(q) = ∑_i=1^rq(e_i)q(f_i) ∈ F_2.
Exact mathematical passage
(e_i,f_i)_{i=1}^{r}
\FFDefinition passage · 37 passages
Search aliases: FF F mathbb
Definition passage: Quadratic refinement · Read passage · Paper_XI_Traditional_Final.tex, line 1474
[Quadratic refinement] A quadratic refinement of b is a map q:V→ F_2 satisfying q(x+y)-q(x)-q(y)=b(x,y).
Exact mathematical passage
q:V\to\FF_2
\FF_2Definition passage · 10 passages
Search aliases: FF_2 F_2 FF mathbb
Definition passage: Quadratic refinement · Read passage · Paper_XI_Traditional_Final.tex, line 1474
[Quadratic refinement] A quadratic refinement of b is a map q:V→ F_2 satisfying q(x+y)-q(x)-q(y)=b(x,y).
Exact mathematical passage
q:V\to\FF_2
GDefinition passage · 31 passages
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Definition passage: Normalized Gauss phase · Read passage · Paper_XI_Traditional_Final.tex, line 745
[Normalized Gauss phase] For nontrivial χ, define γ(χ,ψ) = q^-1/2G(χ,ψ). Then absγ(χ,ψ)=1.
Exact mathematical passage
\gamma(\chi,\psi) = q^{-1/2}G(\chi,\psi).
gDefinition passage · 61 passages
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Definition passage: Odd finite Weil phase · Read passage · Paper_XI_Traditional_Final.tex, line 1311
[Odd finite Weil phase] For fixed point free g, define Φ_p(g,V) = (frac-1p)^r (fracdet(g-I)p).
Exact mathematical passage
g
\GLDefinition passage · 20 passages
Search aliases: GL mathrm GL operatorname
Definition passage: One break Clifford datum · Read passage · Paper_XI_Traditional_Final.tex, line 1016
[One break Clifford datum] A one break Clifford datum consists of finite groups and representations Ntriangleleft G, A=G/N, Π:G→GL( X), ρ:N→GL(W), with the following properties.
Exact mathematical passage
N\triangleleft G, \qquad A=G/N, \qquad \Pi:G\to\GL(\calX), \qquad \rho:N\to\GL(W),
HDefinition passage · 58 passages
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Definition passage: Clifford cocycle class · Read passage · Paper_XI_Traditional_Final.tex, line 1069
[Clifford cocycle class] Choose scalars c(a,b) such that T_aT_b=c(a,b)T_a+b. The cohomology class [c_ρ]∈ H^2(A, C^×) is the Clifford cocycle class of the datum.
Exact mathematical passage
[c_\rho]\in H^2(A,\CC^{\times})
hDefinition passage · 41 passages
Search aliases: h
Definition passage: Bridge datum · Read passage · Paper_XI_Traditional_Final.tex, line 1201
item a one break Galois envelope (V_ρ,b_ρ) item a type side quotient (V_θ,h_θ) item an equivariant symplectic isometry ι:V_ρ⟶ V_θ for a finite cyclic stabilizer C=⟨ c⟩ item the same nontrivial central character on the two Heisenberg groups item when p=2, compatible quadratic refinements and an isomorphism of the resulting Heisenberg central extensions lifting ι.
Exact mathematical passage
(V_\theta,h_\theta)
H^2Definition passage · 1 passage
Search aliases: H^2
Definition passage: Clifford cocycle class · Read passage · Paper_XI_Traditional_Final.tex, line 1069
[Clifford cocycle class] Choose scalars c(a,b) such that T_aT_b=c(a,b)T_a+b. The cohomology class [c_ρ]∈ H^2(A, C^×) is the Clifford cocycle class of the datum.
Exact mathematical passage
[c_\rho]\in H^2(A,\CC^{\times})
h_\thetaDefinition passage · 5 passages
Search aliases: h_θ theta
Definition passage: Bridge datum · Read passage · Paper_XI_Traditional_Final.tex, line 1201
item a one break Galois envelope (V_ρ,b_ρ) item a type side quotient (V_θ,h_θ) item an equivariant symplectic isometry ι:V_ρ⟶ V_θ for a finite cyclic stabilizer C=⟨ c⟩ item the same nontrivial central character on the two Heisenberg groups item when p=2, compatible quadratic refinements and an isomorphism of the resulting Heisenberg central extensions lifting ι.
Exact mathematical passage
(V_\theta,h_\theta)
\HomDefinition passage · 2 passages
Search aliases: Hom operatorname
Definition passage: One break Clifford datum · Read passage · Paper_XI_Traditional_Final.tex, line 1036
The multiplicity space is M_ρ=Hom_N(W, X).
Exact mathematical passage
M_\rho=\Hom_N(W,\calX).
\Hom_NDefinition passage · 1 passage
Search aliases: Hom_N Hom operatorname
Definition passage: One break Clifford datum · Read passage · Paper_XI_Traditional_Final.tex, line 1036
The multiplicity space is M_ρ=Hom_N(W, X).
Exact mathematical passage
M_\rho=\Hom_N(W,\calX).
IDefinition passage · 51 passages
Search aliases: I
Definition passage: Odd finite Weil phase · Read passage · Paper_XI_Traditional_Final.tex, line 1312
[Odd finite Weil phase] For fixed point free g, define Φ_p(g,V) = (frac-1p)^r (fracdet(g-I)p).
Exact mathematical passage
\Phi_p(g,V) = \left(\frac{-1}{p}\right)^r \left(\frac{\det(g-I)}{p}\right).
iDefinition passage · 47 passages
Search aliases: i
Definition passage: Arf invariant · Read passage · Paper_XI_Traditional_Final.tex, line 1492
[Arf invariant] Choose a symplectic basis (e_i,f_i)_i=1^r. Define Arf(q) = ∑_i=1^rq(e_i)q(f_i) ∈ F_2.
Exact mathematical passage
(e_i,f_i)_{i=1}^{r}
\imDefinition passage · 6 passages
Search aliases: im operatorname
Definition passage: Moving phase · Read passage · Paper_XI_Traditional_Final.tex, line 1407
[Moving phase] Under the hypotheses of [source reference], define Φ_p^mov(g,V) = Φ_p(g|_im(g-I),im(g-I)).
Exact mathematical passage
\Phi_p^{\mathrm{mov}}(g,V) = \Phi_p\left(g|_{\im(g-I)},\im(g-I)\right).
jDefinition passage · 65 passages
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Definition passage: Monodromy packet chain · Read passage · Paper_XI_Traditional_Final.tex, line 251
[Monodromy packet chain] Let V_0,…,V_m-1 be copies of the space of τ. Put V=⊕_j=0^m-1V_j. A monodromy packet chain is a nilpotent operator N with N(V_j)⊆ V_j-1 for j>0, with N(V_0)=0, together with a Frobenius operator whose conjugation scales N by q^-1.
Exact mathematical passage
V=\bigoplus_{j=0}^{m-1}V_j.
MDefinition passage · 39 passages
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Definition passage: One break Clifford datum · Read passage · Paper_XI_Traditional_Final.tex, line 1036
The multiplicity space is M_ρ=Hom_N(W, X).
Exact mathematical passage
M_\rho=\Hom_N(W,\calX).
mDefinition passage · 71 passages
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Definition passage: Monodromy packet chain · Read passage · Paper_XI_Traditional_Final.tex, line 250
[Monodromy packet chain] Let V_0,…,V_m-1 be copies of the space of τ. Put V=⊕_j=0^m-1V_j. A monodromy packet chain is a nilpotent operator N with N(V_j)⊆ V_j-1 for j>0, with N(V_0)=0, together with a Frobenius operator whose conjugation scales N by q^-1.
Exact mathematical passage
V_0,\ldots,V_{m-1}
M_\rhoDefinition passage · 10 passages
Search aliases: M_ρ rho
Definition passage: One break Clifford datum · Read passage · Paper_XI_Traditional_Final.tex, line 1036
The multiplicity space is M_ρ=Hom_N(W, X).
Exact mathematical passage
M_\rho=\Hom_N(W,\calX).
\mathrm{mov}Definition passage · 1 passage
Search aliases: mov mathrm
Definition passage: Moving phase · Read passage · Paper_XI_Traditional_Final.tex, line 1407
[Moving phase] Under the hypotheses of [source reference], define Φ_p^mov(g,V) = Φ_p(g|_im(g-I),im(g-I)).
Exact mathematical passage
\Phi_p^{\mathrm{mov}}(g,V) = \Phi_p\left(g|_{\im(g-I)},\im(g-I)\right).
NDefinition passage · 78 passages
Search aliases: N
Definition passage: Monodromy packet chain · Read passage · Paper_XI_Traditional_Final.tex, line 254
[Monodromy packet chain] Let V_0,…,V_m-1 be copies of the space of τ. Put V=⊕_j=0^m-1V_j. A monodromy packet chain is a nilpotent operator N with N(V_j)⊆ V_j-1 for j>0, with N(V_0)=0, together with a Frobenius operator whose conjugation scales N by q^-1.
Exact mathematical passage
N
pDefinition passage · 162 passages
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Definition passage: One break Clifford datum · Read passage · Paper_XI_Traditional_Final.tex, line 1028
item The quotient A is a finite elementary abelian p group. item The representations Π and ρ are irreducible. item The isomorphism class of ρ is fixed by G. item The restriction of Π to N is ρ isotypic. item The quotient records one declared ramification break.
Exact mathematical passage
p
\perpbigDefinition passage · 4 passages
Search aliases: perpbig mathop⊥limits mathop perp limits
Definition passage: Orthogonal break decomposition · Read passage · Paper_XI_Traditional_Final.tex, line 1756
[Orthogonal break decomposition] An orthogonal break decomposition is a symplectic direct sum V=mathop⊥limits_s∈ SV_s that is preserved by the declared tame stabilizer. Each V_s is the finite symplectic packet assigned to one break.
Exact mathematical passage
V=\perpbig_{s\in S}V_s
\perpbig_{s\in S}Definition passage · 1 passage
Search aliases: perpbig_s∈ S mathop⊥limits_s∈ S perpbig in mathop perp limits in
Definition passage: Orthogonal break decomposition · Read passage · Paper_XI_Traditional_Final.tex, line 1756
[Orthogonal break decomposition] An orthogonal break decomposition is a symplectic direct sum V=mathop⊥limits_s∈ SV_s that is preserved by the declared tame stabilizer. Each V_s is the finite symplectic packet assigned to one break.
Exact mathematical passage
V=\perpbig_{s\in S}V_s
qDefinition passage · 123 passages
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Definition passage: Monodromy packet chain · Read passage · Paper_XI_Traditional_Final.tex, line 258
[Monodromy packet chain] Let V_0,…,V_m-1 be copies of the space of τ. Put V=⊕_j=0^m-1V_j. A monodromy packet chain is a nilpotent operator N with N(V_j)⊆ V_j-1 for j>0, with N(V_0)=0, together with a Frobenius operator whose conjugation scales N by q^-1.
Exact mathematical passage
q^{-1}
q^{-1}Definition passage · 7 passages
Search aliases: q^-1
Definition passage: Monodromy packet chain · Read passage · Paper_XI_Traditional_Final.tex, line 258
[Monodromy packet chain] Let V_0,…,V_m-1 be copies of the space of τ. Put V=⊕_j=0^m-1V_j. A monodromy packet chain is a nilpotent operator N with N(V_j)⊆ V_j-1 for j>0, with N(V_0)=0, together with a Frobenius operator whose conjugation scales N by q^-1.
Exact mathematical passage
q^{-1}
q^{-1/2}Definition passage · 2 passages
Search aliases: q^-1/2
Definition passage: Normalized Gauss phase · Read passage · Paper_XI_Traditional_Final.tex, line 745
[Normalized Gauss phase] For nontrivial χ, define γ(χ,ψ) = q^-1/2G(χ,ψ). Then absγ(χ,ψ)=1.
Exact mathematical passage
\gamma(\chi,\psi) = q^{-1/2}G(\chi,\psi).
rDefinition passage · 99 passages
Search aliases: r
Definition passage: Odd finite Weil phase · Read passage · Paper_XI_Traditional_Final.tex, line 1312
[Odd finite Weil phase] For fixed point free g, define Φ_p(g,V) = (frac-1p)^r (fracdet(g-I)p).
Exact mathematical passage
\Phi_p(g,V) = \left(\frac{-1}{p}\right)^r \left(\frac{\det(g-I)}{p}\right).
\radDefinition passage · 2 passages
Search aliases: rad operatorname
Definition passage: One break symplectic envelope · Read passage · Paper_XI_Traditional_Final.tex, line 1094
[One break symplectic envelope] The one break symplectic envelope is V_ρ=A/rad b_ρ.
Exact mathematical passage
\boxed{ V_\rho=A/\rad b_\rho. }
SDefinition passage · 7 passages
Search aliases: S
Definition passage: Orthogonal break decomposition · Read passage · Paper_XI_Traditional_Final.tex, line 1756
[Orthogonal break decomposition] An orthogonal break decomposition is a symplectic direct sum V=mathop⊥limits_s∈ SV_s that is preserved by the declared tame stabilizer. Each V_s is the finite symplectic packet assigned to one break.
Exact mathematical passage
V=\perpbig_{s\in S}V_s
sDefinition passage · 41 passages
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Definition passage: Orthogonal break decomposition · Read passage · Paper_XI_Traditional_Final.tex, line 1756
[Orthogonal break decomposition] An orthogonal break decomposition is a symplectic direct sum V=mathop⊥limits_s∈ SV_s that is preserved by the declared tame stabilizer. Each V_s is the finite symplectic packet assigned to one break.
Exact mathematical passage
V=\perpbig_{s\in S}V_s
TDefinition passage · 29 passages
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Definition passage: Clifford cocycle class · Read passage · Paper_XI_Traditional_Final.tex, line 1065
[Clifford cocycle class] Choose scalars c(a,b) such that T_aT_b=c(a,b)T_a+b. The cohomology class [c_ρ]∈ H^2(A, C^×) is the Clifford cocycle class of the datum.
Exact mathematical passage
T_aT_b=c(a,b)T_{a+b}.
T_aDefinition passage · 13 passages
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Definition passage: Clifford cocycle class · Read passage · Paper_XI_Traditional_Final.tex, line 1065
[Clifford cocycle class] Choose scalars c(a,b) such that T_aT_b=c(a,b)T_a+b. The cohomology class [c_ρ]∈ H^2(A, C^×) is the Clifford cocycle class of the datum.
Exact mathematical passage
T_aT_b=c(a,b)T_{a+b}.
T_{a+b}Definition passage · 2 passages
Search aliases: T_a+b
Definition passage: Clifford cocycle class · Read passage · Paper_XI_Traditional_Final.tex, line 1065
[Clifford cocycle class] Choose scalars c(a,b) such that T_aT_b=c(a,b)T_a+b. The cohomology class [c_ρ]∈ H^2(A, C^×) is the Clifford cocycle class of the datum.
Exact mathematical passage
T_aT_b=c(a,b)T_{a+b}.
T_bDefinition passage · 2 passages
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Definition passage: Clifford cocycle class · Read passage · Paper_XI_Traditional_Final.tex, line 1065
[Clifford cocycle class] Choose scalars c(a,b) such that T_aT_b=c(a,b)T_a+b. The cohomology class [c_ρ]∈ H^2(A, C^×) is the Clifford cocycle class of the datum.
Exact mathematical passage
T_aT_b=c(a,b)T_{a+b}.
\triangleleftDefinition passage · 4 passages
Search aliases: triangleleft
Definition passage: One break Clifford datum · Read passage · Paper_XI_Traditional_Final.tex, line 1016
[One break Clifford datum] A one break Clifford datum consists of finite groups and representations Ntriangleleft G, A=G/N, Π:G→GL( X), ρ:N→GL(W), with the following properties.
Exact mathematical passage
N\triangleleft G, \qquad A=G/N, \qquad \Pi:G\to\GL(\calX), \qquad \rho:N\to\GL(W),
uDefinition passage · 62 passages
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Definition passage: Schrödinger operators · Read passage · Paper_XI_Traditional_Final.tex, line 816
[Schrödinger operators] On H= C^X, define (ρ(x,ξ,z)f)(u) = ψ(z+ξ(u)+frac12ξ(x))f(u+x).
Exact mathematical passage
\bigl(\rho(x,\xi,z)f\bigr)(u) = \psi\left(z+\xi(u)+\frac12\xi(x)\right)f(u+x).
VDefinition passage · 99 passages
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Definition passage: Monodromy packet chain · Read passage · Paper_XI_Traditional_Final.tex, line 250
[Monodromy packet chain] Let V_0,…,V_m-1 be copies of the space of τ. Put V=⊕_j=0^m-1V_j. A monodromy packet chain is a nilpotent operator N with N(V_j)⊆ V_j-1 for j>0, with N(V_0)=0, together with a Frobenius operator whose conjugation scales N by q^-1.
Exact mathematical passage
V_0,\ldots,V_{m-1}
vDefinition passage · 53 passages
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Definition passage: Path transport · Read passage · Paper_XI_Traditional_Final.tex, line 1621
[Path transport] For an oriented path γ=(v_0→ v_1→⋯→ v_m), define A_γ=A_v_m v_m-1⋯ A_v_1v_0, B_γ=B_v_m v_m-1⋯ B_v_1v_0. For a loop based at r, these are the based holonomies.
Exact mathematical passage
\gamma=(v_0\to v_1\to\cdots\to v_m),
V_0Definition passage · 3 passages
Search aliases: V_0
Definition passage: Monodromy packet chain · Read passage · Paper_XI_Traditional_Final.tex, line 250
[Monodromy packet chain] Let V_0,…,V_m-1 be copies of the space of τ. Put V=⊕_j=0^m-1V_j. A monodromy packet chain is a nilpotent operator N with N(V_j)⊆ V_j-1 for j>0, with N(V_0)=0, together with a Frobenius operator whose conjugation scales N by q^-1.
Exact mathematical passage
V_0,\ldots,V_{m-1}
v_0Definition passage · 2 passages
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Definition passage: Path transport · Read passage · Paper_XI_Traditional_Final.tex, line 1621
[Path transport] For an oriented path γ=(v_0→ v_1→⋯→ v_m), define A_γ=A_v_m v_m-1⋯ A_v_1v_0, B_γ=B_v_m v_m-1⋯ B_v_1v_0. For a loop based at r, these are the based holonomies.
Exact mathematical passage
\gamma=(v_0\to v_1\to\cdots\to v_m),
v_1Definition passage · 2 passages
Search aliases: v_1
Definition passage: Path transport · Read passage · Paper_XI_Traditional_Final.tex, line 1621
[Path transport] For an oriented path γ=(v_0→ v_1→⋯→ v_m), define A_γ=A_v_m v_m-1⋯ A_v_1v_0, B_γ=B_v_m v_m-1⋯ B_v_1v_0. For a loop based at r, these are the based holonomies.
Exact mathematical passage
\gamma=(v_0\to v_1\to\cdots\to v_m),
V_jDefinition passage · 10 passages
Search aliases: V_j
Definition passage: Monodromy packet chain · Read passage · Paper_XI_Traditional_Final.tex, line 251
[Monodromy packet chain] Let V_0,…,V_m-1 be copies of the space of τ. Put V=⊕_j=0^m-1V_j. A monodromy packet chain is a nilpotent operator N with N(V_j)⊆ V_j-1 for j>0, with N(V_0)=0, together with a Frobenius operator whose conjugation scales N by q^-1.
Exact mathematical passage
V=\bigoplus_{j=0}^{m-1}V_j.
V_{j-1}Definition passage · 4 passages
Search aliases: V_j-1
Definition passage: Monodromy packet chain · Read passage · Paper_XI_Traditional_Final.tex, line 255
[Monodromy packet chain] Let V_0,…,V_m-1 be copies of the space of τ. Put V=⊕_j=0^m-1V_j. A monodromy packet chain is a nilpotent operator N with N(V_j)⊆ V_j-1 for j>0, with N(V_0)=0, together with a Frobenius operator whose conjugation scales N by q^-1.
Exact mathematical passage
N(V_j)\subseteq V_{j-1}
v_mDefinition passage · 2 passages
Search aliases: v_m
Definition passage: Path transport · Read passage · Paper_XI_Traditional_Final.tex, line 1621
[Path transport] For an oriented path γ=(v_0→ v_1→⋯→ v_m), define A_γ=A_v_m v_m-1⋯ A_v_1v_0, B_γ=B_v_m v_m-1⋯ B_v_1v_0. For a loop based at r, these are the based holonomies.
Exact mathematical passage
\gamma=(v_0\to v_1\to\cdots\to v_m),
V_{m-1}Definition passage · 1 passage
Search aliases: V_m-1
Definition passage: Monodromy packet chain · Read passage · Paper_XI_Traditional_Final.tex, line 250
[Monodromy packet chain] Let V_0,…,V_m-1 be copies of the space of τ. Put V=⊕_j=0^m-1V_j. A monodromy packet chain is a nilpotent operator N with N(V_j)⊆ V_j-1 for j>0, with N(V_0)=0, together with a Frobenius operator whose conjugation scales N by q^-1.
Exact mathematical passage
V_0,\ldots,V_{m-1}
v_{m-1}Definition passage · 1 passage
Search aliases: v_m-1
Definition passage: Path transport · Read passage · Paper_XI_Traditional_Final.tex, line 1625
[Path transport] For an oriented path γ=(v_0→ v_1→⋯→ v_m), define A_γ=A_v_m v_m-1⋯ A_v_1v_0, B_γ=B_v_m v_m-1⋯ B_v_1v_0. For a loop based at r, these are the based holonomies.
Exact mathematical passage
A_\gamma=A_{v_m v_{m-1}}\cdots A_{v_1v_0}, \qquad B_\gamma=B_{v_m v_{m-1}}\cdots B_{v_1v_0}.
V_sDefinition passage · 8 passages
Search aliases: V_s
Definition passage: Orthogonal break decomposition · Read passage · Paper_XI_Traditional_Final.tex, line 1756
[Orthogonal break decomposition] An orthogonal break decomposition is a symplectic direct sum V=mathop⊥limits_s∈ SV_s that is preserved by the declared tame stabilizer. Each V_s is the finite symplectic packet assigned to one break.
Exact mathematical passage
V=\perpbig_{s\in S}V_s
V_\thetaDefinition passage · 10 passages
Search aliases: V_θ theta
Definition passage: Bridge datum · Read passage · Paper_XI_Traditional_Final.tex, line 1201
item a one break Galois envelope (V_ρ,b_ρ) item a type side quotient (V_θ,h_θ) item an equivariant symplectic isometry ι:V_ρ⟶ V_θ for a finite cyclic stabilizer C=⟨ c⟩ item the same nontrivial central character on the two Heisenberg groups item when p=2, compatible quadratic refinements and an isomorphism of the resulting Heisenberg central extensions lifting ι.
Exact mathematical passage
(V_\theta,h_\theta)
V_\rhoDefinition passage · 7 passages
Search aliases: V_ρ rho
Definition passage: One break symplectic envelope · Read passage · Paper_XI_Traditional_Final.tex, line 1094
[One break symplectic envelope] The one break symplectic envelope is V_ρ=A/rad b_ρ.
Exact mathematical passage
\boxed{ V_\rho=A/\rad b_\rho. }
WDefinition passage · 47 passages
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Definition passage: One break Clifford datum · Read passage · Paper_XI_Traditional_Final.tex, line 1016
[One break Clifford datum] A one break Clifford datum consists of finite groups and representations Ntriangleleft G, A=G/N, Π:G→GL( X), ρ:N→GL(W), with the following properties.
Exact mathematical passage
N\triangleleft G, \qquad A=G/N, \qquad \Pi:G\to\GL(\calX), \qquad \rho:N\to\GL(W),
XDefinition passage · 35 passages
Search aliases: X
Definition passage: Schrödinger operators · Read passage · Paper_XI_Traditional_Final.tex, line 815
[Schrödinger operators] On H= C^X, define (ρ(x,ξ,z)f)(u) = ψ(z+ξ(u)+frac12ξ(x))f(u+x).
Exact mathematical passage
\calH=\CC^X
xDefinition passage · 36 passages
Search aliases: x
Definition passage: Schrödinger operators · Read passage · Paper_XI_Traditional_Final.tex, line 816
[Schrödinger operators] On H= C^X, define (ρ(x,ξ,z)f)(u) = ψ(z+ξ(u)+frac12ξ(x))f(u+x).
Exact mathematical passage
\bigl(\rho(x,\xi,z)f\bigr)(u) = \psi\left(z+\xi(u)+\frac12\xi(x)\right)f(u+x).
yDefinition passage · 15 passages
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Definition passage: Quadratic refinement · Read passage · Paper_XI_Traditional_Final.tex, line 1478
[Quadratic refinement] A quadratic refinement of b is a map q:V→ F_2 satisfying q(x+y)-q(x)-q(y)=b(x,y).
Exact mathematical passage
q(x+y)-q(x)-q(y)=b(x,y).
zDefinition passage · 32 passages
Search aliases: z
Definition passage: Schrödinger operators · Read passage · Paper_XI_Traditional_Final.tex, line 816
[Schrödinger operators] On H= C^X, define (ρ(x,ξ,z)f)(u) = ψ(z+ξ(u)+frac12ξ(x))f(u+x).
Exact mathematical passage
\bigl(\rho(x,\xi,z)f\bigr)(u) = \psi\left(z+\xi(u)+\frac12\xi(x)\right)f(u+x).
|Definition passage · 26 passages
Search aliases: |
Definition passage: Moving phase · Read passage · Paper_XI_Traditional_Final.tex, line 1407
[Moving phase] Under the hypotheses of [source reference], define Φ_p^mov(g,V) = Φ_p(g|_im(g-I),im(g-I)).
Exact mathematical passage
\Phi_p^{\mathrm{mov}}(g,V) = \Phi_p\left(g|_{\im(g-I)},\im(g-I)\right).
|_{\im(g-I)}Definition passage · 1 passage
Search aliases: |_im(g-I) im operatorname
Definition passage: Moving phase · Read passage · Paper_XI_Traditional_Final.tex, line 1407
[Moving phase] Under the hypotheses of [source reference], define Φ_p^mov(g,V) = Φ_p(g|_im(g-I),im(g-I)).
Exact mathematical passage
\Phi_p^{\mathrm{mov}}(g,V) = \Phi_p\left(g|_{\im(g-I)},\im(g-I)\right).
\timesDefinition passage · 66 passages
Search aliases: × times
Definition passage: Clifford cocycle class · Read passage · Paper_XI_Traditional_Final.tex, line 1069
[Clifford cocycle class] Choose scalars c(a,b) such that T_aT_b=c(a,b)T_a+b. The cohomology class [c_ρ]∈ H^2(A, C^×) is the Clifford cocycle class of the datum.
Exact mathematical passage
[c_\rho]\in H^2(A,\CC^{\times})
\gammaDefinition passage · 16 passages
Search aliases: γ gamma
Definition passage: Normalized Gauss phase · Read passage · Paper_XI_Traditional_Final.tex, line 745
[Normalized Gauss phase] For nontrivial χ, define γ(χ,ψ) = q^-1/2G(χ,ψ). Then absγ(χ,ψ)=1.
Exact mathematical passage
\gamma(\chi,\psi) = q^{-1/2}G(\chi,\psi).
\thetaDefinition passage · 80 passages
Search aliases: θ theta
Definition passage: Bridge datum · Read passage · Paper_XI_Traditional_Final.tex, line 1201
item a one break Galois envelope (V_ρ,b_ρ) item a type side quotient (V_θ,h_θ) item an equivariant symplectic isometry ι:V_ρ⟶ V_θ for a finite cyclic stabilizer C=⟨ c⟩ item the same nontrivial central character on the two Heisenberg groups item when p=2, compatible quadratic refinements and an isomorphism of the resulting Heisenberg central extensions lifting ι.
Exact mathematical passage
(V_\theta,h_\theta)
\iotaDefinition passage · 11 passages
Search aliases: ι iota
Definition passage: Bridge datum · Read passage · Paper_XI_Traditional_Final.tex, line 1203
item a one break Galois envelope (V_ρ,b_ρ) item a type side quotient (V_θ,h_θ) item an equivariant symplectic isometry ι:V_ρ⟶ V_θ for a finite cyclic stabilizer C=⟨ c⟩ item the same nontrivial central character on the two Heisenberg groups item when p=2, compatible quadratic refinements and an isomorphism of the resulting Heisenberg central extensions lifting ι.
Exact mathematical passage
\iota:V_\rho\longrightarrow V_\theta
\xiDefinition passage · 24 passages
Search aliases: ξ xi
Definition passage: Schrödinger operators · Read passage · Paper_XI_Traditional_Final.tex, line 816
[Schrödinger operators] On H= C^X, define (ρ(x,ξ,z)f)(u) = ψ(z+ξ(u)+frac12ξ(x))f(u+x).
Exact mathematical passage
\bigl(\rho(x,\xi,z)f\bigr)(u) = \psi\left(z+\xi(u)+\frac12\xi(x)\right)f(u+x).
\PiDefinition passage · 8 passages
Search aliases: Π Pi
Definition passage: One break Clifford datum · Read passage · Paper_XI_Traditional_Final.tex, line 1016
[One break Clifford datum] A one break Clifford datum consists of finite groups and representations Ntriangleleft G, A=G/N, Π:G→GL( X), ρ:N→GL(W), with the following properties.
Exact mathematical passage
N\triangleleft G, \qquad A=G/N, \qquad \Pi:G\to\GL(\calX), \qquad \rho:N\to\GL(W),
\rhoDefinition passage · 62 passages
Search aliases: ρ rho
Definition passage: Schrödinger operators · Read passage · Paper_XI_Traditional_Final.tex, line 816
[Schrödinger operators] On H= C^X, define (ρ(x,ξ,z)f)(u) = ψ(z+ξ(u)+frac12ξ(x))f(u+x).
Exact mathematical passage
\bigl(\rho(x,\xi,z)f\bigr)(u) = \psi\left(z+\xi(u)+\frac12\xi(x)\right)f(u+x).
\tauDefinition passage · 28 passages
Search aliases: τ tau
Definition passage: Monodromy packet chain · Read passage · Paper_XI_Traditional_Final.tex, line 250
[Monodromy packet chain] Let V_0,…,V_m-1 be copies of the space of τ. Put V=⊕_j=0^m-1V_j. A monodromy packet chain is a nilpotent operator N with N(V_j)⊆ V_j-1 for j>0, with N(V_0)=0, together with a Frobenius operator whose conjugation scales N by q^-1.
Exact mathematical passage
\tau
\PhiDefinition passage · 38 passages
Search aliases: Φ Phi
Definition passage: Odd finite Weil phase · Read passage · Paper_XI_Traditional_Final.tex, line 1312
[Odd finite Weil phase] For fixed point free g, define Φ_p(g,V) = (frac-1p)^r (fracdet(g-I)p).
Exact mathematical passage
\Phi_p(g,V) = \left(\frac{-1}{p}\right)^r \left(\frac{\det(g-I)}{p}\right).
\Phi_pDefinition passage · 10 passages
Search aliases: Φ_p Phi
Definition passage: Odd finite Weil phase · Read passage · Paper_XI_Traditional_Final.tex, line 1312
[Odd finite Weil phase] For fixed point free g, define Φ_p(g,V) = (frac-1p)^r (fracdet(g-I)p).
Exact mathematical passage
\Phi_p(g,V) = \left(\frac{-1}{p}\right)^r \left(\frac{\det(g-I)}{p}\right).
\Phi_p^{\mathrm{mov}}Definition passage · 1 passage
Search aliases: Φ_p^mov Phi mathrm
Definition passage: Moving phase · Read passage · Paper_XI_Traditional_Final.tex, line 1407
[Moving phase] Under the hypotheses of [source reference], define Φ_p^mov(g,V) = Φ_p(g|_im(g-I),im(g-I)).
Exact mathematical passage
\Phi_p^{\mathrm{mov}}(g,V) = \Phi_p\left(g|_{\im(g-I)},\im(g-I)\right).
\chiDefinition passage · 32 passages
Search aliases: χ chi
Definition passage: Normalized Gauss phase · Read passage · Paper_XI_Traditional_Final.tex, line 744
[Normalized Gauss phase] For nontrivial χ, define γ(χ,ψ) = q^-1/2G(χ,ψ). Then absγ(χ,ψ)=1.
Exact mathematical passage
\chi
\psiDefinition passage · 36 passages
Search aliases: ψ psi
Definition passage: Normalized Gauss phase · Read passage · Paper_XI_Traditional_Final.tex, line 745
[Normalized Gauss phase] For nontrivial χ, define γ(χ,ψ) = q^-1/2G(χ,ψ). Then absγ(χ,ψ)=1.
Exact mathematical passage
\gamma(\chi,\psi) = q^{-1/2}G(\chi,\psi).
\ldotsDefinition passage · 15 passages
Search aliases: … ldots
Definition passage: Monodromy packet chain · Read passage · Paper_XI_Traditional_Final.tex, line 250
[Monodromy packet chain] Let V_0,…,V_m-1 be copies of the space of τ. Put V=⊕_j=0^m-1V_j. A monodromy packet chain is a nilpotent operator N with N(V_j)⊆ V_j-1 for j>0, with N(V_0)=0, together with a Frobenius operator whose conjugation scales N by q^-1.
Exact mathematical passage
V_0,\ldots,V_{m-1}
\toDefinition passage · 26 passages
Search aliases: → to
Definition passage: One break Clifford datum · Read passage · Paper_XI_Traditional_Final.tex, line 1016
[One break Clifford datum] A one break Clifford datum consists of finite groups and representations Ntriangleleft G, A=G/N, Π:G→GL( X), ρ:N→GL(W), with the following properties.
Exact mathematical passage
N\triangleleft G, \qquad A=G/N, \qquad \Pi:G\to\GL(\calX), \qquad \rho:N\to\GL(W),
\inDefinition passage · 58 passages
Search aliases: ∈ in
Definition passage: Clifford cocycle class · Read passage · Paper_XI_Traditional_Final.tex, line 1069
[Clifford cocycle class] Choose scalars c(a,b) such that T_aT_b=c(a,b)T_a+b. The cohomology class [c_ρ]∈ H^2(A, C^×) is the Clifford cocycle class of the datum.
Exact mathematical passage
[c_\rho]\in H^2(A,\CC^{\times})
\sumDefinition passage · 35 passages
Search aliases: ∑ sum
Definition passage: Arf invariant · Read passage · Paper_XI_Traditional_Final.tex, line 1493
[Arf invariant] Choose a symplectic basis (e_i,f_i)_i=1^r. Define Arf(q) = ∑_i=1^rq(e_i)q(f_i) ∈ F_2.
Exact mathematical passage
\Arf(q) = \sum_{i=1}^{r}q(e_i)q(f_i) \in\FF_2.
\sum_{i=1}^{r}Definition passage · 1 passage
Search aliases: ∑_i=1^r sum
Definition passage: Arf invariant · Read passage · Paper_XI_Traditional_Final.tex, line 1493
[Arf invariant] Choose a symplectic basis (e_i,f_i)_i=1^r. Define Arf(q) = ∑_i=1^rq(e_i)q(f_i) ∈ F_2.
Exact mathematical passage
\Arf(q) = \sum_{i=1}^{r}q(e_i)q(f_i) \in\FF_2.
\subseteqDefinition passage · 6 passages
Search aliases: ⊆ subseteq
Definition passage: Monodromy packet chain · Read passage · Paper_XI_Traditional_Final.tex, line 255
[Monodromy packet chain] Let V_0,…,V_m-1 be copies of the space of τ. Put V=⊕_j=0^m-1V_j. A monodromy packet chain is a nilpotent operator N with N(V_j)⊆ V_j-1 for j>0, with N(V_0)=0, together with a Frobenius operator whose conjugation scales N by q^-1.
Exact mathematical passage
N(V_j)\subseteq V_{j-1}
\bigoplusDefinition passage · 5 passages
Search aliases: ⊕ bigoplus
Definition passage: Monodromy packet chain · Read passage · Paper_XI_Traditional_Final.tex, line 251
[Monodromy packet chain] Let V_0,…,V_m-1 be copies of the space of τ. Put V=⊕_j=0^m-1V_j. A monodromy packet chain is a nilpotent operator N with N(V_j)⊆ V_j-1 for j>0, with N(V_0)=0, together with a Frobenius operator whose conjugation scales N by q^-1.
Exact mathematical passage
V=\bigoplus_{j=0}^{m-1}V_j.
\bigoplus_{j=0}^{m-1}Definition passage · 2 passages
Search aliases: ⊕_j=0^m-1 bigoplus
Definition passage: Monodromy packet chain · Read passage · Paper_XI_Traditional_Final.tex, line 251
[Monodromy packet chain] Let V_0,…,V_m-1 be copies of the space of τ. Put V=⊕_j=0^m-1V_j. A monodromy packet chain is a nilpotent operator N with N(V_j)⊆ V_j-1 for j>0, with N(V_0)=0, together with a Frobenius operator whose conjugation scales N by q^-1.
Exact mathematical passage
V=\bigoplus_{j=0}^{m-1}V_j.
\cdotsDefinition passage · 10 passages
Search aliases: ⋯ cdots
Definition passage: Path transport · Read passage · Paper_XI_Traditional_Final.tex, line 1621
[Path transport] For an oriented path γ=(v_0→ v_1→⋯→ v_m), define A_γ=A_v_m v_m-1⋯ A_v_1v_0, B_γ=B_v_m v_m-1⋯ B_v_1v_0. For a loop based at r, these are the based holonomies.
Exact mathematical passage
\gamma=(v_0\to v_1\to\cdots\to v_m),
\langleDefinition passage · 4 passages
Search aliases: ⟨ langle
Definition passage: Bridge datum · Read passage · Paper_XI_Traditional_Final.tex, line 1206
item a one break Galois envelope (V_ρ,b_ρ) item a type side quotient (V_θ,h_θ) item an equivariant symplectic isometry ι:V_ρ⟶ V_θ for a finite cyclic stabilizer C=⟨ c⟩ item the same nontrivial central character on the two Heisenberg groups item when p=2, compatible quadratic refinements and an isomorphism of the resulting Heisenberg central extensions lifting ι.
Exact mathematical passage
C=\langle c\rangle
\rangleDefinition passage · 4 passages
Search aliases: ⟩ rangle
Definition passage: Bridge datum · Read passage · Paper_XI_Traditional_Final.tex, line 1206
item a one break Galois envelope (V_ρ,b_ρ) item a type side quotient (V_θ,h_θ) item an equivariant symplectic isometry ι:V_ρ⟶ V_θ for a finite cyclic stabilizer C=⟨ c⟩ item the same nontrivial central character on the two Heisenberg groups item when p=2, compatible quadratic refinements and an isomorphism of the resulting Heisenberg central extensions lifting ι.
Exact mathematical passage
C=\langle c\rangle
\longrightarrowDefinition passage · 7 passages
Search aliases: ⟶ longrightarrow
Definition passage: Bridge datum · Read passage · Paper_XI_Traditional_Final.tex, line 1203
item a one break Galois envelope (V_ρ,b_ρ) item a type side quotient (V_θ,h_θ) item an equivariant symplectic isometry ι:V_ρ⟶ V_θ for a finite cyclic stabilizer C=⟨ c⟩ item the same nontrivial central character on the two Heisenberg groups item when p=2, compatible quadratic refinements and an isomorphism of the resulting Heisenberg central extensions lifting ι.
Exact mathematical passage
\iota:V_\rho\longrightarrow V_\theta
dIntroduction passage · 113 passages
Search aliases: d
Introduction passage · Read passage · Paper_XI_Traditional_Final.tex, line 2163
For the same candidates the constructions give c|c|c · p odd · p=2 dim_ F_p(J^1/H^1) · p^d-1 · 2^d-2 Swan(π) · p^d-1(p+1)h_d · 3·2^d-1h_d and torsion number one. Here Swan denotes the Swan conductor, and a below denotes the ordinary pair conductor. An epsilon exponent for a specified additive character is a different quantity. The separate progress paper, Sections 22--24, reconciles these conventions and the actual-order hypotheses. Their self-pair conductor formulas are a(π^vee×π) · =p^2d-1+∑_i…
Exact mathematical passage
\begin{array}{c|c|c} & p\text{ odd} & p=2\\ \hline \dim_{\FF_p}(J^1/H^1) & p^d-1 & 2^d-2\\ \Swan(\pi)&p^{d-1}(p+1)h_d&3\cdot2^{d-1}h_d \end{array}
\dimIntroduction passage · 26 passages
Search aliases: dim
Introduction passage · Read passage · Paper_XI_Traditional_Final.tex, line 2163
For the same candidates the constructions give c|c|c · p odd · p=2 dim_ F_p(J^1/H^1) · p^d-1 · 2^d-2 Swan(π) · p^d-1(p+1)h_d · 3·2^d-1h_d and torsion number one. Here Swan denotes the Swan conductor, and a below denotes the ordinary pair conductor. An epsilon exponent for a specified additive character is a different quantity. The separate progress paper, Sections 22--24, reconciles these conventions and the actual-order hypotheses. Their self-pair conductor formulas are a(π^vee×π) · =p^2d-1+∑_i…
Exact mathematical passage
\begin{array}{c|c|c} & p\text{ odd} & p=2\\ \hline \dim_{\FF_p}(J^1/H^1) & p^d-1 & 2^d-2\\ \Swan(\pi)&p^{d-1}(p+1)h_d&3\cdot2^{d-1}h_d \end{array}
\dim_{\FF_p}Introduction passage · 1 passage
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Introduction passage · Read passage · Paper_XI_Traditional_Final.tex, line 2163
For the same candidates the constructions give c|c|c · p odd · p=2 dim_ F_p(J^1/H^1) · p^d-1 · 2^d-2 Swan(π) · p^d-1(p+1)h_d · 3·2^d-1h_d and torsion number one. Here Swan denotes the Swan conductor, and a below denotes the ordinary pair conductor. An epsilon exponent for a specified additive character is a different quantity. The separate progress paper, Sections 22--24, reconciles these conventions and the actual-order hypotheses. Their self-pair conductor formulas are a(π^vee×π) · =p^2d-1+∑_i…
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\begin{array}{c|c|c} & p\text{ odd} & p=2\\ \hline \dim_{\FF_p}(J^1/H^1) & p^d-1 & 2^d-2\\ \Swan(\pi)&p^{d-1}(p+1)h_d&3\cdot2^{d-1}h_d \end{array}
FIntroduction passage · 68 passages
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Introduction passage · Read passage · Paper_XI_Traditional_Final.tex, line 2011
The later constructions provide actual arithmetic candidates and exact finite packet recovery in substantial restricted families. They do not establish [source reference] for every admitted source. Their detailed ordinary proofs are typeset in the appendices, and the original reviewed Markdown files accompany this edition. In these appendices F denotes the base local field, while K=F[β] denotes a parameter field; these are local notation conventions. All references to ``claim 66'' in the retaine…
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Introduction passage · Paper_XI_Visual_Guide_Final.tex, line 58
The symbol K denotes the original local field in the finite-theory discussion. In the explicit arithmetic constructions, F= F_p((t)) is the base field and K=F[β] is its totally ramified parameter field. A symplectic quotient dimension and a Heisenberg representation dimension are different quantities.
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F=\F_p((t))
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Introduction passage · Paper_XI_Visual_Guide_Final.tex, line 58
The symbol K denotes the original local field in the finite-theory discussion. In the explicit arithmetic constructions, F= F_p((t)) is the base field and K=F[β] is its totally ramified parameter field. A symplectic quotient dimension and a Heisenberg representation dimension are different quantities.
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F=\F_p((t))
\FF_pIntroduction passage · 14 passages
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Introduction passage · Read passage · Paper_XI_Traditional_Final.tex, line 2163
For the same candidates the constructions give c|c|c · p odd · p=2 dim_ F_p(J^1/H^1) · p^d-1 · 2^d-2 Swan(π) · p^d-1(p+1)h_d · 3·2^d-1h_d and torsion number one. Here Swan denotes the Swan conductor, and a below denotes the ordinary pair conductor. An epsilon exponent for a specified additive character is a different quantity. The separate progress paper, Sections 22--24, reconciles these conventions and the actual-order hypotheses. Their self-pair conductor formulas are a(π^vee×π) · =p^2d-1+∑_i…
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\begin{array}{c|c|c} & p\text{ odd} & p=2\\ \hline \dim_{\FF_p}(J^1/H^1) & p^d-1 & 2^d-2\\ \Swan(\pi)&p^{d-1}(p+1)h_d&3\cdot2^{d-1}h_d \end{array}
\FF_p^{\times}Introduction passage · 1 passage
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Introduction passage · Read passage · Paper_XI_Traditional_Final.tex, line 1307
[Fixed point free Weil trace] Under the declared normalization, Trω(g) = (frac-1p)^r (fracdet(g-I)p). Here (frac·p) denotes the quadratic character of F_p^×.
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\FF_p^{\times}
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Introduction passage · Paper_XI_Visual_Guide_Final.tex, line 147
lessonPART I / FINITE TAME COORDINATESTensor refinement is a diagonal orbit problemOriginal Theorem 5.1 plate05_tensor_gridThe 4×6 grid separates into two diagonal translation orbits, labelled 0 and 1. The simultaneous translation action on Z/n Z× Z/m Z has d=gcd(n,m) orbits, each of length L=lcm(n,m). Mackey decomposition then gives Ind_W_K_n^W_Kξ⊗Ind_W_K_m^W_Kη ≃⊕_i=0^d-1 Ind_W_K_L^W_K(ξ_L,nη_L,m^Φ^i). Here the subscripts denote norm lifts and the Frobenius superscript follows the fixed source…
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d=\gcd(n,m)\quad\text{orbits, each of length}\quad L=\operatorname{lcm}(n,m).
H^1Introduction passage · 26 passages
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Introduction passage · Read passage · Paper_XI_Traditional_Final.tex, line 2163
For the same candidates the constructions give c|c|c · p odd · p=2 dim_ F_p(J^1/H^1) · p^d-1 · 2^d-2 Swan(π) · p^d-1(p+1)h_d · 3·2^d-1h_d and torsion number one. Here Swan denotes the Swan conductor, and a below denotes the ordinary pair conductor. An epsilon exponent for a specified additive character is a different quantity. The separate progress paper, Sections 22--24, reconciles these conventions and the actual-order hypotheses. Their self-pair conductor formulas are a(π^vee×π) · =p^2d-1+∑_i…
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\begin{array}{c|c|c} & p\text{ odd} & p=2\\ \hline \dim_{\FF_p}(J^1/H^1) & p^d-1 & 2^d-2\\ \Swan(\pi)&p^{d-1}(p+1)h_d&3\cdot2^{d-1}h_d \end{array}
h_dIntroduction passage · 12 passages
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Introduction passage · Read passage · Paper_XI_Traditional_Final.tex, line 2163
For the same candidates the constructions give c|c|c · p odd · p=2 dim_ F_p(J^1/H^1) · p^d-1 · 2^d-2 Swan(π) · p^d-1(p+1)h_d · 3·2^d-1h_d and torsion number one. Here Swan denotes the Swan conductor, and a below denotes the ordinary pair conductor. An epsilon exponent for a specified additive character is a different quantity. The separate progress paper, Sections 22--24, reconciles these conventions and the actual-order hypotheses. Their self-pair conductor formulas are a(π^vee×π) · =p^2d-1+∑_i…
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\begin{array}{c|c|c} & p\text{ odd} & p=2\\ \hline \dim_{\FF_p}(J^1/H^1) & p^d-1 & 2^d-2\\ \Swan(\pi)&p^{d-1}(p+1)h_d&3\cdot2^{d-1}h_d \end{array}
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Introduction passage · Read passage · Paper_XI_Traditional_Final.tex, line 2175
i) · p^d-1(p+1)h_d · 3·2^d-1h_d and torsion number one. Here Swan denotes the Swan conductor, and a below denotes the ordinary pair conductor. An epsilon exponent for a specified additive character is a different quantity. The separate progress paper, Sections 22--24, reconciles these conventions and the actual-order hypotheses. Their self-pair conductor formulas are a(π^vee×π) · =p^2d-1+∑_i=1^d(p^2i-p^2i-2)h_i · · (p odd), a(π^vee×π) · =N^2-1+∑_i=1^d3·4^i-1h_i · · (p=2). These identities do not…
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\begin{aligned} a(\pi^\vee\times\pi) &=p^{2d}-1+\sum_{i=1}^d(p^{2i}-p^{2i-2})h_i &&(p\text{ odd}),\\ a(\pi^\vee\times\pi) &=N^2-1+\sum_{i=1}^d3\cdot4^{i-1}h_i &&(p=2). \end{aligned}
\hlineIntroduction passage · 1 passage
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Introduction passage · Read passage · Paper_XI_Traditional_Final.tex, line 2163
For the same candidates the constructions give c|c|c · p odd · p=2 dim_ F_p(J^1/H^1) · p^d-1 · 2^d-2 Swan(π) · p^d-1(p+1)h_d · 3·2^d-1h_d and torsion number one. Here Swan denotes the Swan conductor, and a below denotes the ordinary pair conductor. An epsilon exponent for a specified additive character is a different quantity. The separate progress paper, Sections 22--24, reconciles these conventions and the actual-order hypotheses. Their self-pair conductor formulas are a(π^vee×π) · =p^2d-1+∑_i…
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\begin{array}{c|c|c} & p\text{ odd} & p=2\\ \hline \dim_{\FF_p}(J^1/H^1) & p^d-1 & 2^d-2\\ \Swan(\pi)&p^{d-1}(p+1)h_d&3\cdot2^{d-1}h_d \end{array}
\IndIntroduction passage · 14 passages
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Introduction passage · Paper_XI_Visual_Guide_Final.tex, line 151
RDINATESTensor refinement is a diagonal orbit problemOriginal Theorem 5.1 plate05_tensor_gridThe 4×6 grid separates into two diagonal translation orbits, labelled 0 and 1. The simultaneous translation action on Z/n Z× Z/m Z has d=gcd(n,m) orbits, each of length L=lcm(n,m). Mackey decomposition then gives Ind_W_K_n^W_Kξ⊗Ind_W_K_m^W_Kη ≃⊕_i=0^d-1 Ind_W_K_L^W_K(ξ_L,nη_L,m^Φ^i). Here the subscripts denote norm lifts and the Frobenius superscript follows the fixed source convention. The dimension ide…
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\Ind_{W_{K_n}}^{W_K}\xi\otimes\Ind_{W_{K_m}}^{W_K}\eta \simeq\bigoplus_{i=0}^{d-1} \Ind_{W_{K_L}}^{W_K}\bigl(\xi_{L,n}\eta_{L,m}^{\Phi^i}\bigr).
\Ind_{W_{K_L}}^{W_K}Introduction passage · 2 passages
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Introduction passage · Paper_XI_Visual_Guide_Final.tex, line 151
RDINATESTensor refinement is a diagonal orbit problemOriginal Theorem 5.1 plate05_tensor_gridThe 4×6 grid separates into two diagonal translation orbits, labelled 0 and 1. The simultaneous translation action on Z/n Z× Z/m Z has d=gcd(n,m) orbits, each of length L=lcm(n,m). Mackey decomposition then gives Ind_W_K_n^W_Kξ⊗Ind_W_K_m^W_Kη ≃⊕_i=0^d-1 Ind_W_K_L^W_K(ξ_L,nη_L,m^Φ^i). Here the subscripts denote norm lifts and the Frobenius superscript follows the fixed source convention. The dimension ide…
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\Ind_{W_{K_n}}^{W_K}\xi\otimes\Ind_{W_{K_m}}^{W_K}\eta \simeq\bigoplus_{i=0}^{d-1} \Ind_{W_{K_L}}^{W_K}\bigl(\xi_{L,n}\eta_{L,m}^{\Phi^i}\bigr).
\Ind_{W_{K_m}}^{W_K}Introduction passage · 2 passages
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Introduction passage · Paper_XI_Visual_Guide_Final.tex, line 151
RDINATESTensor refinement is a diagonal orbit problemOriginal Theorem 5.1 plate05_tensor_gridThe 4×6 grid separates into two diagonal translation orbits, labelled 0 and 1. The simultaneous translation action on Z/n Z× Z/m Z has d=gcd(n,m) orbits, each of length L=lcm(n,m). Mackey decomposition then gives Ind_W_K_n^W_Kξ⊗Ind_W_K_m^W_Kη ≃⊕_i=0^d-1 Ind_W_K_L^W_K(ξ_L,nη_L,m^Φ^i). Here the subscripts denote norm lifts and the Frobenius superscript follows the fixed source convention. The dimension ide…
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\Ind_{W_{K_n}}^{W_K}\xi\otimes\Ind_{W_{K_m}}^{W_K}\eta \simeq\bigoplus_{i=0}^{d-1} \Ind_{W_{K_L}}^{W_K}\bigl(\xi_{L,n}\eta_{L,m}^{\Phi^i}\bigr).
\Ind_{W_{K_n}}^{W_K}Introduction passage · 2 passages
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Introduction passage · Paper_XI_Visual_Guide_Final.tex, line 151
RDINATESTensor refinement is a diagonal orbit problemOriginal Theorem 5.1 plate05_tensor_gridThe 4×6 grid separates into two diagonal translation orbits, labelled 0 and 1. The simultaneous translation action on Z/n Z× Z/m Z has d=gcd(n,m) orbits, each of length L=lcm(n,m). Mackey decomposition then gives Ind_W_K_n^W_Kξ⊗Ind_W_K_m^W_Kη ≃⊕_i=0^d-1 Ind_W_K_L^W_K(ξ_L,nη_L,m^Φ^i). Here the subscripts denote norm lifts and the Frobenius superscript follows the fixed source convention. The dimension ide…
Exact mathematical passage
\Ind_{W_{K_n}}^{W_K}\xi\otimes\Ind_{W_{K_m}}^{W_K}\eta \simeq\bigoplus_{i=0}^{d-1} \Ind_{W_{K_L}}^{W_K}\bigl(\xi_{L,n}\eta_{L,m}^{\Phi^i}\bigr).
JIntroduction passage · 74 passages
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Introduction passage · Read passage · Paper_XI_Traditional_Final.tex, line 2163
For the same candidates the constructions give c|c|c · p odd · p=2 dim_ F_p(J^1/H^1) · p^d-1 · 2^d-2 Swan(π) · p^d-1(p+1)h_d · 3·2^d-1h_d and torsion number one. Here Swan denotes the Swan conductor, and a below denotes the ordinary pair conductor. An epsilon exponent for a specified additive character is a different quantity. The separate progress paper, Sections 22--24, reconciles these conventions and the actual-order hypotheses. Their self-pair conductor formulas are a(π^vee×π) · =p^2d-1+∑_i…
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\begin{array}{c|c|c} & p\text{ odd} & p=2\\ \hline \dim_{\FF_p}(J^1/H^1) & p^d-1 & 2^d-2\\ \Swan(\pi)&p^{d-1}(p+1)h_d&3\cdot2^{d-1}h_d \end{array}
J^1Introduction passage · 25 passages
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Introduction passage · Read passage · Paper_XI_Traditional_Final.tex, line 2163
For the same candidates the constructions give c|c|c · p odd · p=2 dim_ F_p(J^1/H^1) · p^d-1 · 2^d-2 Swan(π) · p^d-1(p+1)h_d · 3·2^d-1h_d and torsion number one. Here Swan denotes the Swan conductor, and a below denotes the ordinary pair conductor. An epsilon exponent for a specified additive character is a different quantity. The separate progress paper, Sections 22--24, reconciles these conventions and the actual-order hypotheses. Their self-pair conductor formulas are a(π^vee×π) · =p^2d-1+∑_i…
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\begin{array}{c|c|c} & p\text{ odd} & p=2\\ \hline \dim_{\FF_p}(J^1/H^1) & p^d-1 & 2^d-2\\ \Swan(\pi)&p^{d-1}(p+1)h_d&3\cdot2^{d-1}h_d \end{array}
KIntroduction passage · 85 passages
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Introduction passage · Read passage · Paper_XI_Traditional_Final.tex, line 2011
ater constructions provide actual arithmetic candidates and exact finite packet recovery in substantial restricted families. They do not establish [source reference] for every admitted source. Their detailed ordinary proofs are typeset in the appendices, and the original reviewed Markdown files accompany this edition. In these appendices F denotes the base local field, while K=F[β] denotes a parameter field; these are local notation conventions. All references to ``claim 66'' in the retained not…
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K=F[\beta]
K_LIntroduction passage · 5 passages
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Introduction passage · Paper_XI_Visual_Guide_Final.tex, line 151
RDINATESTensor refinement is a diagonal orbit problemOriginal Theorem 5.1 plate05_tensor_gridThe 4×6 grid separates into two diagonal translation orbits, labelled 0 and 1. The simultaneous translation action on Z/n Z× Z/m Z has d=gcd(n,m) orbits, each of length L=lcm(n,m). Mackey decomposition then gives Ind_W_K_n^W_Kξ⊗Ind_W_K_m^W_Kη ≃⊕_i=0^d-1 Ind_W_K_L^W_K(ξ_L,nη_L,m^Φ^i). Here the subscripts denote norm lifts and the Frobenius superscript follows the fixed source convention. The dimension ide…
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\Ind_{W_{K_n}}^{W_K}\xi\otimes\Ind_{W_{K_m}}^{W_K}\eta \simeq\bigoplus_{i=0}^{d-1} \Ind_{W_{K_L}}^{W_K}\bigl(\xi_{L,n}\eta_{L,m}^{\Phi^i}\bigr).
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Introduction passage · Paper_XI_Visual_Guide_Final.tex, line 151
RDINATESTensor refinement is a diagonal orbit problemOriginal Theorem 5.1 plate05_tensor_gridThe 4×6 grid separates into two diagonal translation orbits, labelled 0 and 1. The simultaneous translation action on Z/n Z× Z/m Z has d=gcd(n,m) orbits, each of length L=lcm(n,m). Mackey decomposition then gives Ind_W_K_n^W_Kξ⊗Ind_W_K_m^W_Kη ≃⊕_i=0^d-1 Ind_W_K_L^W_K(ξ_L,nη_L,m^Φ^i). Here the subscripts denote norm lifts and the Frobenius superscript follows the fixed source convention. The dimension ide…
Exact mathematical passage
\Ind_{W_{K_n}}^{W_K}\xi\otimes\Ind_{W_{K_m}}^{W_K}\eta \simeq\bigoplus_{i=0}^{d-1} \Ind_{W_{K_L}}^{W_K}\bigl(\xi_{L,n}\eta_{L,m}^{\Phi^i}\bigr).
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Introduction passage · Paper_XI_Visual_Guide_Final.tex, line 151
RDINATESTensor refinement is a diagonal orbit problemOriginal Theorem 5.1 plate05_tensor_gridThe 4×6 grid separates into two diagonal translation orbits, labelled 0 and 1. The simultaneous translation action on Z/n Z× Z/m Z has d=gcd(n,m) orbits, each of length L=lcm(n,m). Mackey decomposition then gives Ind_W_K_n^W_Kξ⊗Ind_W_K_m^W_Kη ≃⊕_i=0^d-1 Ind_W_K_L^W_K(ξ_L,nη_L,m^Φ^i). Here the subscripts denote norm lifts and the Frobenius superscript follows the fixed source convention. The dimension ide…
Exact mathematical passage
\Ind_{W_{K_n}}^{W_K}\xi\otimes\Ind_{W_{K_m}}^{W_K}\eta \simeq\bigoplus_{i=0}^{d-1} \Ind_{W_{K_L}}^{W_K}\bigl(\xi_{L,n}\eta_{L,m}^{\Phi^i}\bigr).
LIntroduction passage · 60 passages
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Introduction passage · Paper_XI_Visual_Guide_Final.tex, line 147
lessonPART I / FINITE TAME COORDINATESTensor refinement is a diagonal orbit problemOriginal Theorem 5.1 plate05_tensor_gridThe 4×6 grid separates into two diagonal translation orbits, labelled 0 and 1. The simultaneous translation action on Z/n Z× Z/m Z has d=gcd(n,m) orbits, each of length L=lcm(n,m). Mackey decomposition then gives Ind_W_K_n^W_Kξ⊗Ind_W_K_m^W_Kη ≃⊕_i=0^d-1 Ind_W_K_L^W_K(ξ_L,nη_L,m^Φ^i). Here the subscripts denote norm lifts and the Frobenius superscript follows the fixed source…
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d=\gcd(n,m)\quad\text{orbits, each of length}\quad L=\operatorname{lcm}(n,m).
\operatorname{lcm}Introduction passage · 1 passage
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Introduction passage · Paper_XI_Visual_Guide_Final.tex, line 147
lessonPART I / FINITE TAME COORDINATESTensor refinement is a diagonal orbit problemOriginal Theorem 5.1 plate05_tensor_gridThe 4×6 grid separates into two diagonal translation orbits, labelled 0 and 1. The simultaneous translation action on Z/n Z× Z/m Z has d=gcd(n,m) orbits, each of length L=lcm(n,m). Mackey decomposition then gives Ind_W_K_n^W_Kξ⊗Ind_W_K_m^W_Kη ≃⊕_i=0^d-1 Ind_W_K_L^W_K(ξ_L,nη_L,m^Φ^i). Here the subscripts denote norm lifts and the Frobenius superscript follows the fixed source…
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d=\gcd(n,m)\quad\text{orbits, each of length}\quad L=\operatorname{lcm}(n,m).
nIntroduction passage · 101 passages
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Introduction passage · Paper_XI_Visual_Guide_Final.tex, line 146
lessonPART I / FINITE TAME COORDINATESTensor refinement is a diagonal orbit problemOriginal Theorem 5.1 plate05_tensor_gridThe 4×6 grid separates into two diagonal translation orbits, labelled 0 and 1. The simultaneous translation action on Z/n Z× Z/m Z has d=gcd(n,m) orbits, each of length L=lcm(n,m). Mackey decomposition then gives Ind_W_K_n^W_Kξ⊗Ind_W_K_m^W_Kη ≃⊕_i=0^d-1 Ind_W_K_L^W_K(ξ_L,nη_L,m^Φ^i). Here the subscripts denote norm lifts and the Frobenius superscript follows the fixed source…
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\mathbb Z/n\mathbb Z\times\mathbb Z/m\mathbb Z
N^2Introduction passage · 3 passages
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Introduction passage · Read passage · Paper_XI_Traditional_Final.tex, line 2175
i) · p^d-1(p+1)h_d · 3·2^d-1h_d and torsion number one. Here Swan denotes the Swan conductor, and a below denotes the ordinary pair conductor. An epsilon exponent for a specified additive character is a different quantity. The separate progress paper, Sections 22--24, reconciles these conventions and the actual-order hypotheses. Their self-pair conductor formulas are a(π^vee×π) · =p^2d-1+∑_i=1^d(p^2i-p^2i-2)h_i · · (p odd), a(π^vee×π) · =N^2-1+∑_i=1^d3·4^i-1h_i · · (p=2). These identities do not…
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\begin{aligned} a(\pi^\vee\times\pi) &=p^{2d}-1+\sum_{i=1}^d(p^{2i}-p^{2i-2})h_i &&(p\text{ odd}),\\ a(\pi^\vee\times\pi) &=N^2-1+\sum_{i=1}^d3\cdot4^{i-1}h_i &&(p=2). \end{aligned}
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Introduction passage · Read passage · dyadic_character.tex, line 373
Let omega_F be any smooth quasicharacter trivial on U_F 1. Since O_F^×=U_F^1, it is determined by its value omega_F(t). The actual maximal type extends to K times J. Concretely, (K times J)/J is infinite cyclic, generated by a K-uniformizer. An intertwiner for its conjugation action on eta gives an extension Lambda_0. Denote its central character by omega_0.
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\OO_F^\times=U_F^1
\OO_F^\timesIntroduction passage · 1 passage
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Introduction passage · Read passage · dyadic_character.tex, line 373
Let omega_F be any smooth quasicharacter trivial on U_F 1. Since O_F^×=U_F^1, it is determined by its value omega_F(t). The actual maximal type extends to K times J. Concretely, (K times J)/J is infinite cyclic, generated by a K-uniformizer. An intertwiner for its conjugation action on eta gives an extension Lambda_0. Denote its central character by omega_0.
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\OO_F^\times=U_F^1
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Introduction passage · Read passage · rank2.tex, line 23
Write R=rad(A), U_A^a=1+R^a for atextgreater=1, and U_K^a=1+ p_K^a. Let theta be a full simple character. Since q is odd and the residue field is F2, the actual maximal simple type group and character groups coincide:
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U_K^a=1+\mathfrak p_K^a
p^{2d}Introduction passage · 3 passages
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Introduction passage · Read passage · Paper_XI_Traditional_Final.tex, line 2175
i) · p^d-1(p+1)h_d · 3·2^d-1h_d and torsion number one. Here Swan denotes the Swan conductor, and a below denotes the ordinary pair conductor. An epsilon exponent for a specified additive character is a different quantity. The separate progress paper, Sections 22--24, reconciles these conventions and the actual-order hypotheses. Their self-pair conductor formulas are a(π^vee×π) · =p^2d-1+∑_i=1^d(p^2i-p^2i-2)h_i · · (p odd), a(π^vee×π) · =N^2-1+∑_i=1^d3·4^i-1h_i · · (p=2). These identities do not…
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\begin{aligned} a(\pi^\vee\times\pi) &=p^{2d}-1+\sum_{i=1}^d(p^{2i}-p^{2i-2})h_i &&(p\text{ odd}),\\ a(\pi^\vee\times\pi) &=N^2-1+\sum_{i=1}^d3\cdot4^{i-1}h_i &&(p=2). \end{aligned}
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Introduction passage · Read passage · Paper_XI_Traditional_Final.tex, line 2175
i) · p^d-1(p+1)h_d · 3·2^d-1h_d and torsion number one. Here Swan denotes the Swan conductor, and a below denotes the ordinary pair conductor. An epsilon exponent for a specified additive character is a different quantity. The separate progress paper, Sections 22--24, reconciles these conventions and the actual-order hypotheses. Their self-pair conductor formulas are a(π^vee×π) · =p^2d-1+∑_i=1^d(p^2i-p^2i-2)h_i · · (p odd), a(π^vee×π) · =N^2-1+∑_i=1^d3·4^i-1h_i · · (p=2). These identities do not…
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\begin{aligned} a(\pi^\vee\times\pi) &=p^{2d}-1+\sum_{i=1}^d(p^{2i}-p^{2i-2})h_i &&(p\text{ odd}),\\ a(\pi^\vee\times\pi) &=N^2-1+\sum_{i=1}^d3\cdot4^{i-1}h_i &&(p=2). \end{aligned}
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Introduction passage · Read passage · Paper_XI_Traditional_Final.tex, line 2175
i) · p^d-1(p+1)h_d · 3·2^d-1h_d and torsion number one. Here Swan denotes the Swan conductor, and a below denotes the ordinary pair conductor. An epsilon exponent for a specified additive character is a different quantity. The separate progress paper, Sections 22--24, reconciles these conventions and the actual-order hypotheses. Their self-pair conductor formulas are a(π^vee×π) · =p^2d-1+∑_i=1^d(p^2i-p^2i-2)h_i · · (p odd), a(π^vee×π) · =N^2-1+∑_i=1^d3·4^i-1h_i · · (p=2). These identities do not…
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\begin{aligned} a(\pi^\vee\times\pi) &=p^{2d}-1+\sum_{i=1}^d(p^{2i}-p^{2i-2})h_i &&(p\text{ odd}),\\ a(\pi^\vee\times\pi) &=N^2-1+\sum_{i=1}^d3\cdot4^{i-1}h_i &&(p=2). \end{aligned}
p^dIntroduction passage · 10 passages
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Introduction passage · Read passage · Paper_XI_Traditional_Final.tex, line 2163
For the same candidates the constructions give c|c|c · p odd · p=2 dim_ F_p(J^1/H^1) · p^d-1 · 2^d-2 Swan(π) · p^d-1(p+1)h_d · 3·2^d-1h_d and torsion number one. Here Swan denotes the Swan conductor, and a below denotes the ordinary pair conductor. An epsilon exponent for a specified additive character is a different quantity. The separate progress paper, Sections 22--24, reconciles these conventions and the actual-order hypotheses. Their self-pair conductor formulas are a(π^vee×π) · =p^2d-1+∑_i…
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\begin{array}{c|c|c} & p\text{ odd} & p=2\\ \hline \dim_{\FF_p}(J^1/H^1) & p^d-1 & 2^d-2\\ \Swan(\pi)&p^{d-1}(p+1)h_d&3\cdot2^{d-1}h_d \end{array}
p^{d-1}Introduction passage · 9 passages
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Introduction passage · Read passage · Paper_XI_Traditional_Final.tex, line 2163
For the same candidates the constructions give c|c|c · p odd · p=2 dim_ F_p(J^1/H^1) · p^d-1 · 2^d-2 Swan(π) · p^d-1(p+1)h_d · 3·2^d-1h_d and torsion number one. Here Swan denotes the Swan conductor, and a below denotes the ordinary pair conductor. An epsilon exponent for a specified additive character is a different quantity. The separate progress paper, Sections 22--24, reconciles these conventions and the actual-order hypotheses. Their self-pair conductor formulas are a(π^vee×π) · =p^2d-1+∑_i…
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\begin{array}{c|c|c} & p\text{ odd} & p=2\\ \hline \dim_{\FF_p}(J^1/H^1) & p^d-1 & 2^d-2\\ \Swan(\pi)&p^{d-1}(p+1)h_d&3\cdot2^{d-1}h_d \end{array}
\mathfrak p_K^aIntroduction passage · 1 passage
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Introduction passage · Read passage · rank2.tex, line 23
Write R=rad(A), U_A^a=1+R^a for atextgreater=1, and U_K^a=1+ p_K^a. Let theta be a full simple character. Since q is odd and the residue field is F2, the actual maximal simple type group and character groups coincide:
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U_K^a=1+\mathfrak p_K^a
RIntroduction passage · 15 passages
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Introduction passage · Read passage · rank2.tex, line 22
Write R=rad(A), U_A^a=1+R^a for atextgreater=1, and U_K^a=1+ p_K^a. Let theta be a full simple character. Since q is odd and the residue field is F2, the actual maximal simple type group and character groups coincide:
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U_A^a=1+R^a
R^aIntroduction passage · 1 passage
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Introduction passage · Read passage · rank2.tex, line 22
Write R=rad(A), U_A^a=1+R^a for atextgreater=1, and U_K^a=1+ p_K^a. Let theta be a full simple character. Since q is odd and the residue field is F2, the actual maximal simple type group and character groups coincide:
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U_A^a=1+R^a
\SwanIntroduction passage · 4 passages
Search aliases: Swan operatorname
Introduction passage · Read passage · Paper_XI_Traditional_Final.tex, line 2163
For the same candidates the constructions give c|c|c · p odd · p=2 dim_ F_p(J^1/H^1) · p^d-1 · 2^d-2 Swan(π) · p^d-1(p+1)h_d · 3·2^d-1h_d and torsion number one. Here Swan denotes the Swan conductor, and a below denotes the ordinary pair conductor. An epsilon exponent for a specified additive character is a different quantity. The separate progress paper, Sections 22--24, reconciles these conventions and the actual-order hypotheses. Their self-pair conductor formulas are a(π^vee×π) · =p^2d-1+∑_i…
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\begin{array}{c|c|c} & p\text{ odd} & p=2\\ \hline \dim_{\FF_p}(J^1/H^1) & p^d-1 & 2^d-2\\ \Swan(\pi)&p^{d-1}(p+1)h_d&3\cdot2^{d-1}h_d \end{array}
tIntroduction passage · 50 passages
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Introduction passage · Paper_XI_Visual_Guide_Final.tex, line 58
The symbol K denotes the original local field in the finite-theory discussion. In the explicit arithmetic constructions, F= F_p((t)) is the base field and K=F[β] is its totally ramified parameter field. A symplectic quotient dimension and a Heisenberg representation dimension are different quantities.
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F=\F_p((t))
\TrIntroduction passage · 8 passages
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Introduction passage · Read passage · Paper_XI_Traditional_Final.tex, line 1299
[Fixed point free Weil trace] Under the declared normalization, Trω(g) = (frac-1p)^r (fracdet(g-I)p). Here (frac·p) denotes the quadratic character of F_p^×.
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\boxed{ \Tr\omega(g) = \left(\frac{-1}{p}\right)^r \left(\frac{\det(g-I)}{p}\right). }
UIntroduction passage · 101 passages
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Introduction passage · Read passage · dyadic_character.tex, line 373
Let omega_F be any smooth quasicharacter trivial on U_F 1. Since O_F^×=U_F^1, it is determined by its value omega_F(t). The actual maximal type extends to K times J. Concretely, (K times J)/J is infinite cyclic, generated by a K-uniformizer. An intertwiner for its conjugation action on eta gives an extension Lambda_0. Denote its central character by omega_0.
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\OO_F^\times=U_F^1
U_A^aIntroduction passage · 2 passages
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Introduction passage · Read passage · rank2.tex, line 22
Write R=rad(A), U_A^a=1+R^a for atextgreater=1, and U_K^a=1+ p_K^a. Let theta be a full simple character. Since q is odd and the residue field is F2, the actual maximal simple type group and character groups coincide:
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U_A^a=1+R^a
U_F^1Introduction passage · 20 passages
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Introduction passage · Read passage · dyadic_character.tex, line 373
Let omega_F be any smooth quasicharacter trivial on U_F 1. Since O_F^×=U_F^1, it is determined by its value omega_F(t). The actual maximal type extends to K times J. Concretely, (K times J)/J is infinite cyclic, generated by a K-uniformizer. An intertwiner for its conjugation action on eta gives an extension Lambda_0. Denote its central character by omega_0.
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\OO_F^\times=U_F^1
U_K^aIntroduction passage · 1 passage
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Introduction passage · Read passage · rank2.tex, line 23
Write R=rad(A), U_A^a=1+R^a for atextgreater=1, and U_K^a=1+ p_K^a. Let theta be a full simple character. Since q is odd and the residue field is F2, the actual maximal simple type group and character groups coincide:
Exact mathematical passage
U_K^a=1+\mathfrak p_K^a
\veeIntroduction passage · 7 passages
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Introduction passage · Read passage · Paper_XI_Traditional_Final.tex, line 2175
i) · p^d-1(p+1)h_d · 3·2^d-1h_d and torsion number one. Here Swan denotes the Swan conductor, and a below denotes the ordinary pair conductor. An epsilon exponent for a specified additive character is a different quantity. The separate progress paper, Sections 22--24, reconciles these conventions and the actual-order hypotheses. Their self-pair conductor formulas are a(π^vee×π) · =p^2d-1+∑_i=1^d(p^2i-p^2i-2)h_i · · (p odd), a(π^vee×π) · =N^2-1+∑_i=1^d3·4^i-1h_i · · (p=2). These identities do not…
Exact mathematical passage
\begin{aligned} a(\pi^\vee\times\pi) &=p^{2d}-1+\sum_{i=1}^d(p^{2i}-p^{2i-2})h_i &&(p\text{ odd}),\\ a(\pi^\vee\times\pi) &=N^2-1+\sum_{i=1}^d3\cdot4^{i-1}h_i &&(p=2). \end{aligned}
W_KIntroduction passage · 17 passages
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Introduction passage · Paper_XI_Visual_Guide_Final.tex, line 151
RDINATESTensor refinement is a diagonal orbit problemOriginal Theorem 5.1 plate05_tensor_gridThe 4×6 grid separates into two diagonal translation orbits, labelled 0 and 1. The simultaneous translation action on Z/n Z× Z/m Z has d=gcd(n,m) orbits, each of length L=lcm(n,m). Mackey decomposition then gives Ind_W_K_n^W_Kξ⊗Ind_W_K_m^W_Kη ≃⊕_i=0^d-1 Ind_W_K_L^W_K(ξ_L,nη_L,m^Φ^i). Here the subscripts denote norm lifts and the Frobenius superscript follows the fixed source convention. The dimension ide…
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\Ind_{W_{K_n}}^{W_K}\xi\otimes\Ind_{W_{K_m}}^{W_K}\eta \simeq\bigoplus_{i=0}^{d-1} \Ind_{W_{K_L}}^{W_K}\bigl(\xi_{L,n}\eta_{L,m}^{\Phi^i}\bigr).
W_{K_L}Introduction passage · 4 passages
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Introduction passage · Paper_XI_Visual_Guide_Final.tex, line 151
RDINATESTensor refinement is a diagonal orbit problemOriginal Theorem 5.1 plate05_tensor_gridThe 4×6 grid separates into two diagonal translation orbits, labelled 0 and 1. The simultaneous translation action on Z/n Z× Z/m Z has d=gcd(n,m) orbits, each of length L=lcm(n,m). Mackey decomposition then gives Ind_W_K_n^W_Kξ⊗Ind_W_K_m^W_Kη ≃⊕_i=0^d-1 Ind_W_K_L^W_K(ξ_L,nη_L,m^Φ^i). Here the subscripts denote norm lifts and the Frobenius superscript follows the fixed source convention. The dimension ide…
Exact mathematical passage
\Ind_{W_{K_n}}^{W_K}\xi\otimes\Ind_{W_{K_m}}^{W_K}\eta \simeq\bigoplus_{i=0}^{d-1} \Ind_{W_{K_L}}^{W_K}\bigl(\xi_{L,n}\eta_{L,m}^{\Phi^i}\bigr).
W_{K_m}Introduction passage · 3 passages
Search aliases: W_K_m
Introduction passage · Paper_XI_Visual_Guide_Final.tex, line 151
RDINATESTensor refinement is a diagonal orbit problemOriginal Theorem 5.1 plate05_tensor_gridThe 4×6 grid separates into two diagonal translation orbits, labelled 0 and 1. The simultaneous translation action on Z/n Z× Z/m Z has d=gcd(n,m) orbits, each of length L=lcm(n,m). Mackey decomposition then gives Ind_W_K_n^W_Kξ⊗Ind_W_K_m^W_Kη ≃⊕_i=0^d-1 Ind_W_K_L^W_K(ξ_L,nη_L,m^Φ^i). Here the subscripts denote norm lifts and the Frobenius superscript follows the fixed source convention. The dimension ide…
Exact mathematical passage
\Ind_{W_{K_n}}^{W_K}\xi\otimes\Ind_{W_{K_m}}^{W_K}\eta \simeq\bigoplus_{i=0}^{d-1} \Ind_{W_{K_L}}^{W_K}\bigl(\xi_{L,n}\eta_{L,m}^{\Phi^i}\bigr).
W_{K_n}Introduction passage · 3 passages
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Introduction passage · Paper_XI_Visual_Guide_Final.tex, line 151
RDINATESTensor refinement is a diagonal orbit problemOriginal Theorem 5.1 plate05_tensor_gridThe 4×6 grid separates into two diagonal translation orbits, labelled 0 and 1. The simultaneous translation action on Z/n Z× Z/m Z has d=gcd(n,m) orbits, each of length L=lcm(n,m). Mackey decomposition then gives Ind_W_K_n^W_Kξ⊗Ind_W_K_m^W_Kη ≃⊕_i=0^d-1 Ind_W_K_L^W_K(ξ_L,nη_L,m^Φ^i). Here the subscripts denote norm lifts and the Frobenius superscript follows the fixed source convention. The dimension ide…
Exact mathematical passage
\Ind_{W_{K_n}}^{W_K}\xi\otimes\Ind_{W_{K_m}}^{W_K}\eta \simeq\bigoplus_{i=0}^{d-1} \Ind_{W_{K_L}}^{W_K}\bigl(\xi_{L,n}\eta_{L,m}^{\Phi^i}\bigr).
\mathbb ZIntroduction passage · 4 passages
Search aliases: Z mathbb
Introduction passage · Paper_XI_Visual_Guide_Final.tex, line 146
lessonPART I / FINITE TAME COORDINATESTensor refinement is a diagonal orbit problemOriginal Theorem 5.1 plate05_tensor_gridThe 4×6 grid separates into two diagonal translation orbits, labelled 0 and 1. The simultaneous translation action on Z/n Z× Z/m Z has d=gcd(n,m) orbits, each of length L=lcm(n,m). Mackey decomposition then gives Ind_W_K_n^W_Kξ⊗Ind_W_K_m^W_Kη ≃⊕_i=0^d-1 Ind_W_K_L^W_K(ξ_L,nη_L,m^Φ^i). Here the subscripts denote norm lifts and the Frobenius superscript follows the fixed source…
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\mathbb Z/n\mathbb Z\times\mathbb Z/m\mathbb Z
\cdotIntroduction passage · 8 passages
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Introduction passage · Read passage · Paper_XI_Traditional_Final.tex, line 1307
[Fixed point free Weil trace] Under the declared normalization, Trω(g) = (frac-1p)^r (fracdet(g-I)p). Here (frac·p) denotes the quadratic character of F_p^×.
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(\frac{\cdot}{p})
\betaIntroduction passage · 9 passages
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Introduction passage · Read passage · Paper_XI_Traditional_Final.tex, line 2011
ater constructions provide actual arithmetic candidates and exact finite packet recovery in substantial restricted families. They do not establish [source reference] for every admitted source. Their detailed ordinary proofs are typeset in the appendices, and the original reviewed Markdown files accompany this edition. In these appendices F denotes the base local field, while K=F[β] denotes a parameter field; these are local notation conventions. All references to ``claim 66'' in the retained not…
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K=F[\beta]
\etaIntroduction passage · 29 passages
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Introduction passage · Paper_XI_Visual_Guide_Final.tex, line 151
RDINATESTensor refinement is a diagonal orbit problemOriginal Theorem 5.1 plate05_tensor_gridThe 4×6 grid separates into two diagonal translation orbits, labelled 0 and 1. The simultaneous translation action on Z/n Z× Z/m Z has d=gcd(n,m) orbits, each of length L=lcm(n,m). Mackey decomposition then gives Ind_W_K_n^W_Kξ⊗Ind_W_K_m^W_Kη ≃⊕_i=0^d-1 Ind_W_K_L^W_K(ξ_L,nη_L,m^Φ^i). Here the subscripts denote norm lifts and the Frobenius superscript follows the fixed source convention. The dimension ide…
Exact mathematical passage
\Ind_{W_{K_n}}^{W_K}\xi\otimes\Ind_{W_{K_m}}^{W_K}\eta \simeq\bigoplus_{i=0}^{d-1} \Ind_{W_{K_L}}^{W_K}\bigl(\xi_{L,n}\eta_{L,m}^{\Phi^i}\bigr).
\eta_{L,m}^{\Phi^i}Introduction passage · 2 passages
Search aliases: η_L,m^Φ^i eta Phi
Introduction passage · Paper_XI_Visual_Guide_Final.tex, line 151
RDINATESTensor refinement is a diagonal orbit problemOriginal Theorem 5.1 plate05_tensor_gridThe 4×6 grid separates into two diagonal translation orbits, labelled 0 and 1. The simultaneous translation action on Z/n Z× Z/m Z has d=gcd(n,m) orbits, each of length L=lcm(n,m). Mackey decomposition then gives Ind_W_K_n^W_Kξ⊗Ind_W_K_m^W_Kη ≃⊕_i=0^d-1 Ind_W_K_L^W_K(ξ_L,nη_L,m^Φ^i). Here the subscripts denote norm lifts and the Frobenius superscript follows the fixed source convention. The dimension ide…
Exact mathematical passage
\Ind_{W_{K_n}}^{W_K}\xi\otimes\Ind_{W_{K_m}}^{W_K}\eta \simeq\bigoplus_{i=0}^{d-1} \Ind_{W_{K_L}}^{W_K}\bigl(\xi_{L,n}\eta_{L,m}^{\Phi^i}\bigr).
\xi_{L,n}Introduction passage · 3 passages
Search aliases: ξ_L,n xi
Introduction passage · Paper_XI_Visual_Guide_Final.tex, line 151
RDINATESTensor refinement is a diagonal orbit problemOriginal Theorem 5.1 plate05_tensor_gridThe 4×6 grid separates into two diagonal translation orbits, labelled 0 and 1. The simultaneous translation action on Z/n Z× Z/m Z has d=gcd(n,m) orbits, each of length L=lcm(n,m). Mackey decomposition then gives Ind_W_K_n^W_Kξ⊗Ind_W_K_m^W_Kη ≃⊕_i=0^d-1 Ind_W_K_L^W_K(ξ_L,nη_L,m^Φ^i). Here the subscripts denote norm lifts and the Frobenius superscript follows the fixed source convention. The dimension ide…
Exact mathematical passage
\Ind_{W_{K_n}}^{W_K}\xi\otimes\Ind_{W_{K_m}}^{W_K}\eta \simeq\bigoplus_{i=0}^{d-1} \Ind_{W_{K_L}}^{W_K}\bigl(\xi_{L,n}\eta_{L,m}^{\Phi^i}\bigr).
\piIntroduction passage · 44 passages
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Introduction passage · Read passage · Paper_XI_Traditional_Final.tex, line 2163
For the same candidates the constructions give c|c|c · p odd · p=2 dim_ F_p(J^1/H^1) · p^d-1 · 2^d-2 Swan(π) · p^d-1(p+1)h_d · 3·2^d-1h_d and torsion number one. Here Swan denotes the Swan conductor, and a below denotes the ordinary pair conductor. An epsilon exponent for a specified additive character is a different quantity. The separate progress paper, Sections 22--24, reconciles these conventions and the actual-order hypotheses. Their self-pair conductor formulas are a(π^vee×π) · =p^2d-1+∑_i…
Exact mathematical passage
\begin{array}{c|c|c} & p\text{ odd} & p=2\\ \hline \dim_{\FF_p}(J^1/H^1) & p^d-1 & 2^d-2\\ \Swan(\pi)&p^{d-1}(p+1)h_d&3\cdot2^{d-1}h_d \end{array}
\pi^\veeIntroduction passage · 5 passages
Search aliases: π^vee pi vee
Introduction passage · Read passage · Paper_XI_Traditional_Final.tex, line 2175
i) · p^d-1(p+1)h_d · 3·2^d-1h_d and torsion number one. Here Swan denotes the Swan conductor, and a below denotes the ordinary pair conductor. An epsilon exponent for a specified additive character is a different quantity. The separate progress paper, Sections 22--24, reconciles these conventions and the actual-order hypotheses. Their self-pair conductor formulas are a(π^vee×π) · =p^2d-1+∑_i=1^d(p^2i-p^2i-2)h_i · · (p odd), a(π^vee×π) · =N^2-1+∑_i=1^d3·4^i-1h_i · · (p=2). These identities do not…
Exact mathematical passage
\begin{aligned} a(\pi^\vee\times\pi) &=p^{2d}-1+\sum_{i=1}^d(p^{2i}-p^{2i-2})h_i &&(p\text{ odd}),\\ a(\pi^\vee\times\pi) &=N^2-1+\sum_{i=1}^d3\cdot4^{i-1}h_i &&(p=2). \end{aligned}
\Phi^iIntroduction passage · 2 passages
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Introduction passage · Paper_XI_Visual_Guide_Final.tex, line 151
RDINATESTensor refinement is a diagonal orbit problemOriginal Theorem 5.1 plate05_tensor_gridThe 4×6 grid separates into two diagonal translation orbits, labelled 0 and 1. The simultaneous translation action on Z/n Z× Z/m Z has d=gcd(n,m) orbits, each of length L=lcm(n,m). Mackey decomposition then gives Ind_W_K_n^W_Kξ⊗Ind_W_K_m^W_Kη ≃⊕_i=0^d-1 Ind_W_K_L^W_K(ξ_L,nη_L,m^Φ^i). Here the subscripts denote norm lifts and the Frobenius superscript follows the fixed source convention. The dimension ide…
Exact mathematical passage
\Ind_{W_{K_n}}^{W_K}\xi\otimes\Ind_{W_{K_m}}^{W_K}\eta \simeq\bigoplus_{i=0}^{d-1} \Ind_{W_{K_L}}^{W_K}\bigl(\xi_{L,n}\eta_{L,m}^{\Phi^i}\bigr).
\omegaIntroduction passage · 15 passages
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Introduction passage · Read passage · Paper_XI_Traditional_Final.tex, line 1299
[Fixed point free Weil trace] Under the declared normalization, Trω(g) = (frac-1p)^r (fracdet(g-I)p). Here (frac·p) denotes the quadratic character of F_p^×.
Exact mathematical passage
\boxed{ \Tr\omega(g) = \left(\frac{-1}{p}\right)^r \left(\frac{\det(g-I)}{p}\right). }
\sum_{i=1}^dIntroduction passage · 5 passages
Search aliases: ∑_i=1^d sum
Introduction passage · Read passage · Paper_XI_Traditional_Final.tex, line 2175
i) · p^d-1(p+1)h_d · 3·2^d-1h_d and torsion number one. Here Swan denotes the Swan conductor, and a below denotes the ordinary pair conductor. An epsilon exponent for a specified additive character is a different quantity. The separate progress paper, Sections 22--24, reconciles these conventions and the actual-order hypotheses. Their self-pair conductor formulas are a(π^vee×π) · =p^2d-1+∑_i=1^d(p^2i-p^2i-2)h_i · · (p odd), a(π^vee×π) · =N^2-1+∑_i=1^d3·4^i-1h_i · · (p=2). These identities do not…
Exact mathematical passage
\begin{aligned} a(\pi^\vee\times\pi) &=p^{2d}-1+\sum_{i=1}^d(p^{2i}-p^{2i-2})h_i &&(p\text{ odd}),\\ a(\pi^\vee\times\pi) &=N^2-1+\sum_{i=1}^d3\cdot4^{i-1}h_i &&(p=2). \end{aligned}
\simeqIntroduction passage · 22 passages
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Introduction passage · Paper_XI_Visual_Guide_Final.tex, line 151
RDINATESTensor refinement is a diagonal orbit problemOriginal Theorem 5.1 plate05_tensor_gridThe 4×6 grid separates into two diagonal translation orbits, labelled 0 and 1. The simultaneous translation action on Z/n Z× Z/m Z has d=gcd(n,m) orbits, each of length L=lcm(n,m). Mackey decomposition then gives Ind_W_K_n^W_Kξ⊗Ind_W_K_m^W_Kη ≃⊕_i=0^d-1 Ind_W_K_L^W_K(ξ_L,nη_L,m^Φ^i). Here the subscripts denote norm lifts and the Frobenius superscript follows the fixed source convention. The dimension ide…
Exact mathematical passage
\Ind_{W_{K_n}}^{W_K}\xi\otimes\Ind_{W_{K_m}}^{W_K}\eta \simeq\bigoplus_{i=0}^{d-1} \Ind_{W_{K_L}}^{W_K}\bigl(\xi_{L,n}\eta_{L,m}^{\Phi^i}\bigr).
\bigoplus_{i=0}^{d-1}Introduction passage · 2 passages
Search aliases: ⊕_i=0^d-1 bigoplus
Introduction passage · Paper_XI_Visual_Guide_Final.tex, line 151
RDINATESTensor refinement is a diagonal orbit problemOriginal Theorem 5.1 plate05_tensor_gridThe 4×6 grid separates into two diagonal translation orbits, labelled 0 and 1. The simultaneous translation action on Z/n Z× Z/m Z has d=gcd(n,m) orbits, each of length L=lcm(n,m). Mackey decomposition then gives Ind_W_K_n^W_Kξ⊗Ind_W_K_m^W_Kη ≃⊕_i=0^d-1 Ind_W_K_L^W_K(ξ_L,nη_L,m^Φ^i). Here the subscripts denote norm lifts and the Frobenius superscript follows the fixed source convention. The dimension ide…
Exact mathematical passage
\Ind_{W_{K_n}}^{W_K}\xi\otimes\Ind_{W_{K_m}}^{W_K}\eta \simeq\bigoplus_{i=0}^{d-1} \Ind_{W_{K_L}}^{W_K}\bigl(\xi_{L,n}\eta_{L,m}^{\Phi^i}\bigr).
\otimesIntroduction passage · 13 passages
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Introduction passage · Paper_XI_Visual_Guide_Final.tex, line 151
RDINATESTensor refinement is a diagonal orbit problemOriginal Theorem 5.1 plate05_tensor_gridThe 4×6 grid separates into two diagonal translation orbits, labelled 0 and 1. The simultaneous translation action on Z/n Z× Z/m Z has d=gcd(n,m) orbits, each of length L=lcm(n,m). Mackey decomposition then gives Ind_W_K_n^W_Kξ⊗Ind_W_K_m^W_Kη ≃⊕_i=0^d-1 Ind_W_K_L^W_K(ξ_L,nη_L,m^Φ^i). Here the subscripts denote norm lifts and the Frobenius superscript follows the fixed source convention. The dimension ide…
Exact mathematical passage
\Ind_{W_{K_n}}^{W_K}\xi\otimes\Ind_{W_{K_m}}^{W_K}\eta \simeq\bigoplus_{i=0}^{d-1} \Ind_{W_{K_L}}^{W_K}\bigl(\xi_{L,n}\eta_{L,m}^{\Phi^i}\bigr).
*Use in source · 11 passages
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Use in source · Read passage · Paper_XI_Traditional_Final.tex, line 790
Let p be odd. Let X be an r dimensional vector space over F_p, and put V=X⊕ X^*. Define the alternating form b((x,ξ),(y,η)) = η(x)-ξ(y). Let Z= F_p, and define the Heisenberg group H(V)=V× Z with multiplication (v,z)(w,t) = (v+w,z+t+frac12 b(v,w)). Fix the additive character ψ(z)= e^2π iz/p.
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V=X\oplus X^*.
\mathbf1Use in source · 1 passage
Search aliases: 1 mathbf
Use in source · Read passage · dyadic_synthesis.tex, line 134
The dyadic floors cannot be replaced by the odd-prime dimension formula. At stage j, the centralizer grade has dimension 2^d-j. Its contribution to J^1/H^1 is (2^d-j+1-2^d-j) 1_n_j even. Every n_j for j<d is even; n_d=q_1 is odd and contributes zero. The sum is exactly N-2. BK nondegeneracy makes this a symplectic space. A whole source packet of d planes fits if and only if N-2≥2d, equivalently d≥3.
Exact mathematical passage
\bigl(2^{d-j+1}-2^{d-j}\bigr)\,\mathbf1_{n_j\text{ even}}.
\mathbf1_{n_j\text{ even}}Use in source · 1 passage
Search aliases: 1_n_j even mathbf text
Use in source · Read passage · dyadic_synthesis.tex, line 134
The dyadic floors cannot be replaced by the odd-prime dimension formula. At stage j, the centralizer grade has dimension 2^d-j. Its contribution to J^1/H^1 is (2^d-j+1-2^d-j) 1_n_j even. Every n_j for j<d is even; n_d=q_1 is odd and contributes zero. The sum is exactly N-2. BK nondegeneracy makes this a symplectic space. A whole source packet of d planes fits if and only if N-2≥2d, equivalently d≥3.
Exact mathematical passage
\bigl(2^{d-j+1}-2^{d-j}\bigr)\,\mathbf1_{n_j\text{ even}}.
\mathcal AUse in source · 6 passages
Search aliases: A mathcal
Use in source · Read passage · Paper_XI_Traditional_Final.tex, line 1839
Fix the paper's reciprocity, additive-character, local-factor, and phase normalizations. Let A be the declared class of irreducible wild Weil atoms under consideration. For ρ∈ A, the supplied assembled finite datum d_ρ consists of the following inputs, historically denoted H1--H4: item an intrinsic finite break decomposition on the Galois side; item the declared equivariant symplectic bridges to the supplied breakwise type-side quotients; item compatible odd or dyadic central-extension and phase…
Exact mathematical passage
\mathcal A
a^{-1}Use in source · 3 passages
Search aliases: a^-1
Use in source · Read passage · Paper_XI_Traditional_Final.tex, line 1420
[Hyperbolic phase formula] Let U be a finite dimensional F_p space, let a∈GL(U) have no eigenvalue one, and let g act on H(U)=U⊕ U^* by g=(a,(a^-1)^*). Then Φ_p(g,H(U)) = (fracdet ap).
Exact mathematical passage
g=(a,(a^{-1})^*).
a^2Use in source · 2 passages
Search aliases: a^2
Use in source · Read passage · dyadic_synthesis.tex, line 182
he scalar-pushout realization contract, with no independently fixed conflicting host marking, choose a labelled symplectic embedding into W. Matching commutators gives a center-fixing isomorphism between the source C^× extension and the corresponding host restriction. Retain the image of the whole marked finite subgroup and every lift. For this isomorphism F_0, F_0(a)^2=F_0(a^2). Thus its square bits, refinement and Gauss sign survive, along with all supplied full character values. They are not …
Exact mathematical passage
F_0(a)^2=F_0(a^2).
a^nUse in source · 2 passages
Search aliases: a^n
Use in source · Paper_XI_Visual_Guide_Final.tex, line 619
tching does not construct realizationBlocks 162, 166--167, profile and capacity audits For compact J⊂GL_n(F), the homomorphism v_F∘det has finite image in Z, hence image zero. An unramified character ν_a(g)=a^v_F(det g) is therefore trivial on J. The compact packet restrictions of π and π⊗ν_a agree, while the central character changes by a^n on a scalar uniformizer. Choosing a^n≠1 gives a different atom.
Exact mathematical passage
a^n
A^{\mathrm{symp}}Use in source · 2 passages
Search aliases: A^symp mathrm
Use in source · Read passage · Paper_XI_Traditional_Final.tex, line 987
[Radical free envelope] For an arbitrary scalar projective action of A, the quotient A^symp=A/rad b_c is the maximal quotient detected by noncommuting projective intertwiners. If A is an elementary abelian p group, then A^symp is a symplectic F_p space.
Exact mathematical passage
\boxed{ A^{\mathrm{symp}}=A/\rad b_c }
a^{v_F(\det g)}Use in source · 1 passage
Search aliases: a^v_F(det g) det
Use in source · Paper_XI_Visual_Guide_Final.tex, line 616
lessonPART IV / REALIZATION AND COMPARISONNumerical matching does not construct realizationBlocks 162, 166--167, profile and capacity audits For compact J⊂GL_n(F), the homomorphism v_F∘det has finite image in Z, hence image zero. An unramified character ν_a(g)=a^v_F(det g) is therefore trivial on J. The compact packet restrictions of π and π⊗ν_a agree, while the central character changes by a^n on a scalar uniformizer. Choosing a^n≠1 gives a different atom.
Exact mathematical passage
\nu_a(g)=a^{v_F(\det g)}
A_0Use in source · 2 passages
Search aliases: A_0
Use in source · Read passage · dyadic_synthesis.tex, line 361
There is also a recovery from compact data explicitly recording that root. With A_0=Λ(ϖ_K) and X=c - Ind_J^Gη, define (Tf)(g)=A_0^-1f(ϖ_Kg). Then X/(T-1)X≃π. After imposing the fixed central character, N/(F^× J) is cyclic of order N, and the finite average N^-1∑_i=0^N-1T^i selects this same extension. These operators act on complex representation spaces, so their scalar division by N is valid. This is a different recovery from the Block159 projector onto a sector of η. The exact kernel and finit…
Exact mathematical passage
A_0=\Lambda(\varpi_K)
a_0Use in source · 3 passages
Search aliases: a_0
Use in source · Read passage · odd_synthesis.tex, line 60
recursive base (L,t_L), introduce E=L(α), α^p-α=t_L^-q_d. This is a separable totally ramified extension of degree p, with v_E(α)=-q_d and Tr_E/Lα=0. Choose 1≤ b<p with q_db≡-1pmod p, set a=(q_db+1)/p, and put z=t_L^aα^b. Then z is an explicit uniformizer. Write its Eisenstein polynomial as f(X)=X^p+⋯+a_0.
Exact mathematical passage
f(X)=X^p+\cdots+a_0
A_0^{-1}Use in source · 2 passages
Search aliases: A_0^-1
Use in source · Read passage · dyadic_synthesis.tex, line 363
There is also a recovery from compact data explicitly recording that root. With A_0=Λ(ϖ_K) and X=c - Ind_J^Gη, define (Tf)(g)=A_0^-1f(ϖ_Kg). Then X/(T-1)X≃π. After imposing the fixed central character, N/(F^× J) is cyclic of order N, and the finite average N^-1∑_i=0^N-1T^i selects this same extension. These operators act on complex representation spaces, so their scalar division by N is valid. This is a different recovery from the Block159 projector onto a sector of η. The exact kernel and finit…
Exact mathematical passage
(Tf)(g)=A_0^{-1}f(\varpi_Kg)
A_eUse in source · 10 passages
Search aliases: A_e
Use in source · Read passage · Paper_XI_Traditional_Final.tex, line 1612
Let Γ be a connected finite graph. At every vertex v, place finite dimensional complex vector spaces V_v and W_v of the same dimension. For every oriented edge e:u→ v, let A_e:V_u⟶ V_v, B_e:W_u⟶ W_v be invertible. The reversed edge carries the inverse map.
Exact mathematical passage
A_e:V_u\longrightarrow V_v, \qquad B_e:W_u\longrightarrow W_v
a_iUse in source · 8 passages
Search aliases: a_i
Use in source · Read passage · Paper_XI_Traditional_Final.tex, line 658
[Tame tensor local factor] The local factor of the tame tensor product is the product of (1-a_i u^r_i)^-1 over precisely those refined packets whose inertia exponent vanishes.
Exact mathematical passage
(1-a_i u^{r_i})^{-1}
a_{p-1}Use in source · 1 passage
Search aliases: a_p-1
Use in source · Read passage · odd_tower.tex, line 59
Then v_E(z)=pa-qb=1, so z is a uniformizer and E=L(z). Its monic minimal polynomial f(X)=X^p+a_p-1X^p-1+⋯+a_0 is Eisenstein. It is an explicit polynomial: compute the product over the p conjugates
Exact mathematical passage
f(X)=X^p+a_{p-1}X^{p-1}+\cdots+a_0
A_{p_u}Use in source · 2 passages
Search aliases: A_p_u
Use in source · Read passage · Paper_XI_Traditional_Final.tex, line 1698
[Exact edge residual from cycle defect] Fix a spanning tree and construct the vertex maps by tree transport from U_r. Let e:u→ v be a non tree edge. Let p_u and p_v be the tree paths from the root to u and v. Define the fundamental cycle defect D_e = U_rA_p_v^-1A_eA_p_u - B_p_v^-1B_eB_p_uU_r. Then the edge residual is exactly U_vA_e-B_eU_u = B_p_vD_eA_p_u^-1.
Exact mathematical passage
D_e = U_rA_{p_v}^{-1}A_eA_{p_u} - B_{p_v}^{-1}B_eB_{p_u}U_r.
A_{p_u}^{-1}Use in source · 6 passages
Search aliases: A_p_u^-1
Use in source · Read passage · Paper_XI_Traditional_Final.tex, line 1706
ct edge residual from cycle defect] Fix a spanning tree and construct the vertex maps by tree transport from U_r. Let e:u→ v be a non tree edge. Let p_u and p_v be the tree paths from the root to u and v. Define the fundamental cycle defect D_e = U_rA_p_v^-1A_eA_p_u - B_p_v^-1B_eB_p_uU_r. Then the edge residual is exactly U_vA_e-B_eU_u = B_p_vD_eA_p_u^-1.
Exact mathematical passage
\boxed{ U_vA_e-B_eU_u = B_{p_v}D_eA_{p_u}^{-1}. }
A_{p_v}^{-1}Use in source · 4 passages
Search aliases: A_p_v^-1
Use in source · Read passage · Paper_XI_Traditional_Final.tex, line 1698
[Exact edge residual from cycle defect] Fix a spanning tree and construct the vertex maps by tree transport from U_r. Let e:u→ v be a non tree edge. Let p_u and p_v be the tree paths from the root to u and v. Define the fundamental cycle defect D_e = U_rA_p_v^-1A_eA_p_u - B_p_v^-1B_eB_p_uU_r. Then the edge residual is exactly U_vA_e-B_eU_u = B_p_vD_eA_p_u^-1.
Exact mathematical passage
D_e = U_rA_{p_v}^{-1}A_eA_{p_u} - B_{p_v}^{-1}B_eB_{p_u}U_r.
\ArtUse in source · 1 passage
Search aliases: Art operatorname
Use in source · Read passage · Paper_XI_Traditional_Final.tex, line 697
[Conductor of a nontrivial tame atom] Let τ be an irreducible tame Weil representation of dimension n>1. Then Swan(τ)=0, dimτ^I_K=0, Art(τ)=n.
Exact mathematical passage
\Swan(\tau)=0, \qquad \dim\tau^{I_K}=0, \qquad \Art(\tau)=n.
\operatorname{Art}Use in source · 2 passages
Search aliases: Art operatorname
Use in source · Read passage · Paper_XI_Traditional_Final.tex, line 1833
The dimension and orbit follow from [source reference]. The cycle polynomial is [source reference]. For rank greater than one, the conductor is [source reference]. In rank one, the Swan conductor is zero, and the Artin conductor is zero for trivial inertia and one otherwise, by the formula Art=dim V-dim V^I_K+Sw. The local polynomial is [source reference]. Tensor refinement is [source reference]. The Gauss sum is its defining finite sum; for a nontrivial multiplicative character its unit phase i…
Exact mathematical passage
\operatorname{Art}=\dim V-\dim V^{I_K}+\operatorname{Sw}
b'Use in source · 1 passage
Search aliases: b'
Use in source · Read passage · Paper_XI_Traditional_Final.tex, line 1322
[Symplectic invariance] If ι:(V,b)⟶(V',b') is a symplectic isometry and g'=ι gι^-1, then Φ_p(g,V)=Φ_p(g',V').
Exact mathematical passage
\iota:(V,b)\longrightarrow(V',b')
B^{[j]}Use in source · 2 passages
Search aliases: B^[j]
Use in source · Read passage · odd_tower.tex, line 256
The field degrees and critical levels of B^[j] are exactly
Exact mathematical passage
B^{[j]}
b_cUse in source · 13 passages
Search aliases: b_c
Use in source · Read passage · Paper_XI_Traditional_Final.tex, line 927
Let A be a finite abelian group, and let c:A× A→ C^× be a normalized two cocycle. The twisted group algebra C^c[A] has basis (u_a)_a∈ A and multiplication u_au_b=c(a,b)u_a+b. Define the commutator bicharacter b_c(a,b)=c(a,b)c(b,a)^-1.
Exact mathematical passage
b_c(a,b)=c(a,b)c(b,a)^{-1}.
B_eUse in source · 10 passages
Search aliases: B_e
Use in source · Read passage · Paper_XI_Traditional_Final.tex, line 1612
Let Γ be a connected finite graph. At every vertex v, place finite dimensional complex vector spaces V_v and W_v of the same dimension. For every oriented edge e:u→ v, let A_e:V_u⟶ V_v, B_e:W_u⟶ W_v be invertible. The reversed edge carries the inverse map.
Exact mathematical passage
A_e:V_u\longrightarrow V_v, \qquad B_e:W_u\longrightarrow W_v
B_{\mathrm{order}}Use in source · 2 passages
Search aliases: B_order mathrm
Use in source · Read passage · odd_capacity.tex, line 191
The right side is the Heisenberg commutator pairing for the minimal simple stratum of gamma. Its quotient J^1(γ)/H^1(γ) is precisely U^m(A)/H^m(γ), because J^1(γ)=U^1(B_order)U^m(A), with U^1(B_order)⊂ H^1(γ). BK's nondegeneracy theorem for simple-character commutators therefore says that the form on S is nondegenerate.
Exact mathematical passage
J^1(\gamma)=U^1(B_{\mathrm{order}})U^m(A)
B_{p_u}Use in source · 3 passages
Search aliases: B_p_u
Use in source · Read passage · Paper_XI_Traditional_Final.tex, line 1698
[Exact edge residual from cycle defect] Fix a spanning tree and construct the vertex maps by tree transport from U_r. Let e:u→ v be a non tree edge. Let p_u and p_v be the tree paths from the root to u and v. Define the fundamental cycle defect D_e = U_rA_p_v^-1A_eA_p_u - B_p_v^-1B_eB_p_uU_r. Then the edge residual is exactly U_vA_e-B_eU_u = B_p_vD_eA_p_u^-1.
Exact mathematical passage
D_e = U_rA_{p_v}^{-1}A_eA_{p_u} - B_{p_v}^{-1}B_eB_{p_u}U_r.
B_{p_v}Use in source · 7 passages
Search aliases: B_p_v
Use in source · Read passage · Paper_XI_Traditional_Final.tex, line 1706
ct edge residual from cycle defect] Fix a spanning tree and construct the vertex maps by tree transport from U_r. Let e:u→ v be a non tree edge. Let p_u and p_v be the tree paths from the root to u and v. Define the fundamental cycle defect D_e = U_rA_p_v^-1A_eA_p_u - B_p_v^-1B_eB_p_uU_r. Then the edge residual is exactly U_vA_e-B_eU_u = B_p_vD_eA_p_u^-1.
Exact mathematical passage
\boxed{ U_vA_e-B_eU_u = B_{p_v}D_eA_{p_u}^{-1}. }
B_{p_v}^{-1}Use in source · 2 passages
Search aliases: B_p_v^-1
Use in source · Read passage · Paper_XI_Traditional_Final.tex, line 1698
[Exact edge residual from cycle defect] Fix a spanning tree and construct the vertex maps by tree transport from U_r. Let e:u→ v be a non tree edge. Let p_u and p_v be the tree paths from the root to u and v. Define the fundamental cycle defect D_e = U_rA_p_v^-1A_eA_p_u - B_p_v^-1B_eB_p_uU_r. Then the edge residual is exactly U_vA_e-B_eU_u = B_p_vD_eA_p_u^-1.
Exact mathematical passage
D_e = U_rA_{p_v}^{-1}A_eA_{p_u} - B_{p_v}^{-1}B_eB_{p_u}U_r.
\backslashUse in source · 1 passage
Search aliases: backslash
Use in source · Read passage · Paper_XI_Traditional_Final.tex, line 585
Apply Mackey decomposition [source reference]. The double cosets W_K_nbackslash W_K/W_K_m are the orbits of simultaneous translation on Z/n Z× Z/m Z. There are d orbits, each of length L. The stabilizer of a point in each orbit is W_K_L. The second character is shifted by the chosen Frobenius representative. Summing the induced terms over the double cosets gives the decomposition.
Exact mathematical passage
W_{K_n}\backslash W_K/W_{K_m}
\bar xUse in source · 2 passages
Search aliases: bar x bar
Use in source · Read passage · Paper_XI_Traditional_Final.tex, line 1162
J_θ. The quotient V_θ=J_θ^1/H_θ^1 is regarded as a finite vector space over F_p, where p is the residue characteristic. Fix a nontrivial additive character ψ_0: F_p→μ_p⊂ C^×, where μ_p is the group of p-th roots of unity. The simple-character commutator takes values in μ_p. Since ψ_0 is injective, the equation ψ_0(h_θ(bar x,bar y)) = θ([x,y]) uniquely defines the F_p-valued commutator pairing h_θ. We use the same identification of μ_p with the additive group of F_p on the Galois side when compar…
Exact mathematical passage
\psi_0\bigl(h_\theta(\bar x,\bar y)\bigr) = \theta([x,y])
\bar yUse in source · 2 passages
Search aliases: bar y bar
Use in source · Read passage · Paper_XI_Traditional_Final.tex, line 1162
J_θ. The quotient V_θ=J_θ^1/H_θ^1 is regarded as a finite vector space over F_p, where p is the residue characteristic. Fix a nontrivial additive character ψ_0: F_p→μ_p⊂ C^×, where μ_p is the group of p-th roots of unity. The simple-character commutator takes values in μ_p. Since ψ_0 is injective, the equation ψ_0(h_θ(bar x,bar y)) = θ([x,y]) uniquely defines the F_p-valued commutator pairing h_θ. We use the same identification of μ_p with the additive group of F_p on the Galois side when compar…
Exact mathematical passage
\psi_0\bigl(h_\theta(\bar x,\bar y)\bigr) = \theta([x,y])
\bmodUse in source · 2 passages
Search aliases: bmod
Use in source · Paper_XI_Visual_Guide_Final.tex, line 160
lessonPART I / FINITE TAME COORDINATESExponent coordinates test inertia trivialityOriginal Proposition 5.3 Write the residue character exponents as abmod(q^n-1) and bbmod(q^m-1). The ith refined packet has exponent c_i=afracq^L-1q^n-1+bq^ifracq^L-1q^m-1 pmodq^L-1. Norm lifting multiplies an exponent by the quotient of group orders. Shifting the second character by Frobenius multiplies its exponent by q^i. Multiplication of characters adds the two terms.
Exact mathematical passage
a\bmod(q^n-1)
\CUse in source · 7 passages
Search aliases: C mathbb
Use in source · Paper_XI_Visual_Guide_Final.tex, line 133
tion 4 Let G_n=GL_n( F_q) and T_n≃ F_q^n^× be an elliptic torus. A regular torus character determines an irreducible cuspidal representation κ_θ by the classical finite-field construction. If its character is χ_θ and degree is d_θ, then e_θ=fracd_θ|G_n|∑_g∈ G_nχ_θ(g^-1)g is its primitive central idempotent in C[G_n].
Exact mathematical passage
\C[G_n]
\mathbb CUse in source · 4 passages
Search aliases: C mathbb
Use in source · Read passage · dyadic_synthesis.tex, line 124
For the final totally ramified field, J/J^1= F_2^×=1, so J=J^1 and the canonical beta extension is η. Extend to K^× J, and choose the noncompact scalar to match ω(t). The needed N-th root exists in C^×. The compact type stays fixed. All choices precede H5. See pathcentral/DYADIC_FULL_CHARACTER167.md for the complete proof.
Exact mathematical passage
\mathbb C^\times
\operatorname{c\!-\!Ind}Use in source · 2 passages
Search aliases: c - Ind operatorname
Use in source · Read passage · dyadic_synthesis.tex, line 349
Consequently M=K^× P is an actual open group compact modulo the center, N=K^× J is normal of index two in M, and Σ=Ind_ N^ MΛ, Σ|_P≃ E, π≃c - Ind_ M^GΣ. Restriction to P is irreducible, so Σ is irreducible. Induction in stages proves the last equality for the original Λ, including its fixed central character and noncompact root. This is a genuine enlarged inducing representation; it changes neither the candidate nor its original canonical beta-extension data.
Exact mathematical passage
\Sigma=\operatorname{Ind}_{\mathcal N}^{\mathcal M}\Lambda, \qquad \Sigma|_P\simeq E, \qquad \boxed{\pi\simeq\operatorname{c\!-\!Ind}_{\mathcal M}^G\Sigma.}
\operatorname{c\!-\!Ind}_{\mathcal M}^GUse in source · 1 passage
Search aliases: c - Ind_ M^G operatorname mathcal
Use in source · Read passage · dyadic_synthesis.tex, line 349
Consequently M=K^× P is an actual open group compact modulo the center, N=K^× J is normal of index two in M, and Σ=Ind_ N^ MΛ, Σ|_P≃ E, π≃c - Ind_ M^GΣ. Restriction to P is irreducible, so Σ is irreducible. Induction in stages proves the last equality for the original Λ, including its fixed central character and noncompact root. This is a genuine enlarged inducing representation; it changes neither the candidate nor its original canonical beta-extension data.
Exact mathematical passage
\Sigma=\operatorname{Ind}_{\mathcal N}^{\mathcal M}\Lambda, \qquad \Sigma|_P\simeq E, \qquad \boxed{\pi\simeq\operatorname{c\!-\!Ind}_{\mathcal M}^G\Sigma.}
\operatorname{c\!-\!Ind}_J^GUse in source · 1 passage
Search aliases: c - Ind_J^G operatorname
Use in source · Read passage · dyadic_synthesis.tex, line 362
There is also a recovery from compact data explicitly recording that root. With A_0=Λ(ϖ_K) and X=c - Ind_J^Gη, define (Tf)(g)=A_0^-1f(ϖ_Kg). Then X/(T-1)X≃π. After imposing the fixed central character, N/(F^× J) is cyclic of order N, and the finite average N^-1∑_i=0^N-1T^i selects this same extension. These operators act on complex representation spaces, so their scalar division by N is valid. This is a different recovery from the Block159 projector onto a sector of η. The exact kernel and finit…
Exact mathematical passage
X=\operatorname{c\!-\!Ind}_J^G\eta
c'Use in source · 2 passages
Search aliases: c'
Use in source · Read passage · Paper_XI_Traditional_Final.tex, line 937
The new cocycle is c'(a,b) = λ_aλ_bλ_a+b^-1c(a,b). The scalar coboundary factor is symmetric in a and b, so it cancels from c'(a,b)c'(b,a)^-1.
Exact mathematical passage
c'(a,b) = \lambda_a\lambda_b\lambda_{a+b}^{-1}c(a,b).
c^mUse in source · 1 passage
Search aliases: c^m
Use in source · Read passage · Paper_XI_Traditional_Final.tex, line 1259
Let C=⟨ c⟩ have order m, and let T be a lift of the projective generator. Since c^m=1, the operator T^m is scalar. Choose λ∈ C^× with λ^mT^m=I. Replacing T by λ T gives an honest cyclic representation.
Exact mathematical passage
c^m=1
\C^XUse in source · 1 passage
Search aliases: C^X C mathbb
Use in source · Paper_XI_Visual_Guide_Final.tex, line 229
lessonPART II / FINITE WILD COMPARISONTranslations and multipliers create Heisenberg packetsOriginal Section 7 plate07_heisenbergThe commutator is central and nontrivial when the pairing is nonzero. For odd p let V=X⊕ X^* and b((x,ξ),(y,η))=η(x)-ξ(y). The group H(V)=V× F_p has product (v,z)(w,t)=(v+w,z+t+tfrac12b(v,w)). On H= C^X the Schrödinger operator is (ρ(x,ξ,z)f)(u) =ψ(z+ξ(u)+tfrac12ξ(x))f(u+x). Translations and character multipliers generate every matrix unit by finite Fourier inversion. …
Exact mathematical passage
\mathcal H=\C^X
\C^\timesUse in source · 4 passages
Search aliases: C^× C times mathbb times
Use in source · Paper_XI_Visual_Guide_Final.tex, line 304
The two pulled-back irreducible Heisenberg representations then have the same central character. Stone--von Neumann gives an invertible U with Uω_ρ(h)=ω_θ(ι_H(h))U. Schur's lemma makes U unique up to a nonzero scalar. After honest cyclic stabilizer linearizations, their discrepancy is a character δ:C→ C^×. Equality at one generator kills this character. boundaryFinite linearization does not identify the stabilizer action with an arithmetic action on a newly constructed common candidate.
Exact mathematical passage
\delta:C\to\C^\times
\mathbb C^\timesUse in source · 4 passages
Search aliases: C^× mathbb times
Use in source · Read passage · dyadic_synthesis.tex, line 124
For the final totally ramified field, J/J^1= F_2^×=1, so J=J^1 and the canonical beta extension is η. Extend to K^× J, and choose the noncompact scalar to match ω(t). The needed N-th root exists in C^×. The compact type stays fixed. All choices precede H5. See pathcentral/DYADIC_FULL_CHARACTER167.md for the complete proof.
Exact mathematical passage
\mathbb C^\times
c_0Use in source · 1 passage
Search aliases: c_0
Use in source · Paper_XI_Visual_Guide_Final.tex, line 168
For q=2,n=4,m=6, the common modulus is 2^12-1=4095. Taking a=b=1, the two packets have c_0=273+65=338, c_1=273+130=403. Both are nonzero modulo 4095, so neither packet contributes inertia invariants.
Exact mathematical passage
c_0=273+65=338,\qquad c_1=273+130=403.
c_1Use in source · 1 passage
Search aliases: c_1
Use in source · Paper_XI_Visual_Guide_Final.tex, line 168
For q=2,n=4,m=6, the common modulus is 2^12-1=4095. Taking a=b=1, the two packets have c_0=273+65=338, c_1=273+130=403. Both are nonzero modulo 4095, so neither packet contributes inertia invariants.
Exact mathematical passage
c_0=273+65=338,\qquad c_1=273+130=403.
\C_cUse in source · 1 passage
Search aliases: C_c C mathbb
Use in source · Paper_XI_Visual_Guide_Final.tex, line 260
The radical consists of a commuting projectively with every b. Its quotient carries a nondegenerate alternating pairing. In the nondegenerate case the twisted group algebra has scalar center and is a full matrix algebra: C_c[A]≃ M_sqrt|A|( C). boundaryThis constructs the finite envelope for a supplied projective action. It does not produce a canonical action for every arbitrary wild inertia atom.
Exact mathematical passage
\C_c[A]\simeq M_{\sqrt{|A|}}(\C).
c_iUse in source · 7 passages
Search aliases: c_i
Use in source · Read passage · Paper_XI_Traditional_Final.tex, line 620
[Refined exponent formula] The ith interaction packet on F_q^L^× has exponent c_i = afracq^L-1q^n-1 +bq^ifracq^L-1q^m-1 pmodq^L-1. It is trivial on tame inertia exactly when c_i=0.
Exact mathematical passage
\boxed{ c_i = a\frac{q^L-1}{q^n-1} +bq^i\frac{q^L-1}{q^m-1} \pmod{q^L-1}. }
c_nUse in source · 2 passages
Search aliases: c_n
Use in source · Read passage · Paper_XI_Traditional_Final.tex, line 414
[Möbius orbit count] The number of regular Frobenius orbits of length n is c_n(q) = frac1n ∑_dmid n μ(d) (q^n/d-1).
Exact mathematical passage
\boxed{ c_n(q) = \frac1n \sum_{d\mid n} \mu(d) \bigl(q^{n/d}-1\bigr). }
\calH_\thetaUse in source · 1 passage
Search aliases: calH_θ H_θ calH theta mathcal theta
Use in source · Read passage · Paper_XI_Traditional_Final.tex, line 1214
[Restricted Heisenberg bridge] A restricted bridge datum determines an isomorphism between the two Heisenberg representations, unique up to a nonzero scalar. More precisely, there is an invertible map U: H_ρ⟶ H_θ such that Uω_ρ(h) = ω_θ(ι_H(h))U for every Heisenberg element h, and every other such map is a scalar multiple of U.
Exact mathematical passage
U:\calH_\rho\longrightarrow\calH_\theta
\calH_\rhoUse in source · 1 passage
Search aliases: calH_ρ H_ρ calH rho mathcal rho
Use in source · Read passage · Paper_XI_Traditional_Final.tex, line 1214
[Restricted Heisenberg bridge] A restricted bridge datum determines an isomorphism between the two Heisenberg representations, unique up to a nonzero scalar. More precisely, there is an invertible map U: H_ρ⟶ H_θ such that Uω_ρ(h) = ω_θ(ι_H(h))U for every Heisenberg element h, and every other such map is a scalar multiple of U.
Exact mathematical passage
U:\calH_\rho\longrightarrow\calH_\theta
\CC^cUse in source · 5 passages
Search aliases: CC^c C^c CC mathbb
Use in source · Read passage · Paper_XI_Traditional_Final.tex, line 922
Let A be a finite abelian group, and let c:A× A→ C^× be a normalized two cocycle. The twisted group algebra C^c[A] has basis (u_a)_a∈ A and multiplication u_au_b=c(a,b)u_a+b. Define the commutator bicharacter b_c(a,b)=c(a,b)c(b,a)^-1.
Exact mathematical passage
\CC^c[A]
\CC^{c_\rho}Use in source · 1 passage
Search aliases: CC^c_ρ C^c_ρ CC rho mathbb rho
Use in source · Read passage · Paper_XI_Traditional_Final.tex, line 1103
[Nondegenerate one break packet] If b_ρ is nondegenerate, then C^c_ρ[A] ≅ End(M_ρ), dim M_ρ=sqrtabs A.
Exact mathematical passage
\CC^{c_\rho}[A] \cong \End(M_\rho), \qquad \dim M_\rho=\sqrt{\abs A}.
\CC^\timesUse in source · 1 passage
Search aliases: CC^× C^× CC times mathbb times
Use in source · Read passage · Paper_XI_Traditional_Final.tex, line 1161
Let θ be a maximal simple character for GL_n(K), with compact open subgroups H_θ^1⊆ J_θ^1⊆ J_θ. The quotient V_θ=J_θ^1/H_θ^1 is regarded as a finite vector space over F_p, where p is the residue characteristic. Fix a nontrivial additive character ψ_0: F_p→μ_p⊂ C^×, where μ_p is the group of p-th roots of unity. The simple-character commutator takes values in μ_p. Since ψ_0 is injective, the equation ψ_0(h_θ(bar x,bar y)) = θ([x,y]) uniquely defines the F_p-valued commutator pairing h_θ. We use t…
Exact mathematical passage
\psi_0:\FF_p\to\mu_p\subset\CC^\times
\cIndUse in source · 3 passages
Search aliases: cInd c - Ind operatorname cInd operatorname
Use in source · Read passage · Paper_XI_Traditional_Final.tex, line 2137
At both ranks K^× normalizes P. Thus N=K^× J is normal of index two in M=K^× P, and Σ=Ind_ N^ MΛ, π≃cInd_ M^GL_N(F)Σ. Induction in stages preserves the full original Λ, its central character, and its noncompact root; see [source reference]. The full marked scalar-pushout comparison applies to these actual enlarged pairs as well as to the original higher-rank types ([source reference]). The full synthesis and scope are in [source reference].
Exact mathematical passage
\Sigma=\Ind_{\mathcal N}^{\mathcal M}\Lambda, \qquad \pi\simeq\cInd_{\mathcal M}^{\GL_N(F)}\Sigma.
\cInd_{\mathcal M}^{G}Use in source · 1 passage
Search aliases: cInd_ M^G c - Ind_ M^G cInd mathcal operatorname mathcal
Use in source · Paper_XI_Visual_Guide_Final.tex, line 563
overy proof plate20_induction_stagesThe noncompact extension Λ is fixed before forming Σ. At both low ranks, K^× normalizes P. Put N=K^× J, M=K^× P, Σ=Ind_ N^ MΛ. The subgroup N is normal of index two, Σ|_P≃ E, and Σ is irreducible. Induction in stages proves π≃c - Ind_ M^GΣ. takeawayThe parameter, full character, original canonical beta-extension data, central character, and noncompact scalar remain those of the original candidate. The inducing carrier changes.
Exact mathematical passage
\pi\simeq\cInd_{\mathcal M}^{G}\Sigma.
\cInd_{\mathcal M}^{\GL_N(F)}Use in source · 1 passage
Search aliases: cInd_ M^GL_N(F) cInd mathcal GL operatorname mathcal operatorname
Use in source · Read passage · Paper_XI_Traditional_Final.tex, line 2137
At both ranks K^× normalizes P. Thus N=K^× J is normal of index two in M=K^× P, and Σ=Ind_ N^ MΛ, π≃cInd_ M^GL_N(F)Σ. Induction in stages preserves the full original Λ, its central character, and its noncompact root; see [source reference]. The full marked scalar-pushout comparison applies to these actual enlarged pairs as well as to the original higher-rank types ([source reference]). The full synthesis and scope are in [source reference].
Exact mathematical passage
\Sigma=\Ind_{\mathcal N}^{\mathcal M}\Lambda, \qquad \pi\simeq\cInd_{\mathcal M}^{\GL_N(F)}\Sigma.
\cInd_J^GUse in source · 1 passage
Search aliases: cInd_J^G c - Ind_J^G cInd operatorname
Use in source · Paper_XI_Visual_Guide_Final.tex, line 569
lessonPART III / ACTUAL ARITHMETIC CONSTRUCTIONCompact recovery must remember the noncompact rootBlock 167, enlarged-carrier contract Let X=c - Ind_J^Gη and retain A_0=Λ(ϖ_K). Define the normalized shift (Tf)(g)=A_0^-1f(ϖ_Kg). The original candidate is the quotient X/(T-1)X≃π. This does not split before central normalization: a vector of X has finite support modulo compact J, whereas a scalar uniformizer shifts determinant valuations. It has no nonzero eigenvector with the required central eigen…
Exact mathematical passage
X=\cInd_J^G\eta
DUse in source · 11 passages
Search aliases: D
Use in source · Read passage · Paper_XI_Traditional_Final.tex, line 1698
[Exact edge residual from cycle defect] Fix a spanning tree and construct the vertex maps by tree transport from U_r. Let e:u→ v be a non tree edge. Let p_u and p_v be the tree paths from the root to u and v. Define the fundamental cycle defect D_e = U_rA_p_v^-1A_eA_p_u - B_p_v^-1B_eB_p_uU_r. Then the edge residual is exactly U_vA_e-B_eU_u = B_p_vD_eA_p_u^-1.
Exact mathematical passage
D_e = U_rA_{p_v}^{-1}A_eA_{p_u} - B_{p_v}^{-1}B_eB_{p_u}U_r.
D_8Use in source · 2 passages
Search aliases: D_8
Use in source · Read passage · Paper_XI_Traditional_Final.tex, line 2158
ter pair. For an independently fixed native quadratic marking, a preassigned embedding must satisfy q_W(f(v))=q_rm src(v). At equal capacity, a free embedding requires equal Arf invariant. Spare capacity for d≥4 admits either source Arf type when the embedding is free. Retaining only the central {±1} would erase the distinction between whole marked finite extensions such as D_8 and Q_8.
Exact mathematical passage
D_8
D_eUse in source · 6 passages
Search aliases: D_e
Use in source · Read passage · Paper_XI_Traditional_Final.tex, line 1698
[Exact edge residual from cycle defect] Fix a spanning tree and construct the vertex maps by tree transport from U_r. Let e:u→ v be a non tree edge. Let p_u and p_v be the tree paths from the root to u and v. Define the fundamental cycle defect D_e = U_rA_p_v^-1A_eA_p_u - B_p_v^-1B_eB_p_uU_r. Then the edge residual is exactly U_vA_e-B_eU_u = B_p_vD_eA_p_u^-1.
Exact mathematical passage
D_e = U_rA_{p_v}^{-1}A_eA_{p_u} - B_{p_v}^{-1}B_eB_{p_u}U_r.
d_\thetaUse in source · 5 passages
Search aliases: d_θ theta
Use in source · Read passage · Paper_XI_Traditional_Final.tex, line 469
[Primitive cuspidal idempotent] Let χ_θ be the irreducible character of κ_θ, and let d_θ=χ_θ(1). Then e_θ = fracd_θabsG_n ∑_g∈ G_n χ_θ(g^-1)g is the primitive central idempotent selecting the κ_θ block in C[G_n].
Exact mathematical passage
d_\theta=\chi_\theta(1)
d_\rhoUse in source · 8 passages
Search aliases: d_ρ rho
Use in source · Read passage · Paper_XI_Traditional_Final.tex, line 1841
Fix the paper's reciprocity, additive-character, local-factor, and phase normalizations. Let A be the declared class of irreducible wild Weil atoms under consideration. For ρ∈ A, the supplied assembled finite datum d_ρ consists of the following inputs, historically denoted H1--H4: item an intrinsic finite break decomposition on the Galois side; item the declared equivariant symplectic bridges to the supplied breakwise type-side quotients; item compatible odd or dyadic central-extension and phase…
Exact mathematical passage
d_\rho
\operatorname{diag}Use in source · 2 passages
Search aliases: diag operatorname
Use in source · Read passage · Paper_XI_Traditional_Final.tex, line 277
[Scalar covariance model] Let F=diag(1,q,q^2,…,q^m-1) and let N_j-1,j=1. Then FNF^-1=q^-1N.
Exact mathematical passage
F=\operatorname{diag}(1,q,q^2,\ldots,q^{m-1})
\dim_{\mathbb F_2}Use in source · 1 passage
Search aliases: dim_ F_2 dim mathbb
Use in source · Read passage · dyadic_synthesis.tex, line 30
For this same constructed candidate, t(π) · =1, · Sw(π) · =3 2^d-1h_d, dim_ F_2W · =N-2, · dimη · =2^N/2-1, W=J^1/H^1, a(π^vee×π) · =N^2-1+∑_i=1^d3 4^i-1h_i. Matching these profiles does not identify the source, its endo-class, or the pair factors in original H5.
Exact mathematical passage
\begin{aligned} t(\pi)&=1,& \operatorname{Sw}(\pi)&=3\,2^{d-1}h_d,\\ \dim_{\mathbb F_2}W&=N-2,& \dim\eta&=2^{N/2-1},\qquad W=J^1/H^1,\\ a(\pi^\vee\times\pi) &=N^2-1+\sum_{i=1}^d3\,4^{i-1}h_i. \end{aligned}
\dim_{\mathbb F_p}Use in source · 2 passages
Search aliases: dim_ F_p dim mathbb
Use in source · Read passage · odd_synthesis.tex, line 39
The same candidate has rank(π) · =N, · t(π) · =1, Sw(π) · =p^d-1(p+1)h_d, · dim_ F_p(J^1/H^1) · =N-1, dimη · =p^(N-1)/2, a(π^vee×π) · =p^2d-1+∑_i=1^d(p^2i-p^2i-2)h_i. These numerical identities do not identify a Weil representation, an endo-class, or the pair epsilon factors required by H5.
Exact mathematical passage
\begin{aligned} \operatorname{rank}(\pi)&=N, &t(\pi)&=1,\\ \operatorname{Sw}(\pi)&=p^{d-1}(p+1)h_d, & \dim_{\mathbb F_p}(J^1/H^1)&=N-1,\\ \dim\eta&=p^{(N-1)/2},\\ a(\pi^\vee\times\pi) &=p^{2d}-1+\sum_{i=1}^d(p^{2i}-p^{2i-2})h_i. \end{aligned}
\dim_{\F_2}Use in source · 1 passage
Search aliases: dim_F_2 dim_ F_2 dim F dim mathbb
Use in source · Paper_XI_Visual_Guide_Final.tex, line 476
a(π^vee×π) · =p^2d-1+∑_i=1^d(p^2i-p^2i-2)h_i. For p=2, Sw(π) · =3 2^d-1h_d, dim_ F_2(J^1/H^1) · =N-2, dimη=2^N/2-1, a(π^vee×π) · =N^2-1+∑_i=1^d3 4^i-1h_i. boundaryThese identities belong to the same constructed candidate. Matching numerical profiles does not identify a source Weil atom, an endo-class, or H5 pair factors.
Exact mathematical passage
\begin{aligned} \Sw(\pi)&=3\,2^{d-1}h_d,\\ \dim_{\F_2}(J^1/H^1)&=N-2,\quad\dim\eta=2^{N/2-1},\\ a(\pi^\vee\times\pi)&=N^2-1+\sum_{i=1}^d3\,4^{i-1}h_i. \end{aligned}
\dim_{\F_p}Use in source · 1 passage
Search aliases: dim_F_p dim_ F_p dim F dim mathbb
Use in source · Paper_XI_Visual_Guide_Final.tex, line 468
For odd p, Sw(π) · =p^d-1(p+1)h_d, dim_ F_p(J^1/H^1) · =N-1, dimη=p^(N-1)/2, a(π^vee×π) · =p^2d-1+∑_i=1^d(p^2i-p^2i-2)h_i. For p=2, Sw(π) · =3 2^d-1h_d,
Exact mathematical passage
\begin{aligned} \Sw(\pi)&=p^{d-1}(p+1)h_d,\\ \dim_{\F_p}(J^1/H^1)&=N-1,\quad\dim\eta=p^{(N-1)/2},\\ a(\pi^\vee\times\pi)&=p^{2d}-1+\sum_{i=1}^d(p^{2i}-p^{2i-2})h_i. \end{aligned}
EUse in source · 42 passages
Search aliases: E
Use in source · Read passage · Paper_XI_Traditional_Final.tex, line 1676
[Scalar and projective gluing] If every based loop acts by a scalar, finite gluing is equivalent to equality of the scalar holonomies on a fundamental cycle basis. The number of independent tests is abs E-abs V+1.
Exact mathematical passage
\boxed{ \abs E-\abs V+1. }
\eUse in source · 2 passages
Search aliases: e mathrm
Use in source · Read passage · Paper_XI_Traditional_Final.tex, line 317
Assume that P_K acts trivially. After passing to a finite quotient, tame inertia is cyclic. Let t generate a quotient of order M, and put ζ_M= e^2π i/M. For j∈ Z/M Z, define P_j = frac1M ∑_a=0^M-1 ζ_M^-jar(t)^a.
Exact mathematical passage
\zeta_M=\e^{2\pi i/M}
\e^{2\pi i/M}Use in source · 1 passage
Search aliases: e^2π i/M e pi mathrm pi
Use in source · Read passage · Paper_XI_Traditional_Final.tex, line 317
Assume that P_K acts trivially. After passing to a finite quotient, tame inertia is cyclic. Let t generate a quotient of order M, and put ζ_M= e^2π i/M. For j∈ Z/M Z, define P_j = frac1M ∑_a=0^M-1 ζ_M^-jar(t)^a.
Exact mathematical passage
\zeta_M=\e^{2\pi i/M}
e^{2\pi i/M}Use in source · 1 passage
Search aliases: e^2π i/M pi
Use in source · Paper_XI_Visual_Guide_Final.tex, line 88
lessonPART I / FINITE TAME COORDINATESFourier projectors isolate tame inertiaOriginal Section 3; geometric Frobenius convention Let t generate a finite tame quotient of order M, with (q,M)=1. For j∈ Z/M Z define P_j=frac1M∑_a=0^M-1ζ_M^-jar(t)^a, ζ_M=e^2π i/M. The finite Fourier identities give P_jP_k=δ_jkP_j and ∑_jP_j=I. Thus P_jV is the ζ_M^j character space for r(t).
Exact mathematical passage
P_j=\frac1M\sum_{a=0}^{M-1}\zeta_M^{-ja}r(t)^a, \qquad \zeta_M=e^{2\pi i/M}.
\e^{2\pi iz/p}Use in source · 1 passage
Search aliases: e^2π iz/p e pi mathrm pi
Use in source · Read passage · Paper_XI_Traditional_Final.tex, line 810
Let p be odd. Let X be an r dimensional vector space over F_p, and put V=X⊕ X^*. Define the alternating form b((x,ξ),(y,η)) = η(x)-ξ(y). Let Z= F_p, and define the Heisenberg group H(V)=V× Z with multiplication (v,z)(w,t) = (v+w,z+t+frac12 b(v,w)). Fix the additive character ψ(z)= e^2π iz/p.
Exact mathematical passage
\psi(z)=\e^{2\pi iz/p}.
E_0Use in source · 6 passages
Search aliases: E_0
Use in source · Read passage · dyadic_synthesis.tex, line 313
Use the actual rank-four outer approximation field E_0/F of degree two and its centralizer B=End_E_0(F^4). The odd lower critical level is r=3h_1=2m+1; the half leading level is a=3h_2>r. Define P=J U_B^m, where J=J^1 is the actual compact type group. The filtered centralizer estimate for the actual final field, without assuming E_0⊂ K, proves that P is a compact group with J normal of index two. For a concrete new coset representative u, λ=θ^u/θ on the actual H=H^1 is nontrivial with values {±1…
Exact mathematical passage
E_0/F
e_0Use in source · 3 passages
Search aliases: e_0
Use in source · Read passage · Paper_XI_Traditional_Final.tex, line 518
The induced tame Weil atom has an elementary cycle invariant. In an orbit basis e_0,…,e_n-1, write τ(Φ)e_j=e_j+1 for j<n-1, and τ(Φ)e_n-1=ae_0.
Exact mathematical passage
e_0,\ldots,e_{n-1}
e_jUse in source · 2 passages
Search aliases: e_j
Use in source · Read passage · Paper_XI_Traditional_Final.tex, line 519
The induced tame Weil atom has an elementary cycle invariant. In an orbit basis e_0,…,e_n-1, write τ(Φ)e_j=e_j+1 for j<n-1, and τ(Φ)e_n-1=ae_0.
Exact mathematical passage
\tau(\Phi)e_j=e_{j+1}
e_{j+1}Use in source · 2 passages
Search aliases: e_j+1
Use in source · Read passage · Paper_XI_Traditional_Final.tex, line 519
The induced tame Weil atom has an elementary cycle invariant. In an orbit basis e_0,…,e_n-1, write τ(Φ)e_j=e_j+1 for j<n-1, and τ(Φ)e_n-1=ae_0.
Exact mathematical passage
\tau(\Phi)e_j=e_{j+1}
e_{n-1}Use in source · 3 passages
Search aliases: e_n-1
Use in source · Read passage · Paper_XI_Traditional_Final.tex, line 518
The induced tame Weil atom has an elementary cycle invariant. In an orbit basis e_0,…,e_n-1, write τ(Φ)e_j=e_j+1 for j<n-1, and τ(Φ)e_n-1=ae_0.
Exact mathematical passage
e_0,\ldots,e_{n-1}
e_UUse in source · 5 passages
Search aliases: e_U
Use in source · Read passage · Paper_XI_Traditional_Final.tex, line 492
[Parabolic annihilation] Let P=LU be a proper parabolic subgroup of G_n, and put e_U=frac1abs U∑_u∈ Uu. If κ_θ is cuspidal, then e_Ue_θ=e_θ e_U=0.
Exact mathematical passage
e_U=\frac1{\abs U}\sum_{u\in U}u.
e_\thetaUse in source · 5 passages
Search aliases: e_θ theta
Use in source · Read passage · Paper_XI_Traditional_Final.tex, line 470
[Primitive cuspidal idempotent] Let χ_θ be the irreducible character of κ_θ, and let d_θ=χ_θ(1). Then e_θ = fracd_θabsG_n ∑_g∈ G_n χ_θ(g^-1)g is the primitive central idempotent selecting the κ_θ block in C[G_n].
Exact mathematical passage
e_\theta = \frac{d_\theta}{\abs{G_n}} \sum_{g\in G_n} \chi_\theta(g^{-1})g
e_\LambdaUse in source · 1 passage
Search aliases: e_Λ Lambda
Use in source · Paper_XI_Visual_Guide_Final.tex, line 580
After imposing the already fixed central character, the relevant quotient is cyclic of order N. Then T^N=1 and the finite average e_Λ=frac1N∑_i=0^N-1T^i selects the same extension. The operator acts on a complex representation space, where division by N is valid. boundaryThis is a different recovery mechanism from a finite sector inside the original η. The root is retained, not reconstructed from the compact packet alone.
Exact mathematical passage
e_\Lambda=\frac1N\sum_{i=0}^{N-1}T^i
\EndUse in source · 1 passage
Search aliases: End operatorname
Use in source · Read passage · Paper_XI_Traditional_Final.tex, line 1103
[Nondegenerate one break packet] If b_ρ is nondegenerate, then C^c_ρ[A] ≅ End(M_ρ), dim M_ρ=sqrtabs A.
Exact mathematical passage
\CC^{c_\rho}[A] \cong \End(M_\rho), \qquad \dim M_\rho=\sqrt{\abs A}.
\operatorname{End}Use in source · 2 passages
Search aliases: End operatorname
Use in source · Read passage · dyadic_synthesis.tex, line 314
Use the actual rank-four outer approximation field E_0/F of degree two and its centralizer B=End_E_0(F^4). The odd lower critical level is r=3h_1=2m+1; the half leading level is a=3h_2>r. Define P=J U_B^m, where J=J^1 is the actual compact type group. The filtered centralizer estimate for the actual final field, without assuming E_0⊂ K, proves that P is a compact group with J normal of index two. For a concrete new coset representative u, λ=θ^u/θ on the actual H=H^1 is nontrivial with values {±1…
Exact mathematical passage
B=\operatorname{End}_{E_0}(F^4)
\operatorname{End}_{E_0}Use in source · 2 passages
Search aliases: End_E_0 operatorname
Use in source · Read passage · dyadic_synthesis.tex, line 314
Use the actual rank-four outer approximation field E_0/F of degree two and its centralizer B=End_E_0(F^4). The odd lower critical level is r=3h_1=2m+1; the half leading level is a=3h_2>r. Define P=J U_B^m, where J=J^1 is the actual compact type group. The filtered centralizer estimate for the actual final field, without assuming E_0⊂ K, proves that P is a compact group with J normal of index two. For a concrete new coset representative u, λ=θ^u/θ on the actual H=H^1 is nontrivial with values {±1…
Exact mathematical passage
B=\operatorname{End}_{E_0}(F^4)
\mathbb FUse in source · 11 passages
Search aliases: F mathbb
Use in source · Read passage · dyadic_synthesis.tex, line 13
Let F= F_2((t)), fix an additive character ψ_F of conductor t O_F, and take increasing odd positive integers h_1<⋯<h_d. Put q_i=3h_i, N=2^d, n_j=2^d-jq_d-j+1. There is an explicit β∈ M_N(F) generating a separable totally ramified degree-N field that normalizes the complete lattice chain. It gives a maximal simple stratum with leading level n_1, final critical exponent -q_1, and successive approximation-field degrees 2,4,…,2^d, with critical levels n_2,…,n_d.
Exact mathematical passage
F=\mathbb F_2((t))
f'Use in source · 2 passages
Search aliases: f'
Use in source · Read passage · odd_synthesis.tex, line 64
Let P_0 extract the constant coefficient in the basis 1,z,…,z^p-1. The normalized corestriction s=κpr, where κ=-f'(z)/z^p-1, satisfies u_0=s(P_0)=a_0/z^p∈ O_E^×, Lift_E(C)=u_0^-1CP_0, s(Lift_E(C))=C. The lift preserves the exact filtration valuation. Its crucial additional identity is tr_L(Lift_E(C)) =Tr_E/L (κ^-1tr_E(C)). Define B(L,t_L;h_1,…,h_d) =α I_p^d-1+ Lift_E(B(E,z;h_1,…,h_d-1)), with the degree-p multiplication matrix as the base case. Induction proves trace zero at every native rank. T…
Exact mathematical passage
\kappa=-f'(z)/z^{p-1}
F^{-1}Use in source · 1 passage
Search aliases: F^-1
Use in source · Read passage · Paper_XI_Traditional_Final.tex, line 281
[Scalar covariance model] Let F=diag(1,q,q^2,…,q^m-1) and let N_j-1,j=1. Then FNF^-1=q^-1N.
Exact mathematical passage
FNF^{-1}=q^{-1}N.
F^4Use in source · 2 passages
Search aliases: F^4
Use in source · Read passage · dyadic_synthesis.tex, line 314
Use the actual rank-four outer approximation field E_0/F of degree two and its centralizer B=End_E_0(F^4). The odd lower critical level is r=3h_1=2m+1; the half leading level is a=3h_2>r. Define P=J U_B^m, where J=J^1 is the actual compact type group. The filtered centralizer estimate for the actual final field, without assuming E_0⊂ K, proves that P is a compact group with J normal of index two. For a concrete new coset representative u, λ=θ^u/θ on the actual H=H^1 is nontrivial with values {±1…
Exact mathematical passage
B=\operatorname{End}_{E_0}(F^4)
F^\timesUse in source · 3 passages
Search aliases: F^× times
Use in source · Read passage · dyadic_synthesis.tex, line 364
There is also a recovery from compact data explicitly recording that root. With A_0=Λ(ϖ_K) and X=c - Ind_J^Gη, define (Tf)(g)=A_0^-1f(ϖ_Kg). Then X/(T-1)X≃π. After imposing the fixed central character, N/(F^× J) is cyclic of order N, and the finite average N^-1∑_i=0^N-1T^i selects this same extension. These operators act on complex representation spaces, so their scalar division by N is valid. This is a different recovery from the Block159 projector onto a sector of η. The exact kernel and finit…
Exact mathematical passage
\mathcal N/(F^\times J)
F_0Use in source · 4 passages
Search aliases: F_0
Use in source · Read passage · dyadic_synthesis.tex, line 182
Under the scalar-pushout realization contract, with no independently fixed conflicting host marking, choose a labelled symplectic embedding into W. Matching commutators gives a center-fixing isomorphism between the source C^× extension and the corresponding host restriction. Retain the image of the whole marked finite subgroup and every lift. For this isomorphism F_0, F_0(a)^2=F_0(a^2). Thus its square bits, refinement and Gauss sign survive, along with all supplied full character values. They a…
Exact mathematical passage
F_0
\F_2Use in source · 4 passages
Search aliases: F_2 F mathbb
Use in source · Paper_XI_Visual_Guide_Final.tex, line 327
lessonPART II / FINITE WILD COMPARISONTwo dyadic refinements can have opposite signsOriginal Section 13 plate10_quadratic_planesThe entries are exact values in F_2; both polar forms are xy'+x'y. A quadratic refinement satisfies q(x+y)+q(x)+q(y)=b(x,y). On a plane, q_0(x,y)=xy, q_1(x,y)=xy+x+y. For q_0 there are three zeros and one one, so its Gauss sum is 2. For q_1 there is one zero and three ones, so its sum is -2. The alternating forms agree but the Arf invariants differ. ∑_x∈ V(-1)^q(x)=(-1)…
Exact mathematical passage
\F_2
\mathbb F_2Use in source · 3 passages
Search aliases: F_2 mathbb
Use in source · Read passage · dyadic_synthesis.tex, line 13
Let F= F_2((t)), fix an additive character ψ_F of conductor t O_F, and take increasing odd positive integers h_1<⋯<h_d. Put q_i=3h_i, N=2^d, n_j=2^d-jq_d-j+1. There is an explicit β∈ M_N(F) generating a separable totally ramified degree-N field that normalizes the complete lattice chain. It gives a maximal simple stratum with leading level n_1, final critical exponent -q_1, and successive approximation-field degrees 2,4,…,2^d, with critical levels n_2,…,n_d.
Exact mathematical passage
F=\mathbb F_2((t))
\F_2^2Use in source · 1 passage
Search aliases: F_2^2 F mathbb
Use in source · Paper_XI_Visual_Guide_Final.tex, line 535
tructions plate19_enlarged_carrierThe new carrier occurs in the original candidate; it is not an abstract extra tensor factor. At rank two, write q=2m+1, m≥1, and use J=U_K^1U_A^m+1, P=U_K^1U_A^m. Then Jtriangleleft P, [P:J]=2, and P∩(K^× J)=J. The conjugate character ratio λ=θ^g/θ is nontrivial and sign-valued. With H_0=kerλ, P/H_0≃ F_2^2, E=Ind_J^Pθ, dim E=2. The represented diagonal sign and off-diagonal generators anticommute, proving the nondegenerate Heisenberg structure. boundaryThe origi…
Exact mathematical passage
P/H_0\simeq\F_2^2,\qquad E=\Ind_J^P\theta,\quad\dim E=2.
\mathbb F_2^2Use in source · 1 passage
Search aliases: F_2^2 mathbb
Use in source · Read passage · dyadic_synthesis.tex, line 296
. Define P=U_K^1U_A^m. This is an actual compact open group, with Jtriangleleft P, [P:J]=2 and P∩(K^× J)=J. For g∈ P∖ J, the actual intertwining theorem makes λ=θ^g/θ nontrivial. The commutator [P,J]⊂ U_A^m+1 lies in the forced additive-character domain, so λ has values {±1}. Put H_0=kerλ. Then P/H_0≃ F_2^2, E=Ind_J^Pθ is irreducible of dimension 2. The actual represented generators are a diagonal sign and an off-diagonal matrix that anticommute. Thus their commutator form is nondegenerate, and …
Exact mathematical passage
P/H_0\simeq\mathbb F_2^2,\qquad E=\operatorname{Ind}_J^P\theta\text{ is irreducible of dimension }2.
\F_2^4Use in source · 1 passage
Search aliases: F_2^4 F mathbb
Use in source · Paper_XI_Visual_Guide_Final.tex, line 549
For a new coset representative u, the ratio λ=θ^u/θ on the actual H^1 is nontrivial and sign-valued. Setting H_0=kerλ gives P/H_0≃ F_2^4, E=Ind_J^Pη, dim E=4. Its represented commutators are scalar; the quotient has exponent two; irreducibility and the exact scalar preimage prove nondegeneracy. The finite scalar center can exceed μ_2. boundaryNo literal extraspecial quotient is claimed. Full character values and marked attachment are retained through the actual finite extension.
Exact mathematical passage
P/H_0\simeq\F_2^4,\qquad E=\Ind_J^P\eta,\quad\dim E=4.
\mathbb F_2^4Use in source · 1 passage
Search aliases: F_2^4 mathbb
Use in source · Read passage · dyadic_synthesis.tex, line 324
) is the actual compact type group. The filtered centralizer estimate for the actual final field, without assuming E_0⊂ K, proves that P is a compact group with J normal of index two. For a concrete new coset representative u, λ=θ^u/θ on the actual H=H^1 is nontrivial with values {±1}. Put H_0=kerλ⊂ H. The calculation proves P/H_0≃ F_2^4, E=Ind_J^Pη is irreducible of dimension 4. The new represented commutators are scalar, its quotient has exponent two, and nondegeneracy follows from its exact s…
Exact mathematical passage
P/H_0\simeq\mathbb F_2^4,\qquad E=\operatorname{Ind}_J^P\eta\text{ is irreducible of dimension }4.
\mathbb F_2^\timesUse in source · 1 passage
Search aliases: F_2^× mathbb times
Use in source · Read passage · dyadic_synthesis.tex, line 121
For the final totally ramified field, J/J^1= F_2^×=1, so J=J^1 and the canonical beta extension is η. Extend to K^× J, and choose the noncompact scalar to match ω(t). The needed N-th root exists in C^×. The compact type stays fixed. All choices precede H5. See pathcentral/DYADIC_FULL_CHARACTER167.md for the complete proof.
Exact mathematical passage
J/J^1=\mathbb F_2^\times=1
\mathbb F_pUse in source · 4 passages
Search aliases: F_p mathbb
Use in source · Read passage · odd_synthesis.tex, line 17
Let p be odd, F= F_p((t)), and 0<h_1<⋯<h_d, pnmid h_i, q_i=(p+1)h_i, N=p^d. For every such tuple there is an explicit matrix β∈ M_N(F) whose field K=F[β] is totally ramified of degree N, normalizes the complete lattice chain, and gives a maximal simple stratum. It has a literal defining sequence of degrees p,p^2,…,p^d, with leading level n_1 and successive critical levels n_2,…,n_d given by n_j=p^d-jq_d-j+1 (1≤ j≤ d). Every native parameter in the construction, every amplified approximation, and…
Exact mathematical passage
F=\mathbb F_p((t))
\mathbb F_p^\timesUse in source · 1 passage
Search aliases: F_p^× mathbb times
Use in source · Read passage · odd_synthesis.tex, line 109
This procedure never substitutes an unjustified extension through a nonmaximal centralizer. It proves full-character compatibility, including shallow units, rather than merely cancellation of a leading trace. The residue factor fixes ω on F_p^×; an N-th-root choice for the noncompact generator fixes ω(t). Every choice is recorded before H5. See pathcentral/TRACE_ZERO_CHARACTER_INDUCTION166.md for the complete induction.
Exact mathematical passage
\mathbb F_p^\times
\F_qUse in source · 2 passages
Search aliases: F_q F mathbb
Use in source · Paper_XI_Visual_Guide_Final.tex, line 129
lessonPART I / FINITE TAME COORDINATESA primitive projector selects a cuspidal blockOriginal Section 4 Let G_n=GL_n( F_q) and T_n≃ F_q^n^× be an elliptic torus. A regular torus character determines an irreducible cuspidal representation κ_θ by the classical finite-field construction. If its character is χ_θ and degree is d_θ, then e_θ=fracd_θ|G_n|∑_g∈ G_nχ_θ(g^-1)g is its primitive central idempotent in C[G_n].
Exact mathematical passage
G_n=\GL_n(\F_q)
\F_{q^n}^{\times}Use in source · 2 passages
Search aliases: F_q^n^× F times mathbb times
Use in source · Paper_XI_Visual_Guide_Final.tex, line 117
lessonPART I / FINITE TAME COORDINATESRegular orbits have exact countsOriginal Proposition 3.6 plate04_orbit_countsExact integer counts; the vertical scale is logarithmic. Characters of F_q^n^× are indexed modulo q^n-1. Characters fixed by a Frobenius power of degree rmid n number q^r-1. Möbius inversion removes those belonging to smaller orbits: c_n(q)=frac1n∑_dmid nμ(d)(q^n/d-1). For q=2,n=4, the regular exponent orbits modulo 15 are (1,2,4,8), (3,6,12,9), (7,14,13,11). Their number is tfrac14…
Exact mathematical passage
\F_{q^n}^{\times}
\F_q^\timesUse in source · 2 passages
Search aliases: F_q^× F times mathbb times
Use in source · Paper_XI_Visual_Guide_Final.tex, line 204
lessonPART I / FINITE TAME COORDINATESFinite Gauss phases are unit scalarsOriginal Section 6 For a nontrivial additive character ψ of F_q and a nontrivial multiplicative character χ of F_q^×, define G(χ,ψ)=∑_x∈ F_q^×χ(x)ψ(x), γ(χ,ψ)=q^-1/2G(χ,ψ). The exact identities are |G(χ,ψ)|^2=q, γ(χ,ψ)γ(χ^-1,ψ)=χ(-1). The magnitude calculation substitutes x=ty in the double sum. The inner sum is q-1 for t=1 and -1 otherwise; multiplicative-character orthogonality supplies the result.
Exact mathematical passage
\F_q^\times
\FF_2^2Use in source · 1 passage
Search aliases: FF_2^2 F_2^2 FF mathbb
Use in source · Read passage · Paper_XI_Traditional_Final.tex, line 2126
At rank two, with q=2m+1, enlarge J=U_K^1U_A^m+1 to P=U_K^1U_A^m. The actual character ratio λ=θ^g/θ for g∉ J is nontrivial and sign-valued. With H_0=kerλ, the quotient is P/H_0≃ F_2^2, and Ind_J^Pθ is an irreducible two-dimensional Heisenberg carrier ([source reference]). At rank four, the actual outer quadratic centralizer gives P=J U_B^m, [P:J]=2, and P/H_0≃ F_2^4, with carrier Ind_J^Pη of dimension four ([source reference]). The finite scalar center can be larger than μ_2; no literal extrasp…
Exact mathematical passage
P/H_0\simeq\FF_2^2
\FF_2^4Use in source · 1 passage
Search aliases: FF_2^4 F_2^4 FF mathbb
Use in source · Read passage · Paper_XI_Traditional_Final.tex, line 2130
(J=U_K^1U_A^m+1 to P=U_K^1U_A^m. The actual character ratio λ=θ^g/θ for g∉ J is nontrivial and sign-valued. With H_0=kerλ, the quotient is P/H_0≃ F_2^2, and Ind_J^Pθ is an irreducible two-dimensional Heisenberg carrier ([source reference]). At rank four, the actual outer quadratic centralizer gives P=J U_B^m, [P:J]=2, and P/H_0≃ F_2^4, with carrier Ind_J^Pη of dimension four ([source reference]). The finite scalar center can be larger than μ_2; no literal extraspecial quotient is claimed.
Exact mathematical passage
P/H_0\simeq\FF_2^4
\FF_2^\timesUse in source · 1 passage
Search aliases: FF_2^× F_2^× FF times mathbb times
Use in source · Read passage · dyadic_character.tex, line 35
The construction retains the actual canonical beta extension and can be completed with any prescribed central quasicharacter omega_F trivial on U_F 1. Because k_F^×= F_2^× is trivial, such an omega_F is unramified. Thus this covers every tamely ramified central quasicharacter in this particular residue-field setting.
Exact mathematical passage
k_F^\times=\FF_2^\times
\FF_3^2Use in source · 1 passage
Search aliases: FF_3^2 F_3^2 FF mathbb
Use in source · Read passage · Paper_XI_Traditional_Final.tex, line 1958
[Rank one Heisenberg group] Take V= F_3^2. The Heisenberg group has order 3^3=27, and its irreducible representation with a fixed nontrivial central character has dimension three. The trace vanishes on every noncentral element. Averaging the translation subgroup gives a rank one projector.
Exact mathematical passage
V=\FF_3^2
\FF_p^2Use in source · 1 passage
Search aliases: FF_p^2 F_p^2 FF mathbb
Use in source · Read passage · Paper_XI_Traditional_Final.tex, line 1962
[Nondegenerate twisted algebra] Let A= F_p^2 with its standard alternating form. A cocycle with that commutator form gives C^c[A]≅ M_p( C). The unique irreducible projective representation has dimension p.
Exact mathematical passage
A=\FF_p^2
\FF_qUse in source · 2 passages
Search aliases: FF_q F_q FF mathbb
Use in source · Read passage · Paper_XI_Traditional_Final.tex, line 455
Let G_n=GL_n( F_q), T_n≅ F_q^n^× with T_n an elliptic maximal torus.
Exact mathematical passage
G_n=\GL_n(\FF_q), \qquad T_n\cong\FF_{q^n}^{\times}
\FF_{q^L}^{\times}Use in source · 2 passages
Search aliases: FF_q^L^× F_q^L^× FF times mathbb times
Use in source · Read passage · Paper_XI_Traditional_Final.tex, line 619
[Refined exponent formula] The ith interaction packet on F_q^L^× has exponent c_i = afracq^L-1q^n-1 +bq^ifracq^L-1q^m-1 pmodq^L-1. It is trivial on tame inertia exactly when c_i=0.
Exact mathematical passage
\FF_{q^L}^{\times}
\FF_{q^n}^{\times}Use in source · 3 passages
Search aliases: FF_q^n^× F_q^n^× FF times mathbb times
Use in source · Read passage · Paper_XI_Traditional_Final.tex, line 410
Characters of F_q^n^× are indexed by exponents modulo q^n-1, and Frobenius acts by multiplication by q.
Exact mathematical passage
\FF_{q^n}^{\times}
\FF_{q^r}^{\times}Use in source · 1 passage
Search aliases: FF_q^r^× F_q^r^× FF times mathbb times
Use in source · Read passage · Paper_XI_Traditional_Final.tex, line 427
For every divisor rmid n, the characters fixed by Frob^r are exactly those descending to F_q^r^×, and there are q^r-1 of them. Möbius inversion gives the number of characters of exact orbit length n. Division by n gives the number of orbits.
Exact mathematical passage
\FF_{q^r}^{\times}
\frakHUse in source · 3 passages
Search aliases: frakH H mathfrak
Use in source · Read passage · Paper_XI_Traditional_Final.tex, line 800
Let p be odd. Let X be an r dimensional vector space over F_p, and put V=X⊕ X^*. Define the alternating form b((x,ξ),(y,η)) = η(x)-ξ(y). Let Z= F_p, and define the Heisenberg group H(V)=V× Z with multiplication (v,z)(w,t) = (v+w,z+t+frac12 b(v,w)). Fix the additive character ψ(z)= e^2π iz/p.
Exact mathematical passage
\frakH(V)=V\times Z
\FrobUse in source · 1 passage
Search aliases: Frob operatorname
Use in source · Read passage · Paper_XI_Traditional_Final.tex, line 427
For every divisor rmid n, the characters fixed by Frob^r are exactly those descending to F_q^r^×, and there are q^r-1 of them. Möbius inversion gives the number of characters of exact orbit length n. Division by n gives the number of orbits.
Exact mathematical passage
\Frob^r
\Frob^rUse in source · 1 passage
Search aliases: Frob^r Frob operatorname
Use in source · Read passage · Paper_XI_Traditional_Final.tex, line 427
For every divisor rmid n, the characters fixed by Frob^r are exactly those descending to F_q^r^×, and there are q^r-1 of them. Möbius inversion gives the number of characters of exact orbit length n. Division by n gives the number of orbits.
Exact mathematical passage
\Frob^r
g'Use in source · 2 passages
Search aliases: g'
Use in source · Read passage · Paper_XI_Traditional_Final.tex, line 1325
[Symplectic invariance] If ι:(V,b)⟶(V',b') is a symplectic isometry and g'=ι gι^-1, then Φ_p(g,V)=Φ_p(g',V').
Exact mathematical passage
g'=\iota g\iota^{-1}
g^{-1}Use in source · 4 passages
Search aliases: g^-1
Use in source · Read passage · Paper_XI_Traditional_Final.tex, line 470
[Primitive cuspidal idempotent] Let χ_θ be the irreducible character of κ_θ, and let d_θ=χ_θ(1). Then e_θ = fracd_θabsG_n ∑_g∈ G_n χ_θ(g^-1)g is the primitive central idempotent selecting the κ_θ block in C[G_n].
Exact mathematical passage
e_\theta = \frac{d_\theta}{\abs{G_n}} \sum_{g\in G_n} \chi_\theta(g^{-1})g
G_nUse in source · 10 passages
Search aliases: G_n
Use in source · Read passage · Paper_XI_Traditional_Final.tex, line 455
Let G_n=GL_n( F_q), T_n≅ F_q^n^× with T_n an elliptic maximal torus.
Exact mathematical passage
G_n=\GL_n(\FF_q), \qquad T_n\cong\FF_{q^n}^{\times}
g_sUse in source · 4 passages
Search aliases: g_s
Use in source · Read passage · Paper_XI_Traditional_Final.tex, line 1784
[Breakwise phase product] In odd residue characteristic, let g∈Sp(V) preserve every break packet and suppose its restriction to M=im(g-I) is fixed point free. Put g_s=g|_V_s and M_s=im(g_s-I). The moving restrictions are symplectic, and Φ_p(g|_M,M) = ∏_s∈ SΦ_p(g_s|_M_s,M_s). In residue characteristic two, the analogous normalized Gauss sign is the product of the signs determined by the breakwise quadratic refinements.
Exact mathematical passage
g_s=g|_{V_s}
\mathrm{GL}Use in source · 1 passage
Search aliases: GL mathrm
Use in source · Read passage · dyadic_synthesis.tex, line 25
The actual stratum admits a full simple character θ trivial on U_F^1. Its canonical beta extension can be completed to an actual irreducible admissible supercuspidal π of GL_N(F) with any prescribed smooth central quasicharacter ω trivial on U_F^1, provided the noncompact completion is free or already compatible. Over this residue field, such ω is unramified.
Exact mathematical passage
\mathrm{GL}_N(F)
\GL_{\dim\rho}Use in source · 3 passages
Search aliases: GL_dimρ GL dim rho mathrm dim rho GL_dimρ GL dim rho operatorname dim rho
Use in source · Read passage · Paper_XI_Traditional_Final.tex, line 1875
begingroup renewcommand6666 [Arithmetic packet realization] For every ρ∈ A with its supplied H1--H4 assembled finite datum d_ρ, there exists an actual admissible supercuspidal representation π of GL_dimρ(K) and an arithmetic realization witness R(d_ρ,π). endgroup
Exact mathematical passage
\GL_{\dim\rho}(K)
\GL_NUse in source · 2 passages
Search aliases: GL_N GL mathrm GL_N GL operatorname
Use in source · Read passage · Paper_XI_Traditional_Final.tex, line 2137
At both ranks K^× normalizes P. Thus N=K^× J is normal of index two in M=K^× P, and Σ=Ind_ N^ MΛ, π≃cInd_ M^GL_N(F)Σ. Induction in stages preserves the full original Λ, its central character, and its noncompact root; see [source reference]. The full marked scalar-pushout comparison applies to these actual enlarged pairs as well as to the original higher-rank types ([source reference]). The full synthesis and scope are in [source reference].
Exact mathematical passage
\Sigma=\Ind_{\mathcal N}^{\mathcal M}\Lambda, \qquad \pi\simeq\cInd_{\mathcal M}^{\GL_N(F)}\Sigma.
\GL_nUse in source · 10 passages
Search aliases: GL_n GL mathrm GL_n GL operatorname
Use in source · Read passage · Paper_XI_Traditional_Final.tex, line 190
The local Langlands correspondence for GL_n(K) is an established theorem. It matches irreducible smooth representations of GL_n(K) with Frobenius semisimple Weil Deligne representations of dimension n. Its proof and normalization draw on global methods, harmonic analysis, types, local factors, and the geometry of Shimura varieties [source reference]. The present article asks a narrower structural question. Which parts of the local correspondence admit finite packet coordinates whose projectors, …
Exact mathematical passage
\GL_n(K)
\mathrm{GL}_NUse in source · 1 passage
Search aliases: GL_N mathrm
Use in source · Read passage · dyadic_synthesis.tex, line 25
The actual stratum admits a full simple character θ trivial on U_F^1. Its canonical beta extension can be completed to an actual irreducible admissible supercuspidal π of GL_N(F) with any prescribed smooth central quasicharacter ω trivial on U_F^1, provided the noncompact completion is free or already compatible. Over this residue field, such ω is unramified.
Exact mathematical passage
\mathrm{GL}_N(F)
\mathcal HUse in source · 1 passage
Search aliases: H mathcal
Use in source · Paper_XI_Visual_Guide_Final.tex, line 229
lessonPART II / FINITE WILD COMPARISONTranslations and multipliers create Heisenberg packetsOriginal Section 7 plate07_heisenbergThe commutator is central and nontrivial when the pairing is nonzero. For odd p let V=X⊕ X^* and b((x,ξ),(y,η))=η(x)-ξ(y). The group H(V)=V× F_p has product (v,z)(w,t)=(v+w,z+t+tfrac12b(v,w)). On H= C^X the Schrödinger operator is (ρ(x,ξ,z)f)(u) =ψ(z+ξ(u)+tfrac12ξ(x))f(u+x). Translations and character multipliers generate every matrix unit by finite Fourier inversion. …
Exact mathematical passage
\mathcal H=\C^X
\mathrm{H5}Use in source · 1 passage
Search aliases: H5 mathrm
Use in source · Paper_XI_Visual_Guide_Final.tex, line 639
There is a logical difference between ∀π [R(d,π)⇒H5(π)] and ∃π [R(d,π)landH5(π)]. The first statement can be vacuous. The second includes an actual witness. Choosing the established reference atom only supplies a profile-matching object; it leaves its recovery of d unproved. takeawayH5 remains a conditional premise about the same independently constructed candidate. It is not renamed as an unconditional factor-comparison theorem.
Exact mathematical passage
\forall\pi\,[R(d,\pi)\Rightarrow\mathrm{H5}(\pi)] \quad\text{and}\quad \exists\pi\,[R(d,\pi)\land\mathrm{H5}(\pi)].
H^aUse in source · 1 passage
Search aliases: H^a
Use in source · Read passage · odd_capacity.tex, line 133
For H, BK 3.1.9(ii) states that H^a(β_i)=H^a(β_i+1) as soon as
Exact mathematical passage
H^a(\beta_i)=H^a(\beta_{i+1})
H^mUse in source · 1 passage
Search aliases: H^m
Use in source · Read passage · odd_capacity.tex, line 190
The right side is the Heisenberg commutator pairing for the minimal simple stratum of gamma. Its quotient J^1(γ)/H^1(γ) is precisely U^m(A)/H^m(γ), because J^1(γ)=U^1(B_order)U^m(A), with U^1(B_order)⊂ H^1(γ). BK's nondegeneracy theorem for simple-character commutators therefore says that the form on S is nondegenerate.
Exact mathematical passage
U^m(A)/H^m(\gamma)
H^{m+1}Use in source · 1 passage
Search aliases: H^m+1
Use in source · Read passage · dyadic_synthesis.tex, line 109
For a later native step, first transfer the full beta-determinant unit character to multiplicity two. BK 3.6.13(ii) and 3.6.14 give, on scalar units, Θ_j(uI)=θ_j(u^2)=1. For odd critical level r, use m=lfloor r/2rfloor and D=H^m+1(β)=H^m+1(γ). The endpoint is valid because 2(m+1)-r=1: the additive-character cross term lies in the radical on which the matrix additive character is trivial. BK 3.3.18 supplies ϑ=Θ_j|_D ψ_β-γ. Integral trace makes it trivial on D∩ U_F^1.
Exact mathematical passage
D=H^{m+1}(\beta)=H^{m+1}(\gamma)
H_0Use in source · 13 passages
Search aliases: H_0
Use in source · Read passage · Paper_XI_Traditional_Final.tex, line 2125
At rank two, with q=2m+1, enlarge J=U_K^1U_A^m+1 to P=U_K^1U_A^m. The actual character ratio λ=θ^g/θ for g∉ J is nontrivial and sign-valued. With H_0=kerλ, the quotient is P/H_0≃ F_2^2, and Ind_J^Pθ is an irreducible two-dimensional Heisenberg carrier ([source reference]). At rank four, the actual outer quadratic centralizer gives P=J U_B^m, [P:J]=2, and P/H_0≃ F_2^4, with carrier Ind_J^Pη of dimension four ([source reference]). The finite scalar center can be larger than μ_2; no literal extrasp…
Exact mathematical passage
H_0=\ker\lambda
h_1Use in source · 11 passages
Search aliases: h_1
Use in source · Read passage · Paper_XI_Traditional_Final.tex, line 2021
[Restricted odd-prime arithmetic realization] Let p be an odd prime, F= F_p((t)), fix d≥1, and let 0<h_1<⋯<h_d, pnmid h_i, q_i=(p+1)h_i, N=p^d. There is an explicit maximal simple stratum in M_N(F) with totally ramified full-degree parameter field, a literal defining sequence of degrees p,p^2,…,p^d, and levels n_j=p^d-jq_d-j+1. Every native parameter and consecutive correction has trace zero over its recursive base field. The stratum admits a full simple character trivial on U_F^1. Its fixed can…
Exact mathematical passage
0<h_1<\cdots<h_d,\qquad p\nmid h_i, \qquad q_i=(p+1)h_i,\qquad N=p^d.
h_2Use in source · 6 passages
Search aliases: h_2
Use in source · Read passage · dyadic_synthesis.tex, line 219
The rank-two and rank-four carrier failures cannot be repaired by changing the maximal simple parameter while keeping the specified Swan exponent and torsion number. At rank two, odd Swan forces a minimal odd-level stratum and J^1=H^1. At rank four, Swan 6h_2 with h_2 odd forces the degree pattern 4→2; the maximal quadratic derived field forces an odd lower critical level. Every such maximal-type carrier has dimension two, below the required four. Scalar twists preserve this capacity.
Exact mathematical passage
6h_2
h_{d-1}Use in source · 2 passages
Search aliases: h_d-1
Use in source · Read passage · dyadic_synthesis.tex, line 62
erse are κ=α^-1, s=κpr, s(P_0)=t_L/z^2, Lift_E(C)=(z^2/t_L)CP_0. They satisfy s(Lift_E(C))=C, preserve the exact filtration valuation, and give the identity tr_LLift_E(C) =Tr_E/L (αtr_E C). The actual recursive matrix is B(L,t_L;h_1,…,h_d) =α I_2^d-1+ Lift_E(B(E,z;h_1,…,h_d-1)), with multiplication by α as the rank-two base case.
Exact mathematical passage
B(L,t_L;h_1,\ldots,h_d) =\alpha I_{2^{d-1}}+ \operatorname{Lift}_E\bigl(B(E,z;h_1,\ldots,h_{d-1})\bigr),
H_i^1Use in source · 2 passages
Search aliases: H_i^1
Use in source · Read passage · odd_synthesis.tex, line 144
Choose a labelled symplectic embedding V_src→ S. For each original full character form P_i=( C^×× J_i^1)/ {(θ_i(h)^-1,h):h∈ H_i^1}. The balanced product and the corresponding host restriction have the same alternating commutator. The center-fixing extension isomorphism, obtained with matched Darboux lifts and complex roots, therefore preserves every supplied value θ_i(h), including the shallow values.
Exact mathematical passage
\mathcal P_i=(\mathbb C^\times\times J_i^1)/ \{(\theta_i(h)^{-1},h):h\in H_i^1\}.
H_\theta^1Use in source · 6 passages
Search aliases: H_θ^1 theta
Use in source · Read passage · Paper_XI_Traditional_Final.tex, line 1154
Let θ be a maximal simple character for GL_n(K), with compact open subgroups H_θ^1⊆ J_θ^1⊆ J_θ. The quotient V_θ=J_θ^1/H_θ^1 is regarded as a finite vector space over F_p, where p is the residue characteristic. Fix a nontrivial additive character ψ_0: F_p→μ_p⊂ C^×, where μ_p is the group of p-th roots of unity. The simple-character commutator takes values in μ_p. Since ψ_0 is injective, the equation ψ_0(h_θ(bar x,bar y)) = θ([x,y]) uniquely defines the F_p-valued commutator pairing h_θ. We use t…
Exact mathematical passage
H_\theta^1\subseteq J_\theta^1\subseteq J_\theta.
\operatorname{Hom}Use in source · 2 passages
Search aliases: Hom operatorname
Use in source · Paper_XI_Visual_Guide_Final.tex, line 267
lessonPART II / FINITE WILD COMPARISONClifford multiplicity is the correct action spaceOriginal Section 9 Supply finite groups Ntriangleleft G, elementary abelian quotient A=G/N, and irreducibles Π and ρ, with ρ fixed up to isomorphism by G and Π|_N isotypic. The multiplicity space is M_ρ=Hom_N(W,X), W⊗ M_ρ≃ X. For a section s:A→ G and intertwiners I_a of ρ, write Π(s(a))=I_a⊗ T_a. The operators T_a give a projective action on M_ρ. Section and intertwiner changes alter it only by scalar changes …
Exact mathematical passage
M_\rho=\operatorname{Hom}_N(W,X),\qquad W\otimes M_\rho\simeq X.
\operatorname{Hom}_NUse in source · 1 passage
Search aliases: Hom_N operatorname
Use in source · Paper_XI_Visual_Guide_Final.tex, line 267
lessonPART II / FINITE WILD COMPARISONClifford multiplicity is the correct action spaceOriginal Section 9 Supply finite groups Ntriangleleft G, elementary abelian quotient A=G/N, and irreducibles Π and ρ, with ρ fixed up to isomorphism by G and Π|_N isotypic. The multiplicity space is M_ρ=Hom_N(W,X), W⊗ M_ρ≃ X. For a section s:A→ G and intertwiners I_a of ρ, write Π(s(a))=I_a⊗ T_a. The operators T_a give a projective action on M_ρ. Section and intertwiner changes alter it only by scalar changes …
Exact mathematical passage
M_\rho=\operatorname{Hom}_N(W,X),\qquad W\otimes M_\rho\simeq X.
\Hom_{W_K}Use in source · 1 passage
Search aliases: Hom_W_K Hom operatorname
Use in source · Read passage · Paper_XI_Traditional_Final.tex, line 683
[Tame Hom test] For irreducible tame atoms, dimHom_W_K(τ_ξ^vee,τ_η) is zero or one, and it is one exactly when the refined tensor packet test in [source reference] finds one trivial packet.
Exact mathematical passage
\dim\Hom_{W_K}(\tau_\xi^\vee,\tau_\eta)
I_{2^{d-1}}Use in source · 1 passage
Search aliases: I_2^d-1
Use in source · Read passage · dyadic_synthesis.tex, line 62
erse are κ=α^-1, s=κpr, s(P_0)=t_L/z^2, Lift_E(C)=(z^2/t_L)CP_0. They satisfy s(Lift_E(C))=C, preserve the exact filtration valuation, and give the identity tr_LLift_E(C) =Tr_E/L (αtr_E C). The actual recursive matrix is B(L,t_L;h_1,…,h_d) =α I_2^d-1+ Lift_E(B(E,z;h_1,…,h_d-1)), with multiplication by α as the rank-two base case.
Exact mathematical passage
B(L,t_L;h_1,\ldots,h_d) =\alpha I_{2^{d-1}}+ \operatorname{Lift}_E\bigl(B(E,z;h_1,\ldots,h_{d-1})\bigr),
I_aUse in source · 5 passages
Search aliases: I_a
Use in source · Read passage · Paper_XI_Traditional_Final.tex, line 1047
The isotypic condition gives an N equivariant evaluation isomorphism W⊗ M_ρ ⟶ X. Choose a section s:A→ G. For every a∈ A, choose an intertwiner I_a from ρ to its conjugate by s(a). Under the evaluation isomorphism, there is a unique operator T_a on M_ρ such that Π(s(a))=I_a⊗ T_a. The operators T_a form a projective representation of A.
Exact mathematical passage
I_a
I_KUse in source · 9 passages
Search aliases: I_K
Use in source · Read passage · Paper_XI_Traditional_Final.tex, line 234
Let W_K be the Weil group of K, let I_K be inertia, let P_K be wild inertia, and choose a geometric Frobenius lift Φ. A Weil Deligne representation is a pair (r,N) with r(w)Nr(w)^-1=absw_K N.
Exact mathematical passage
I_K
I_mUse in source · 2 passages
Search aliases: I_m
Use in source · Paper_XI_Visual_Guide_Final.tex, line 399
STRUCTIONA recursive matrix gives a literal parameterBlocks 166--167, tower notes plate14_recursive_parameterThe recursive lift is an actual filtered right inverse of normalized corestriction. Start over a coefficient field L with uniformizer t_L and form E=L(α), α^p-α=t_L^-q. At a recursive stage, C is an actual matrix over E. The outer matrix is B=α I_m+Lift_E(C). The lift has prescribed corestriction and preserves its exact filtration valuation. The field and simple-stratum proof uses the act…
Exact mathematical passage
B=\alpha I_m+\operatorname{Lift}_E(C).
I_NUse in source · 3 passages
Search aliases: I_N
Use in source · Read passage · odd_synthesis.tex, line 183
There is also a positive result beyond C={1}. Suppose the finite prime-to-p stabilizer acts trivially on the source scalar extension, and its prescribed honest operators on the entire rank-N Weil carrier are χ(c)I_N, each lying in the actual inertia image. Assume the candidate's central character is the determinant of that same source under the fixed rank-one reciprocity convention.
Exact mathematical passage
\chi(c)I_N
I_{p^{d-1}}Use in source · 1 passage
Search aliases: I_p^d-1
Use in source · Read passage · odd_synthesis.tex, line 75
, where κ=-f'(z)/z^p-1, satisfies u_0=s(P_0)=a_0/z^p∈ O_E^×, Lift_E(C)=u_0^-1CP_0, s(Lift_E(C))=C. The lift preserves the exact filtration valuation. Its crucial additional identity is tr_L(Lift_E(C)) =Tr_E/L (κ^-1tr_E(C)). Define B(L,t_L;h_1,…,h_d) =α I_p^d-1+ Lift_E(B(E,z;h_1,…,h_d-1)), with the degree-p multiplication matrix as the base case. Induction proves trace zero at every native rank. This is stronger than the automatic zero trace obtained by multiplying any matrix multiplicity by p in…
Exact mathematical passage
B(L,t_L;h_1,\ldots,h_d) =\alpha I_{p^{d-1}}+ \operatorname{Lift}_E\bigl(B(E,z;h_1,\ldots,h_{d-1})\bigr),
\operatorname{im}Use in source · 1 passage
Search aliases: im operatorname
Use in source · Paper_XI_Visual_Guide_Final.tex, line 320
For a hyperbolic space U⊕ U^* with g=(a,(a^-1)^*) and a-I invertible, Φ_p(g,U⊕ U^*)=(fracdet ap). If g has order prime to p, then V=ker(g-I)⊥im(g-I). The moving summand is nondegenerate and fixed-point-free. Orthogonal block phases multiply. boundaryThe fixed-point-free formula applies to the moving part. The full trace with a fixed part requires its declared normalization data.
Exact mathematical passage
V=\ker(g-I)\perp\operatorname{im}(g-I).
\operatorname{Ind}Use in source · 3 passages
Search aliases: Ind operatorname
Use in source · Read passage · dyadic_synthesis.tex, line 296
. Define P=U_K^1U_A^m. This is an actual compact open group, with Jtriangleleft P, [P:J]=2 and P∩(K^× J)=J. For g∈ P∖ J, the actual intertwining theorem makes λ=θ^g/θ nontrivial. The commutator [P,J]⊂ U_A^m+1 lies in the forced additive-character domain, so λ has values {±1}. Put H_0=kerλ. Then P/H_0≃ F_2^2, E=Ind_J^Pθ is irreducible of dimension 2. The actual represented generators are a diagonal sign and an off-diagonal matrix that anticommute. Thus their commutator form is nondegenerate, and …
Exact mathematical passage
P/H_0\simeq\mathbb F_2^2,\qquad E=\operatorname{Ind}_J^P\theta\text{ is irreducible of dimension }2.
\Ind_{\mathcal N}^{\mathcal M}Use in source · 2 passages
Search aliases: Ind_ N^ M Ind mathcal mathcal operatorname mathcal mathcal
Use in source · Read passage · Paper_XI_Traditional_Final.tex, line 2137
At both ranks K^× normalizes P. Thus N=K^× J is normal of index two in M=K^× P, and Σ=Ind_ N^ MΛ, π≃cInd_ M^GL_N(F)Σ. Induction in stages preserves the full original Λ, its central character, and its noncompact root; see [source reference]. The full marked scalar-pushout comparison applies to these actual enlarged pairs as well as to the original higher-rank types ([source reference]). The full synthesis and scope are in [source reference].
Exact mathematical passage
\Sigma=\Ind_{\mathcal N}^{\mathcal M}\Lambda, \qquad \pi\simeq\cInd_{\mathcal M}^{\GL_N(F)}\Sigma.
\operatorname{Ind}_{\mathcal N}^{\mathcal M}Use in source · 1 passage
Search aliases: Ind_ N^ M operatorname mathcal mathcal
Use in source · Read passage · dyadic_synthesis.tex, line 349
Consequently M=K^× P is an actual open group compact modulo the center, N=K^× J is normal of index two in M, and Σ=Ind_ N^ MΛ, Σ|_P≃ E, π≃c - Ind_ M^GΣ. Restriction to P is irreducible, so Σ is irreducible. Induction in stages proves the last equality for the original Λ, including its fixed central character and noncompact root. This is a genuine enlarged inducing representation; it changes neither the candidate nor its original canonical beta-extension data.
Exact mathematical passage
\Sigma=\operatorname{Ind}_{\mathcal N}^{\mathcal M}\Lambda, \qquad \Sigma|_P\simeq E, \qquad \boxed{\pi\simeq\operatorname{c\!-\!Ind}_{\mathcal M}^G\Sigma.}
\Ind_J^PUse in source · 4 passages
Search aliases: Ind_J^P Ind operatorname
Use in source · Read passage · Paper_XI_Traditional_Final.tex, line 2126
At rank two, with q=2m+1, enlarge J=U_K^1U_A^m+1 to P=U_K^1U_A^m. The actual character ratio λ=θ^g/θ for g∉ J is nontrivial and sign-valued. With H_0=kerλ, the quotient is P/H_0≃ F_2^2, and Ind_J^Pθ is an irreducible two-dimensional Heisenberg carrier ([source reference]). At rank four, the actual outer quadratic centralizer gives P=J U_B^m, [P:J]=2, and P/H_0≃ F_2^4, with carrier Ind_J^Pη of dimension four ([source reference]). The finite scalar center can be larger than μ_2; no literal extrasp…
Exact mathematical passage
\Ind_J^P\theta
\operatorname{Ind}_J^PUse in source · 2 passages
Search aliases: Ind_J^P operatorname
Use in source · Read passage · dyadic_synthesis.tex, line 296
. Define P=U_K^1U_A^m. This is an actual compact open group, with Jtriangleleft P, [P:J]=2 and P∩(K^× J)=J. For g∈ P∖ J, the actual intertwining theorem makes λ=θ^g/θ nontrivial. The commutator [P,J]⊂ U_A^m+1 lies in the forced additive-character domain, so λ has values {±1}. Put H_0=kerλ. Then P/H_0≃ F_2^2, E=Ind_J^Pθ is irreducible of dimension 2. The actual represented generators are a diagonal sign and an off-diagonal matrix that anticommute. Thus their commutator form is nondegenerate, and …
Exact mathematical passage
P/H_0\simeq\mathbb F_2^2,\qquad E=\operatorname{Ind}_J^P\theta\text{ is irreducible of dimension }2.
\Ind_{W_{K_f}}^{W_K}Use in source · 5 passages
Search aliases: Ind_W_K_f^W_K Ind operatorname
Use in source · Read passage · Paper_XI_Traditional_Final.tex, line 339
[Tame orbit induction] Let τ be an irreducible complex representation of W_K with trivial wild inertia. Then the nonzero tame character packets form one transitive Frobenius orbit of length f. If K_f/K is the unramified extension of degree f, there is a character ξ of W_K_f such that τ≅Ind_W_K_f^W_Kξ.
Exact mathematical passage
\boxed{ \tau\cong\Ind_{W_{K_f}}^{W_K}\xi. }
j^{-1}Use in source · 1 passage
Search aliases: j^-1
Use in source · Read passage · rank2.tex, line 139
Finally, if kj∈P with k∈K× and j∈J, then k∈ Pj^-1⊂ U_A^1. Hence k∈U_K 1 and kj∈J. Therefore
Exact mathematical passage
k\in Pj^{-1}\subset U_A^1
J_i^1Use in source · 2 passages
Search aliases: J_i^1
Use in source · Read passage · odd_synthesis.tex, line 144
Choose a labelled symplectic embedding V_src→ S. For each original full character form P_i=( C^×× J_i^1)/ {(θ_i(h)^-1,h):h∈ H_i^1}. The balanced product and the corresponding host restriction have the same alternating commutator. The center-fixing extension isomorphism, obtained with matched Darboux lifts and complex roots, therefore preserves every supplied value θ_i(h), including the shallow values.
Exact mathematical passage
\mathcal P_i=(\mathbb C^\times\times J_i^1)/ \{(\theta_i(h)^{-1},h):h\in H_i^1\}.
J_\thetaUse in source · 5 passages
Search aliases: J_θ theta
Use in source · Read passage · Paper_XI_Traditional_Final.tex, line 1154
Let θ be a maximal simple character for GL_n(K), with compact open subgroups H_θ^1⊆ J_θ^1⊆ J_θ. The quotient V_θ=J_θ^1/H_θ^1 is regarded as a finite vector space over F_p, where p is the residue characteristic. Fix a nontrivial additive character ψ_0: F_p→μ_p⊂ C^×, where μ_p is the group of p-th roots of unity. The simple-character commutator takes values in μ_p. Since ψ_0 is injective, the equation ψ_0(h_θ(bar x,bar y)) = θ([x,y]) uniquely defines the F_p-valued commutator pairing h_θ. We use t…
Exact mathematical passage
H_\theta^1\subseteq J_\theta^1\subseteq J_\theta.
J_\theta^1Use in source · 10 passages
Search aliases: J_θ^1 theta
Use in source · Read passage · Paper_XI_Traditional_Final.tex, line 1154
Let θ be a maximal simple character for GL_n(K), with compact open subgroups H_θ^1⊆ J_θ^1⊆ J_θ. The quotient V_θ=J_θ^1/H_θ^1 is regarded as a finite vector space over F_p, where p is the residue characteristic. Fix a nontrivial additive character ψ_0: F_p→μ_p⊂ C^×, where μ_p is the group of p-th roots of unity. The simple-character commutator takes values in μ_p. Since ψ_0 is injective, the equation ψ_0(h_θ(bar x,bar y)) = θ([x,y]) uniquely defines the F_p-valued commutator pairing h_θ. We use t…
Exact mathematical passage
H_\theta^1\subseteq J_\theta^1\subseteq J_\theta.
kUse in source · 18 passages
Search aliases: k
Use in source · Read passage · Paper_XI_Traditional_Final.tex, line 715
Let k= F_q, let ψ:k→ C^× be a nontrivial additive character, and let χ:k^×→ C^× be a multiplicative character. Define G(χ,ψ) = ∑_x∈ k^×χ(x)ψ(x). The standard theory of Gauss and Jacobi sums will be used in the form recorded in [source reference].
Exact mathematical passage
k=\FF_q
K^\timesUse in source · 14 passages
Search aliases: K^× times
Use in source · Read passage · Paper_XI_Traditional_Final.tex, line 2134
At both ranks K^× normalizes P. Thus N=K^× J is normal of index two in M=K^× P, and Σ=Ind_ N^ MΛ, π≃cInd_ M^GL_N(F)Σ. Induction in stages preserves the full original Λ, its central character, and its noncompact root; see [source reference]. The full marked scalar-pushout comparison applies to these actual enlarged pairs as well as to the original higher-rank types ([source reference]). The full synthesis and scope are in [source reference].
Exact mathematical passage
K^\times
K^{\times}Use in source · 1 passage
Search aliases: K^× times
Use in source · Read passage · Paper_XI_Traditional_Final.tex, line 1816
Local class field theory identifies one dimensional Weil characters with characters of K^× [source reference]. The tame induction theorem extends this explicit picture to higher dimensional tame atoms. The local polynomial and conductor calculations in [source reference] therefore give explicit finite invariants for the declared tame packets. Finite Gauss sums give the residue field phase. The full local factor, however, also includes ramification depth, Haar measure normalization, the additive …
Exact mathematical passage
K^{\times}
k^\timesUse in source · 1 passage
Search aliases: k^× times
Use in source · Read passage · odd_synthesis.tex, line 194
Indeed, for residue cardinality q, let B=ω(Teich(k^×)). The actual inertia hypothesis gives χ(C)^N⊂ B. Since N is a power of p and χ(C) has order prime to p, this implies χ(C)⊂ B. Choose k_0 inverse to N modulo q-1, and set δ(u)=ω(Teich(u))^k_0, widetilde C={(c,u):χ(c)=δ(u)}. The matrix Teich(u^k_0)I_N acts by χ(c). The kernel acts trivially on the entire sector induction and ambient compact induction, because these matrices are central. This preserves the fixed Λ, its canonical beta-extension f…
Exact mathematical passage
B=\omega(\operatorname{Teich}(k^\times))
k^{\times}Use in source · 4 passages
Search aliases: k^× times
Use in source · Read passage · Paper_XI_Traditional_Final.tex, line 715
Let k= F_q, let ψ:k→ C^× be a nontrivial additive character, and let χ:k^×→ C^× be a multiplicative character. Define G(χ,ψ) = ∑_x∈ k^×χ(x)ψ(x). The standard theory of Gauss and Jacobi sums will be used in the form recorded in [source reference].
Exact mathematical passage
\chi:k^{\times}\to\CC^{\times}
k_0Use in source · 4 passages
Search aliases: k_0
Use in source · Read passage · odd_synthesis.tex, line 86
For the inductive field step d≥2, put n=p^d-1q_d and R=p^d-2q_d-1. The derived matrix is the recursively constructed maximal-field parameter, with k_0=-q_1. The inequalities n>pR>2R imply -2R>-n-q_1. Bushnell--Kutzko Remark 2.2.7 therefore applies to the actual displayed matrix. It supplies its maximal field, normalization, and critical exponent. Valuation preservation and literal truncation of the recursion supply the whole defining sequence. The full details, quotient count, and conductor depe…
Exact mathematical passage
k_0=-q_1
K_fUse in source · 10 passages
Search aliases: K_f
Use in source · Read passage · Paper_XI_Traditional_Final.tex, line 338
[Tame orbit induction] Let τ be an irreducible complex representation of W_K with trivial wild inertia. Then the nonzero tame character packets form one transitive Frobenius orbit of length f. If K_f/K is the unramified extension of degree f, there is a character ξ of W_K_f such that τ≅Ind_W_K_f^W_Kξ.
Exact mathematical passage
K_f/K
k_F^\timesUse in source · 1 passage
Search aliases: k_F^× times
Use in source · Read passage · dyadic_character.tex, line 35
The construction retains the actual canonical beta extension and can be completed with any prescribed central quasicharacter omega_F trivial on U_F 1. Because k_F^×= F_2^× is trivial, such an omega_F is unramified. Thus this covers every tamely ramified central quasicharacter in this particular residue-field setting.
Exact mathematical passage
k_F^\times=\FF_2^\times
k_rUse in source · 1 passage
Search aliases: k_r
Use in source · Read passage · Paper_XI_Traditional_Final.tex, line 773
[Hasse and Davenport lifting] Let k_r/k be the extension of degree r. Lift χ by the norm and ψ by the trace. Then the lifted Gauss sum satisfies the Hasse and Davenport product relation, with a sign determined by the chosen additive character normalization.
Exact mathematical passage
k_r/k
\kerUse in source · 10 passages
Search aliases: ker
Use in source · Read passage · Paper_XI_Traditional_Final.tex, line 1387
[Fixed and moving decomposition] Assume that g∈Sp(V) has order prime to p. Then V=ker(g-I)mathop⊥limitsim(g-I). The restriction of the symplectic form to im(g-I) is nondegenerate, and g-I is invertible on that moving summand.
Exact mathematical passage
V=\ker(g-I)\perpbig\im(g-I).
\landUse in source · 1 passage
Search aliases: land
Use in source · Paper_XI_Visual_Guide_Final.tex, line 639
There is a logical difference between ∀π [R(d,π)⇒H5(π)] and ∃π [R(d,π)landH5(π)]. The first statement can be vacuous. The second includes an actual witness. Choosing the established reference atom only supplies a profile-matching object; it leaves its recovery of d unproved. takeawayH5 remains a conditional premise about the same independently constructed candidate. It is not renamed as an unconditional factor-comparison theorem.
Exact mathematical passage
\forall\pi\,[R(d,\pi)\Rightarrow\mathrm{H5}(\pi)] \quad\text{and}\quad \exists\pi\,[R(d,\pi)\land\mathrm{H5}(\pi)].
\lcmUse in source · 2 passages
Search aliases: lcm operatorname
Use in source · Read passage · Paper_XI_Traditional_Final.tex, line 561
Let τ_ξ=Ind_W_K_n^W_Kξ, τ_η=Ind_W_K_m^W_Kη, where K_n/K and K_m/K are unramified. Put d=gcd(n,m), L=lcm(n,m).
Exact mathematical passage
d=\gcd(n,m), \qquad L=\lcm(n,m).
\lfloorUse in source · 3 passages
Search aliases: lfloor
Use in source · Read passage · Paper_XI_Traditional_Final.tex, line 2117
ently each native correction δ satisfies v_F(trδ)+s≥1 (s=1), which makes its additive scalar correction trivial on U_F^1. The actual field and defining-sequence proof is [source reference]. Full beta-determinant transfer and character extension at each new maximal native field prove the full-character claim, including the odd-critical endpoint 2(lfloor r/2rfloor+1)-r=1; see [source reference]. The exact original quotient has dimension 2^d-2, which accommodates the required 2d source dimensions p…
Exact mathematical passage
2(\lfloor r/2\rfloor+1)-r=1
\operatorname{Lift}Use in source · 11 passages
Search aliases: Lift operatorname
Use in source · Read passage · dyadic_synthesis.tex, line 53
er t_L, put E=L(α), α^2-α=t_L^-q, a=(q+1)/2, z=t_L^aα. The extension is separable and totally ramified of degree two; z is a uniformizer and z^2+t_L^az+t_L=0, Tr_E/L(α)=1. For constant-coefficient projection P_0 in the basis 1,z, the normalized trace corestriction and exact filtered right inverse are κ=α^-1, s=κpr, s(P_0)=t_L/z^2, Lift_E(C)=(z^2/t_L)CP_0. They satisfy s(Lift_E(C))=C, preserve the exact filtration valuation, and give the identity tr_LLift_E(C) =Tr_E/L (αtr_E C). The actual recurs…
Exact mathematical passage
\kappa=\alpha^{-1},\qquad s=\kappa\operatorname{pr},\qquad s(P_0)=t_L/z^2,\qquad \operatorname{Lift}_E(C)=(z^2/t_L)CP_0.
\operatorname{Lift}_EUse in source · 11 passages
Search aliases: Lift_E operatorname
Use in source · Read passage · dyadic_synthesis.tex, line 53
er t_L, put E=L(α), α^2-α=t_L^-q, a=(q+1)/2, z=t_L^aα. The extension is separable and totally ramified of degree two; z is a uniformizer and z^2+t_L^az+t_L=0, Tr_E/L(α)=1. For constant-coefficient projection P_0 in the basis 1,z, the normalized trace corestriction and exact filtered right inverse are κ=α^-1, s=κpr, s(P_0)=t_L/z^2, Lift_E(C)=(z^2/t_L)CP_0. They satisfy s(Lift_E(C))=C, preserve the exact filtration valuation, and give the identity tr_LLift_E(C) =Tr_E/L (αtr_E C). The actual recurs…
Exact mathematical passage
\kappa=\alpha^{-1},\qquad s=\kappa\operatorname{pr},\qquad s(P_0)=t_L/z^2,\qquad \operatorname{Lift}_E(C)=(z^2/t_L)CP_0.
\LLCUse in source · 2 passages
Search aliases: LLC operatorname
Use in source · Paper_XI_Visual_Guide_Final.tex, line 632
lessonPART IV / REALIZATION AND COMPARISONH5 compares this same actual candidateContract audit 162; conditional comparison proposition Suppose π is an actual admissible supercuspidal representation of the correct rank and R(d_ρ,π) is supplied. If H5 gives equality of its required pair local factors with those of LLC(ρ) on the stipulated characterizing twist family, with the fixed normalizations, then π≃LLC(ρ). The proof applies the supplied local-factor characterization to these two actual repre…
Exact mathematical passage
\LLC(\rho)
\LongrightarrowUse in source · 1 passage
Search aliases: Longrightarrow
Use in source · Paper_XI_Visual_Guide_Final.tex, line 430
lessonPART III / ACTUAL ARITHMETIC CONSTRUCTIONDyadic trace one is deep enoughBlock 167, depth invariant plate16_scalar_depthThe scalar correction disappears when its argument lies in t O_F. Fix ψ_F of conductor t O_F. On a scalar unit u∈ U_F^s, the additive correction sees (u-1)trδ. Hence v_F(trδ)+s≥1 Longrightarrow ψ_F((u-1)trδ)=1. The dyadic native corrections have trace 1. Its valuation is 0, so s=1 already satisfies the inequality. Thus trace zero is sufficient but unnecessary. boundaryEule…
Exact mathematical passage
v_F(\tr\delta)+s\ge1 \quad\Longrightarrow\quad \psi_F((u-1)\tr\delta)=1.
\mathcal MUse in source · 7 passages
Search aliases: M mathcal
Use in source · Read passage · Paper_XI_Traditional_Final.tex, line 2136
At both ranks K^× normalizes P. Thus N=K^× J is normal of index two in M=K^× P, and Σ=Ind_ N^ MΛ, π≃cInd_ M^GL_N(F)Σ. Induction in stages preserves the full original Λ, its central character, and its noncompact root; see [source reference]. The full marked scalar-pushout comparison applies to these actual enlarged pairs as well as to the original higher-rank types ([source reference]). The full synthesis and scope are in [source reference].
Exact mathematical passage
\mathcal M=K^\times P
M^\perpUse in source · 1 passage
Search aliases: M^⊥ perp
Use in source · Read passage · Paper_XI_Traditional_Final.tex, line 1796
The invariant orthogonal decomposition gives M=mathop⊥limits_s∈ SM_s. Symplectic invariance gives M^⊥=ker(g-I). Fixed point freeness on M therefore makes the restricted form nondegenerate on M and on each M_s. Each restriction of g_s-I to M_s is invertible, so [source reference] applies to the moving spaces. A zero moving space contributes one. In characteristic two, apply [source reference].
Exact mathematical passage
M^\perp=\ker(g-I)
M_NUse in source · 3 passages
Search aliases: M_N
Use in source · Read passage · Paper_XI_Traditional_Final.tex, line 2025
[Restricted odd-prime arithmetic realization] Let p be an odd prime, F= F_p((t)), fix d≥1, and let 0<h_1<⋯<h_d, pnmid h_i, q_i=(p+1)h_i, N=p^d. There is an explicit maximal simple stratum in M_N(F) with totally ramified full-degree parameter field, a literal defining sequence of degrees p,p^2,…,p^d, and levels n_j=p^d-jq_d-j+1. Every native parameter and consecutive correction has trace zero over its recursive base field. The stratum admits a full simple character trivial on U_F^1. Its fixed can…
Exact mathematical passage
M_N(F)
M_pUse in source · 1 passage
Search aliases: M_p
Use in source · Read passage · Paper_XI_Traditional_Final.tex, line 1963
[Nondegenerate twisted algebra] Let A= F_p^2 with its standard alternating form. A cocycle with that commutator form gives C^c[A]≅ M_p( C). The unique irreducible projective representation has dimension p.
Exact mathematical passage
\CC^c[A]\cong M_p(\CC).
M_sUse in source · 5 passages
Search aliases: M_s
Use in source · Read passage · Paper_XI_Traditional_Final.tex, line 1784
[Breakwise phase product] In odd residue characteristic, let g∈Sp(V) preserve every break packet and suppose its restriction to M=im(g-I) is fixed point free. Put g_s=g|_V_s and M_s=im(g_s-I). The moving restrictions are symplectic, and Φ_p(g|_M,M) = ∏_s∈ SΦ_p(g_s|_M_s,M_s). In residue characteristic two, the analogous normalized Gauss sign is the product of the signs determined by the breakwise quadratic refinements.
Exact mathematical passage
M_s=\im(g_s-I)
M_{\sqrt{\abs A}}Use in source · 1 passage
Search aliases: M_sqrtabs A sqrt abs
Use in source · Read passage · Paper_XI_Traditional_Final.tex, line 959
[Nondegenerate twisted group algebra] Assume that b_c is nondegenerate. Then abs A is a square, and C^c[A] ≅ M_sqrtabs A( C). Consequently, C^c[A] has one irreducible module up to isomorphism, of dimension sqrtabs A.
Exact mathematical passage
\boxed{ \CC^c[A] \cong M_{\sqrt{\abs A}}(\CC). }
M_{\sqrt{|A|}}Use in source · 1 passage
Search aliases: M_sqrt|A| sqrt
Use in source · Paper_XI_Visual_Guide_Final.tex, line 260
The radical consists of a commuting projectively with every b. Its quotient carries a nondegenerate alternating pairing. In the nondegenerate case the twisted group algebra has scalar center and is a full matrix algebra: C_c[A]≃ M_sqrt|A|( C). boundaryThis constructs the finite envelope for a supplied projective action. It does not produce a canonical action for every arbitrary wild inertia atom.
Exact mathematical passage
\C_c[A]\simeq M_{\sqrt{|A|}}(\C).
\mathopUse in source · 1 passage
Search aliases: mathop
Use in source · Read passage · Paper_XI_Traditional_Final.tex, line 1796
The invariant orthogonal decomposition gives M=mathop⊥limits_s∈ SM_s. Symplectic invariance gives M^⊥=ker(g-I). Fixed point freeness on M therefore makes the restricted form nondegenerate on M and on each M_s. Each restriction of g_s-I to M_s is invertible, so [source reference] applies to the moving spaces. A zero moving space contributes one. In characteristic two, apply [source reference].
Exact mathematical passage
M=\mathop{\perp}\limits_{s\in S}M_s
\midUse in source · 8 passages
Search aliases: mid
Use in source · Read passage · Paper_XI_Traditional_Final.tex, line 414
[Möbius orbit count] The number of regular Frobenius orbits of length n is c_n(q) = frac1n ∑_dmid n μ(d) (q^n/d-1).
Exact mathematical passage
\boxed{ c_n(q) = \frac1n \sum_{d\mid n} \mu(d) \bigl(q^{n/d}-1\bigr). }
\mathcal NUse in source · 7 passages
Search aliases: N mathcal
Use in source · Read passage · Paper_XI_Traditional_Final.tex, line 2135
At both ranks K^× normalizes P. Thus N=K^× J is normal of index two in M=K^× P, and Σ=Ind_ N^ MΛ, π≃cInd_ M^GL_N(F)Σ. Induction in stages preserves the full original Λ, its central character, and its noncompact root; see [source reference]. The full marked scalar-pushout comparison applies to these actual enlarged pairs as well as to the original higher-rank types ([source reference]). The full synthesis and scope are in [source reference].
Exact mathematical passage
\mathcal N=K^\times J
N^{-1}Use in source · 1 passage
Search aliases: N^-1
Use in source · Read passage · dyadic_synthesis.tex, line 366
There is also a recovery from compact data explicitly recording that root. With A_0=Λ(ϖ_K) and X=c - Ind_J^Gη, define (Tf)(g)=A_0^-1f(ϖ_Kg). Then X/(T-1)X≃π. After imposing the fixed central character, N/(F^× J) is cyclic of order N, and the finite average N^-1∑_i=0^N-1T^i selects this same extension. These operators act on complex representation spaces, so their scalar division by N is valid. This is a different recovery from the Block159 projector onto a sector of η. The exact kernel and finit…
Exact mathematical passage
N^{-1}\sum_{i=0}^{N-1}T^i
N^mUse in source · 3 passages
Search aliases: N^m
Use in source · Read passage · Paper_XI_Traditional_Final.tex, line 263
[Exact block reconstruction] Let P_j be the projector onto V_j. Then N=∑_j=1^m-1P_j-1NP_j, N^m=0. Every power N^r is the sum of the packet paths of length r along the chain.
Exact mathematical passage
N=\sum_{j=1}^{m-1}P_{j-1}NP_j, \qquad N^m=0.
N^rUse in source · 2 passages
Search aliases: N^r
Use in source · Read passage · Paper_XI_Traditional_Final.tex, line 268
[Exact block reconstruction] Let P_j be the projector onto V_j. Then N=∑_j=1^m-1P_j-1NP_j, N^m=0. Every power N^r is the sum of the packet paths of length r along the chain.
Exact mathematical passage
N^r
n_1Use in source · 4 passages
Search aliases: n_1
Use in source · Read passage · dyadic_synthesis.tex, line 19
((t)), fix an additive character ψ_F of conductor t O_F, and take increasing odd positive integers h_1<⋯<h_d. Put q_i=3h_i, N=2^d, n_j=2^d-jq_d-j+1. There is an explicit β∈ M_N(F) generating a separable totally ramified degree-N field that normalizes the complete lattice chain. It gives a maximal simple stratum with leading level n_1, final critical exponent -q_1, and successive approximation-field degrees 2,4,…,2^d, with critical levels n_2,…,n_d.
Exact mathematical passage
n_1
n_2Use in source · 3 passages
Search aliases: n_2
Use in source · Read passage · dyadic_synthesis.tex, line 21
cdots<h_d. Put q_i=3h_i, N=2^d, n_j=2^d-jq_d-j+1. There is an explicit β∈ M_N(F) generating a separable totally ramified degree-N field that normalizes the complete lattice chain. It gives a maximal simple stratum with leading level n_1, final critical exponent -q_1, and successive approximation-field degrees 2,4,…,2^d, with critical levels n_2,…,n_d.
Exact mathematical passage
n_2,\ldots,n_d
n_dUse in source · 4 passages
Search aliases: n_d
Use in source · Read passage · dyadic_synthesis.tex, line 21
cdots<h_d. Put q_i=3h_i, N=2^d, n_j=2^d-jq_d-j+1. There is an explicit β∈ M_N(F) generating a separable totally ramified degree-N field that normalizes the complete lattice chain. It gives a maximal simple stratum with leading level n_1, final critical exponent -q_1, and successive approximation-field degrees 2,4,…,2^d, with critical levels n_2,…,n_d.
Exact mathematical passage
n_2,\ldots,n_d
n_jUse in source · 7 passages
Search aliases: n_j
Use in source · Read passage · Paper_XI_Traditional_Final.tex, line 2028
[Restricted odd-prime arithmetic realization] Let p be an odd prime, F= F_p((t)), fix d≥1, and let 0<h_1<⋯<h_d, pnmid h_i, q_i=(p+1)h_i, N=p^d. There is an explicit maximal simple stratum in M_N(F) with totally ramified full-degree parameter field, a literal defining sequence of degrees p,p^2,…,p^d, and levels n_j=p^d-jq_d-j+1. Every native parameter and consecutive correction has trace zero over its recursive base field. The stratum admits a full simple character trivial on U_F^1. Its fixed can…
Exact mathematical passage
n_j=p^{d-j}q_{d-j+1}
N_{j-1,j}Use in source · 1 passage
Search aliases: N_j-1,j
Use in source · Read passage · Paper_XI_Traditional_Final.tex, line 280
[Scalar covariance model] Let F=diag(1,q,q^2,…,q^m-1) and let N_j-1,j=1. Then FNF^-1=q^-1N.
Exact mathematical passage
N_{j-1,j}=1
\nmidUse in source · 3 passages
Search aliases: nmid
Use in source · Read passage · Paper_XI_Traditional_Final.tex, line 2021
[Restricted odd-prime arithmetic realization] Let p be an odd prime, F= F_p((t)), fix d≥1, and let 0<h_1<⋯<h_d, pnmid h_i, q_i=(p+1)h_i, N=p^d. There is an explicit maximal simple stratum in M_N(F) with totally ramified full-degree parameter field, a literal defining sequence of degrees p,p^2,…,p^d, and levels n_j=p^d-jq_d-j+1. Every native parameter and consecutive correction has trace zero over its recursive base field. The stratum admits a full simple character trivial on U_F^1. Its fixed can…
Exact mathematical passage
0<h_1<\cdots<h_d,\qquad p\nmid h_i, \qquad q_i=(p+1)h_i,\qquad N=p^d.
\normUse in source · 1 passage
Search aliases: norm
Use in source · Read passage · Paper_XI_Traditional_Final.tex, line 1727
[Quantitative finite gluing] With the notation of [source reference], normU_vA_e-B_eU_u ≤ normB_p_vnormD_enormA_p_u^-1. Thus small fundamental cycle defects give small non tree edge residuals whenever the tree transports are uniformly conditioned.
Exact mathematical passage
\norm{U_vA_e-B_eU_u} \leq \norm{B_{p_v}}\norm{D_e}\norm{A_{p_u}^{-1}}.
OUse in source · 3 passages
Search aliases: O
Use in source · Read passage · Paper_XI_Traditional_Final.tex, line 1591
Every refinement is q_v for a unique v∈ V, because b identifies V with V^*. By [source reference], the Arf invariant changes according to q(v). Let Z and O be the numbers of zeros and ones of q. Then Z+O=2^2r and [source reference] gives Z-O=(-1)^Arf(q)2^r. Solving gives the two displayed numbers, with their roles exchanged when the starting refinement has Arf invariant one. Since translation runs through every refinement, the final counts are independent of the starting choice.
Exact mathematical passage
O
\mathcal OUse in source · 3 passages
Search aliases: O mathcal
Use in source · Read passage · dyadic_synthesis.tex, line 14
Let F= F_2((t)), fix an additive character ψ_F of conductor t O_F, and take increasing odd positive integers h_1<⋯<h_d. Put q_i=3h_i, N=2^d, n_j=2^d-jq_d-j+1. There is an explicit β∈ M_N(F) generating a separable totally ramified degree-N field that normalizes the complete lattice chain. It gives a maximal simple stratum with leading level n_1, final critical exponent -q_1, and successive approximation-field degrees 2,4,…,2^d, with critical levels n_2,…,n_d.
Exact mathematical passage
t\mathcal O_F
\mathcal O_E^\timesUse in source · 1 passage
Search aliases: O_E^× mathcal times
Use in source · Read passage · odd_synthesis.tex, line 65
Let P_0 extract the constant coefficient in the basis 1,z,…,z^p-1. The normalized corestriction s=κpr, where κ=-f'(z)/z^p-1, satisfies u_0=s(P_0)=a_0/z^p∈ O_E^×, Lift_E(C)=u_0^-1CP_0, s(Lift_E(C))=C. The lift preserves the exact filtration valuation. Its crucial additional identity is tr_L(Lift_E(C)) =Tr_E/L (κ^-1tr_E(C)). Define B(L,t_L;h_1,…,h_d) =α I_p^d-1+ Lift_E(B(E,z;h_1,…,h_d-1)), with the degree-p multiplication matrix as the base case. Induction proves trace zero at every native rank. T…
Exact mathematical passage
u_0=s(P_0)=a_0/z^p\in\mathcal O_E^\times, \qquad \operatorname{Lift}_E(C)=u_0^{-1}CP_0, \qquad s(\operatorname{Lift}_E(C))=C.
\mathcal O_FUse in source · 2 passages
Search aliases: O_F mathcal
Use in source · Read passage · dyadic_synthesis.tex, line 14
Let F= F_2((t)), fix an additive character ψ_F of conductor t O_F, and take increasing odd positive integers h_1<⋯<h_d. Put q_i=3h_i, N=2^d, n_j=2^d-jq_d-j+1. There is an explicit β∈ M_N(F) generating a separable totally ramified degree-N field that normalizes the complete lattice chain. It gives a maximal simple stratum with leading level n_1, final critical exponent -q_1, and successive approximation-field degrees 2,4,…,2^d, with critical levels n_2,…,n_d.
Exact mathematical passage
t\mathcal O_F
\OO_FUse in source · 3 passages
Search aliases: OO_F O_F OO mathcal
Use in source · Read passage · Paper_XI_Traditional_Final.tex, line 2082
[Restricted dyadic marked assembly realization] Let F= F_2((t)), fix an additive character of conductor t O_F, fix d≥1, and let h_1<⋯<h_d be positive odd integers. Put q_i=3h_i, N=2^d, and n_j=2^d-jq_d-j+1. The explicit recursive matrix generates a separable totally ramified degree-N maximal parameter field, normalizes the complete lattice chain, and has the literal defining sequence of degrees 2,4,…,2^d. It admits a full simple character trivial on U_F^1 and an actual supercuspidal completion w…
Exact mathematical passage
t\OO_F
\OO_KUse in source · 1 passage
Search aliases: OO_K O_K OO mathcal
Use in source · Read passage · Paper_XI_Traditional_Final.tex, line 516
The depth zero construction inflates a finite cuspidal representation to GL_n( O_K), extends it to the standard compact modulo center subgroup, and then compactly induces it to GL_n(K). This is classical type theory [source reference].
Exact mathematical passage
\GL_n(\OO_K)
\OO_K^\timesUse in source · 1 passage
Search aliases: OO_K^× O_K^× OO times mathcal times
Use in source · Read passage · dyadic_character.tex, line 360
Here O_K^×=U_K^1 because the residue field is F_2. Since U_K 1 is contained in J 1, we have
Exact mathematical passage
\OO_K^\times=U_K^1
\mathrm{order}Use in source · 2 passages
Search aliases: order mathrm
Use in source · Read passage · odd_capacity.tex, line 191
The right side is the Heisenberg commutator pairing for the minimal simple stratum of gamma. Its quotient J^1(γ)/H^1(γ) is precisely U^m(A)/H^m(γ), because J^1(γ)=U^1(B_order)U^m(A), with U^1(B_order)⊂ H^1(γ). BK's nondegeneracy theorem for simple-character commutators therefore says that the form on S is nondegenerate.
Exact mathematical passage
J^1(\gamma)=U^1(B_{\mathrm{order}})U^m(A)
\overline PUse in source · 2 passages
Search aliases: overline P overline
Use in source · Read passage · Paper_XI_Traditional_Final.tex, line 979
Let z=∑_a∈ Az_au_a be central. Commuting with u_x gives z_ab_c(x,a)=z_a for every x. Nondegeneracy forces z_a=0 for every nonzero a. Thus the center is C. To prove semisimplicity directly, choose a linear projection P onto a submodule of a finite dimensional C^c[A]-module and form overline P=frac1abs A∑_a∈ Au_a P u_a^-1. Every summand maps into the submodule and is the identity there. Conjugation by any u_x permutes the summands: the cocycle scalars cancel between each operator and its inverse. …
Exact mathematical passage
\overline P=\frac{1}{\abs A}\sum_{a\in A}u_a P u_a^{-1}.
PUse in source · 79 passages
Search aliases: P
Use in source · Read passage · Paper_XI_Traditional_Final.tex, line 234
Let W_K be the Weil group of K, let I_K be inertia, let P_K be wild inertia, and choose a geometric Frobenius lift Φ. A Weil Deligne representation is a pair (r,N) with r(w)Nr(w)^-1=absw_K N.
Exact mathematical passage
P_K
\mathcal PUse in source · 2 passages
Search aliases: P mathcal
Use in source · Read passage · odd_synthesis.tex, line 144
Choose a labelled symplectic embedding V_src→ S. For each original full character form P_i=( C^×× J_i^1)/ {(θ_i(h)^-1,h):h∈ H_i^1}. The balanced product and the corresponding host restriction have the same alternating commutator. The center-fixing extension isomorphism, obtained with matched Darboux lifts and complex roots, therefore preserves every supplied value θ_i(h), including the shallow values.
Exact mathematical passage
\mathcal P_i=(\mathbb C^\times\times J_i^1)/ \{(\theta_i(h)^{-1},h):h\in H_i^1\}.
p^{(N-1)/2}Use in source · 2 passages
Search aliases: p^(N-1)/2
Use in source · Read passage · odd_synthesis.tex, line 39
The same candidate has rank(π) · =N, · t(π) · =1, Sw(π) · =p^d-1(p+1)h_d, · dim_ F_p(J^1/H^1) · =N-1, dimη · =p^(N-1)/2, a(π^vee×π) · =p^2d-1+∑_i=1^d(p^2i-p^2i-2)h_i. These numerical identities do not identify a Weil representation, an endo-class, or the pair epsilon factors required by H5.
Exact mathematical passage
\begin{aligned} \operatorname{rank}(\pi)&=N, &t(\pi)&=1,\\ \operatorname{Sw}(\pi)&=p^{d-1}(p+1)h_d, & \dim_{\mathbb F_p}(J^1/H^1)&=N-1,\\ \dim\eta&=p^{(N-1)/2},\\ a(\pi^\vee\times\pi) &=p^{2d}-1+\sum_{i=1}^d(p^{2i}-p^{2i-2})h_i. \end{aligned}
p^2Use in source · 3 passages
Search aliases: p^2
Use in source · Read passage · Paper_XI_Traditional_Final.tex, line 2027
[Restricted odd-prime arithmetic realization] Let p be an odd prime, F= F_p((t)), fix d≥1, and let 0<h_1<⋯<h_d, pnmid h_i, q_i=(p+1)h_i, N=p^d. There is an explicit maximal simple stratum in M_N(F) with totally ramified full-degree parameter field, a literal defining sequence of degrees p,p^2,…,p^d, and levels n_j=p^d-jq_d-j+1. Every native parameter and consecutive correction has trace zero over its recursive base field. The stratum admits a full simple character trivial on U_F^1. Its fixed can…
Exact mathematical passage
p,p^2,\ldots,p^d
P^aUse in source · 1 passage
Search aliases: P^a
Use in source · Read passage · odd_capacity.tex, line 73
Put P=rad(A), U^a(A)=1+P^a for atextgreater=1, and m=N/2.
Exact mathematical passage
U^a(A)=1+P^a
p^{d-2}Use in source · 1 passage
Search aliases: p^d-2
Use in source · Read passage · odd_synthesis.tex, line 85
For the inductive field step d≥2, put n=p^d-1q_d and R=p^d-2q_d-1. The derived matrix is the recursively constructed maximal-field parameter, with k_0=-q_1. The inequalities n>pR>2R imply -2R>-n-q_1. Bushnell--Kutzko Remark 2.2.7 therefore applies to the actual displayed matrix. It supplies its maximal field, normalization, and critical exponent. Valuation preservation and literal truncation of the recursion supply the whole defining sequence. The full details, quotient count, and conductor depe…
Exact mathematical passage
R=p^{d-2}q_{d-1}
p^{d-j}Use in source · 3 passages
Search aliases: p^d-j
Use in source · Read passage · Paper_XI_Traditional_Final.tex, line 2028
[Restricted odd-prime arithmetic realization] Let p be an odd prime, F= F_p((t)), fix d≥1, and let 0<h_1<⋯<h_d, pnmid h_i, q_i=(p+1)h_i, N=p^d. There is an explicit maximal simple stratum in M_N(F) with totally ramified full-degree parameter field, a literal defining sequence of degrees p,p^2,…,p^d, and levels n_j=p^d-jq_d-j+1. Every native parameter and consecutive correction has trace zero over its recursive base field. The stratum admits a full simple character trivial on U_F^1. Its fixed can…
Exact mathematical passage
n_j=p^{d-j}q_{d-j+1}
p^jUse in source · 1 passage
Search aliases: p^j
Use in source · Read passage · odd_synthesis.tex, line 96
The full-character argument uses a specific order of operations. First construct a full character at native rank p^j, trivial on U_F^1. Transfer its full beta-determinant unit character to multiplicity p, using BK 3.6.13(ii) and 3.6.14. On a scalar u∈ U_F^1, this gives θ_j(u^p)=1. Next add the actual trace-zero correction, using BK 3.3.18. It remains trivial on the scalar intersection of its domain. Finally extend at the next maximal native parameter field: the forced deep character and the triv…
Exact mathematical passage
p^j
P^rUse in source · 3 passages
Search aliases: P^r
Use in source · Read passage · dyadic_tower.tex, line 120
B-bimodularity gives s(Lift_E(C))=C. The lift preserves exact valuation: u_0 (-1) is an E-unit, and expansion in 1,z gives coefficient valuations distinct modulo 2. Each coefficient entry has the same grade in the interleaved L-chain and the E-chain. In particular s(P^r)=Q^r.
Exact mathematical passage
s(P^r)=Q^r
p^rUse in source · 8 passages
Search aliases: p^r
Use in source · Read passage · Paper_XI_Traditional_Final.tex, line 825
[Finite Schrödinger representation] The map ρ: H(V)→GL( H) is a representation. It is irreducible, its center acts by ψ, and dim H=p^r=sqrtabs V.
Exact mathematical passage
\dim\calH=p^r=\sqrt{\abs V}.
P_0Use in source · 8 passages
Search aliases: P_0
Use in source · Read passage · dyadic_synthesis.tex, line 52
ursive coefficient field L of characteristic two, with residue field F_2 and uniformizer t_L, put E=L(α), α^2-α=t_L^-q, a=(q+1)/2, z=t_L^aα. The extension is separable and totally ramified of degree two; z is a uniformizer and z^2+t_L^az+t_L=0, Tr_E/L(α)=1. For constant-coefficient projection P_0 in the basis 1,z, the normalized trace corestriction and exact filtered right inverse are κ=α^-1, s=κpr, s(P_0)=t_L/z^2, Lift_E(C)=(z^2/t_L)CP_0. They satisfy s(Lift_E(C))=C, preserve the exact filtrati…
Exact mathematical passage
P_0
P_A^jUse in source · 1 passage
Search aliases: P_A^j
Use in source · Read passage · rank4.tex, line 58
Use U_A^j=1+P_A^j and U_B^j=1+Q^j. BK3.1.7--9 give the actual groups
Exact mathematical passage
U_A^j=1+P_A^j
P_A^{n-r}Use in source · 1 passage
Search aliases: P_A^n-r
Use in source · Read passage · dyadic_synthesis.tex, line 342
At both low ranks, K^× normalizes the new P. In rank two this follows directly from normalization of A and its radical. In rank four, a K-uniformizer is approximated by the derived chain shift R times a unit of B, with relative error in P_A^n-r. Its conjugation error on U_B^m lies in U_A^n-r+m⊂ J, which proves normalization. The exact argument is pathlowrank/RANK4_INDUCTION_BY_STAGES167.md.
Exact mathematical passage
P_A^{n-r}
P_gUse in source · 5 passages
Search aliases: P_g
Use in source · Read passage · Paper_XI_Traditional_Final.tex, line 1363
[Characteristic polynomial certificate] Let P_g(X)=det(XI-gmid V). Then Φ_p(g,V) = (frac-1p)^r (fracP_g(1)p).
Exact mathematical passage
P_g(X)=\det(XI-g\mid V).
\mathcal P_iUse in source · 2 passages
Search aliases: P_i mathcal
Use in source · Read passage · odd_synthesis.tex, line 144
Choose a labelled symplectic embedding V_src→ S. For each original full character form P_i=( C^×× J_i^1)/ {(θ_i(h)^-1,h):h∈ H_i^1}. The balanced product and the corresponding host restriction have the same alternating commutator. The center-fixing extension isomorphism, obtained with matched Darboux lifts and complex roots, therefore preserves every supplied value θ_i(h), including the shallow values.
Exact mathematical passage
\mathcal P_i=(\mathbb C^\times\times J_i^1)/ \{(\theta_i(h)^{-1},h):h\in H_i^1\}.
P_jUse in source · 13 passages
Search aliases: P_j
Use in source · Read passage · Paper_XI_Traditional_Final.tex, line 262
[Exact block reconstruction] Let P_j be the projector onto V_j. Then N=∑_j=1^m-1P_j-1NP_j, N^m=0. Every power N^r is the sum of the packet paths of length r along the chain.
Exact mathematical passage
P_j
P_{j-1}Use in source · 2 passages
Search aliases: P_j-1
Use in source · Read passage · Paper_XI_Traditional_Final.tex, line 263
[Exact block reconstruction] Let P_j be the projector onto V_j. Then N=∑_j=1^m-1P_j-1NP_j, N^m=0. Every power N^r is the sum of the packet paths of length r along the chain.
Exact mathematical passage
N=\sum_{j=1}^{m-1}P_{j-1}NP_j, \qquad N^m=0.
P_KUse in source · 2 passages
Search aliases: P_K
Use in source · Read passage · Paper_XI_Traditional_Final.tex, line 234
Let W_K be the Weil group of K, let I_K be inertia, let P_K be wild inertia, and choose a geometric Frobenius lift Φ. A Weil Deligne representation is a pair (r,N) with r(w)Nr(w)^-1=absw_K N.
Exact mathematical passage
P_K
P_kUse in source · 1 passage
Search aliases: P_k
Use in source · Paper_XI_Visual_Guide_Final.tex, line 92
lessonPART I / FINITE TAME COORDINATESFourier projectors isolate tame inertiaOriginal Section 3; geometric Frobenius convention Let t generate a finite tame quotient of order M, with (q,M)=1. For j∈ Z/M Z define P_j=frac1M∑_a=0^M-1ζ_M^-jar(t)^a, ζ_M=e^2π i/M. The finite Fourier identities give P_jP_k=δ_jkP_j and ∑_jP_j=I. Thus P_jV is the ζ_M^j character space for r(t).
Exact mathematical passage
P_jP_k=\delta_{jk}P_j
\mathfrak p_K^mUse in source · 1 passage
Search aliases: p_K^m mathfrak
Use in source · Read passage · rank2.tex, line 132
For if ku∈U_A m with k∈U_K 1 and u∈U_A (m+1), then k∈ K^×∩ U_A^m=U_K^m. The reverse inclusion is immediate. The complete period-two A-grade R^m/R^m+1 has F2-dimension2, and its field grade p_K^m/ p_K^m+1 has dimension1. Thus
Exact mathematical passage
\mathfrak p_K^m/\mathfrak p_K^{m+1}
\mathfrak p_K^{m+1}Use in source · 1 passage
Search aliases: p_K^m+1 mathfrak
Use in source · Read passage · rank2.tex, line 132
For if ku∈U_A m with k∈U_K 1 and u∈U_A (m+1), then k∈ K^×∩ U_A^m=U_K^m. The reverse inclusion is immediate. The complete period-two A-grade R^m/R^m+1 has F2-dimension2, and its field grade p_K^m/ p_K^m+1 has dimension1. Thus
Exact mathematical passage
\mathfrak p_K^m/\mathfrak p_K^{m+1}
P_LUse in source · 5 passages
Search aliases: P_L
Use in source · Read passage · Paper_XI_Traditional_Final.tex, line 874
[Isotropic averaging projector] The operator P_L = frac1abs L ∑_ℓ∈widetilde L ρ(ℓ) is an orthogonal projector. In the Schrödinger model with L=X, it has rank one and selects the constant functions.
Exact mathematical passage
P_L = \frac1{\abs L} \sum_{\ell\in\widetilde L} \rho(\ell)
P_{qj}Use in source · 3 passages
Search aliases: P_qj
Use in source · Read passage · Paper_XI_Traditional_Final.tex, line 328
[Tame character projectors] The operators P_j are pairwise orthogonal idempotents whose sum is the identity. Frobenius permutes them by Φ P_jΦ^-1=P_qj.
Exact mathematical passage
\Phi P_j\Phi^{-1}=P_{qj}.
p_uUse in source · 9 passages
Search aliases: p_u
Use in source · Read passage · Paper_XI_Traditional_Final.tex, line 1697
[Exact edge residual from cycle defect] Fix a spanning tree and construct the vertex maps by tree transport from U_r. Let e:u→ v be a non tree edge. Let p_u and p_v be the tree paths from the root to u and v. Define the fundamental cycle defect D_e = U_rA_p_v^-1A_eA_p_u - B_p_v^-1B_eB_p_uU_r. Then the edge residual is exactly U_vA_e-B_eU_u = B_p_vD_eA_p_u^-1.
Exact mathematical passage
p_u
p_vUse in source · 11 passages
Search aliases: p_v
Use in source · Read passage · Paper_XI_Traditional_Final.tex, line 1697
[Exact edge residual from cycle defect] Fix a spanning tree and construct the vertex maps by tree transport from U_r. Let e:u→ v be a non tree edge. Let p_u and p_v be the tree paths from the root to u and v. Define the fundamental cycle defect D_e = U_rA_p_v^-1A_eA_p_u - B_p_v^-1B_eB_p_uU_r. Then the edge residual is exactly U_vA_e-B_eU_u = B_p_vD_eA_p_u^-1.
Exact mathematical passage
p_v
\pmodUse in source · 5 passages
Search aliases: pmod
Use in source · Read passage · Paper_XI_Traditional_Final.tex, line 612
Represent the residue characters by exponents apmodq^n-1, bpmodq^m-1.
Exact mathematical passage
a\pmod{q^n-1}, \qquad b\pmod{q^m-1}.
\operatorname{pr}Use in source · 2 passages
Search aliases: pr operatorname
Use in source · Read passage · dyadic_synthesis.tex, line 53
er t_L, put E=L(α), α^2-α=t_L^-q, a=(q+1)/2, z=t_L^aα. The extension is separable and totally ramified of degree two; z is a uniformizer and z^2+t_L^az+t_L=0, Tr_E/L(α)=1. For constant-coefficient projection P_0 in the basis 1,z, the normalized trace corestriction and exact filtered right inverse are κ=α^-1, s=κpr, s(P_0)=t_L/z^2, Lift_E(C)=(z^2/t_L)CP_0. They satisfy s(Lift_E(C))=C, preserve the exact filtration valuation, and give the identity tr_LLift_E(C) =Tr_E/L (αtr_E C). The actual recurs…
Exact mathematical passage
\kappa=\alpha^{-1},\qquad s=\kappa\operatorname{pr},\qquad s(P_0)=t_L/z^2,\qquad \operatorname{Lift}_E(C)=(z^2/t_L)CP_0.
QUse in source · 6 passages
Search aliases: Q
Use in source · Read passage · Paper_XI_Traditional_Final.tex, line 2158
r an independently fixed native quadratic marking, a preassigned embedding must satisfy q_W(f(v))=q_rm src(v). At equal capacity, a free embedding requires equal Arf invariant. Spare capacity for d≥4 admits either source Arf type when the embedding is free. Retaining only the central {±1} would erase the distinction between whole marked finite extensions such as D_8 and Q_8.
Exact mathematical passage
Q_8
q^{-j}Use in source · 2 passages
Search aliases: q^-j
Use in source · Read passage · Paper_XI_Traditional_Final.tex, line 287
The only nonzero entry in column j and row j-1 is multiplied by q^j-1q^-j=q^-1 under conjugation.
Exact mathematical passage
q^{j-1}q^{-j}=q^{-1}
q^{1/2}Use in source · 1 passage
Search aliases: q^1/2
Use in source · Read passage · Paper_XI_Traditional_Final.tex, line 725
[Gauss magnitude] If χ is nontrivial, then absG(χ,ψ)=q^1/2.
Exact mathematical passage
\boxed{ \abs{G(\chi,\psi)}=q^{1/2}. }
q^2Use in source · 1 passage
Search aliases: q^2
Use in source · Read passage · Paper_XI_Traditional_Final.tex, line 277
[Scalar covariance model] Let F=diag(1,q,q^2,…,q^m-1) and let N_j-1,j=1. Then FNF^-1=q^-1N.
Exact mathematical passage
F=\operatorname{diag}(1,q,q^2,\ldots,q^{m-1})
q^iUse in source · 4 passages
Search aliases: q^i
Use in source · Read passage · Paper_XI_Traditional_Final.tex, line 620
[Refined exponent formula] The ith interaction packet on F_q^L^× has exponent c_i = afracq^L-1q^n-1 +bq^ifracq^L-1q^m-1 pmodq^L-1. It is trivial on tame inertia exactly when c_i=0.
Exact mathematical passage
\boxed{ c_i = a\frac{q^L-1}{q^n-1} +bq^i\frac{q^L-1}{q^m-1} \pmod{q^L-1}. }
Q^jUse in source · 1 passage
Search aliases: Q^j
Use in source · Read passage · rank4.tex, line 58
Use U_A^j=1+P_A^j and U_B^j=1+Q^j. BK3.1.7--9 give the actual groups
Exact mathematical passage
U_B^j=1+Q^j
q^{j-1}Use in source · 2 passages
Search aliases: q^j-1
Use in source · Read passage · Paper_XI_Traditional_Final.tex, line 287
The only nonzero entry in column j and row j-1 is multiplied by q^j-1q^-j=q^-1 under conjugation.
Exact mathematical passage
q^{j-1}q^{-j}=q^{-1}
q^LUse in source · 6 passages
Search aliases: q^L
Use in source · Read passage · Paper_XI_Traditional_Final.tex, line 619
[Refined exponent formula] The ith interaction packet on F_q^L^× has exponent c_i = afracq^L-1q^n-1 +bq^ifracq^L-1q^m-1 pmodq^L-1. It is trivial on tame inertia exactly when c_i=0.
Exact mathematical passage
\FF_{q^L}^{\times}
q^mUse in source · 4 passages
Search aliases: q^m
Use in source · Read passage · Paper_XI_Traditional_Final.tex, line 612
Represent the residue characters by exponents apmodq^n-1, bpmodq^m-1.
Exact mathematical passage
a\pmod{q^n-1}, \qquad b\pmod{q^m-1}.
q^{m-1}Use in source · 2 passages
Search aliases: q^m-1
Use in source · Read passage · Paper_XI_Traditional_Final.tex, line 277
[Scalar covariance model] Let F=diag(1,q,q^2,…,q^m-1) and let N_j-1,j=1. Then FNF^-1=q^-1N.
Exact mathematical passage
F=\operatorname{diag}(1,q,q^2,\ldots,q^{m-1})
q^nUse in source · 12 passages
Search aliases: q^n
Use in source · Read passage · Paper_XI_Traditional_Final.tex, line 410
Characters of F_q^n^× are indexed by exponents modulo q^n-1, and Frobenius acts by multiplication by q.
Exact mathematical passage
\FF_{q^n}^{\times}
q^{n/d}Use in source · 2 passages
Search aliases: q^n/d
Use in source · Read passage · Paper_XI_Traditional_Final.tex, line 414
[Möbius orbit count] The number of regular Frobenius orbits of length n is c_n(q) = frac1n ∑_dmid n μ(d) (q^n/d-1).
Exact mathematical passage
\boxed{ c_n(q) = \frac1n \sum_{d\mid n} \mu(d) \bigl(q^{n/d}-1\bigr). }
Q^rUse in source · 3 passages
Search aliases: Q^r
Use in source · Read passage · dyadic_tower.tex, line 120
B-bimodularity gives s(Lift_E(C))=C. The lift preserves exact valuation: u_0 (-1) is an E-unit, and expansion in 1,z gives coefficient valuations distinct modulo 2. Each coefficient entry has the same grade in the interleaved L-chain and the E-chain. In particular s(P^r)=Q^r.
Exact mathematical passage
s(P^r)=Q^r
q^rUse in source · 3 passages
Search aliases: q^r
Use in source · Read passage · Paper_XI_Traditional_Final.tex, line 427
For every divisor rmid n, the characters fixed by Frob^r are exactly those descending to F_q^r^×, and there are q^r-1 of them. Möbius inversion gives the number of characters of exact orbit length n. Division by n gives the number of orbits.
Exact mathematical passage
\FF_{q^r}^{\times}
q_0Use in source · 4 passages
Search aliases: q_0
Use in source · Read passage · Paper_XI_Traditional_Final.tex, line 1970
[Dyadic phase ambiguity] On a symplectic plane over F_2, choose refinements q_0 and q_1 with Arf invariants zero and one. Their polar forms are the same, while ∑_x(-1)^q_0(x)=2, ∑_x(-1)^q_1(x)=-2. Thus the alternating packet alone cannot select the dyadic sign.
Exact mathematical passage
q_0
q_1Use in source · 11 passages
Search aliases: q_1
Use in source · Read passage · Paper_XI_Traditional_Final.tex, line 1524
[Orthogonal dyadic multiplicativity] For an orthogonal sum (V,q)=(V_1,q_1)mathop⊥limits(V_2,q_2), we have Arf(q)=Arf(q_1)+Arf(q_2) and the normalized Gauss sign is the product of the two normalized signs.
Exact mathematical passage
(V,q)=(V_1,q_1)\perpbig(V_2,q_2),
q_2Use in source · 2 passages
Search aliases: q_2
Use in source · Read passage · Paper_XI_Traditional_Final.tex, line 1524
[Orthogonal dyadic multiplicativity] For an orthogonal sum (V,q)=(V_1,q_1)mathop⊥limits(V_2,q_2), we have Arf(q)=Arf(q_1)+Arf(q_2) and the normalized Gauss sign is the product of the two normalized signs.
Exact mathematical passage
(V,q)=(V_1,q_1)\perpbig(V_2,q_2),
Q_8Use in source · 2 passages
Search aliases: Q_8
Use in source · Read passage · Paper_XI_Traditional_Final.tex, line 2158
r an independently fixed native quadratic marking, a preassigned embedding must satisfy q_W(f(v))=q_rm src(v). At equal capacity, a free embedding requires equal Arf invariant. Spare capacity for d≥4 admits either source Arf type when the embedding is free. Retaining only the central {±1} would erase the distinction between whole marked finite extensions such as D_8 and Q_8.
Exact mathematical passage
Q_8
q_dUse in source · 6 passages
Search aliases: q_d
Use in source · Read passage · dyadic_synthesis.tex, line 75
At an inductive field step d≥2, set n=2^d-1q_d, R=2^d-2q_d-1. Then n>2R and -2R>-n-q_1. BK Remark 2.2.7 gives the maximal field generated by the actual displayed matrix. Literal truncation and the filtered lift give the defining sequence. Nonzero native trace also proves separability: the irreducible characteristic polynomial has a nonzero coefficient of X^N-1, so its derivative is nonzero. See pathtower/DYADIC_AS_TOWER167.md for all field and filtration details.
Exact mathematical passage
n=2^{d-1}q_d
q_{d-1}Use in source · 2 passages
Search aliases: q_d-1
Use in source · Read passage · dyadic_synthesis.tex, line 76
At an inductive field step d≥2, set n=2^d-1q_d, R=2^d-2q_d-1. Then n>2R and -2R>-n-q_1. BK Remark 2.2.7 gives the maximal field generated by the actual displayed matrix. Literal truncation and the filtered lift give the defining sequence. Nonzero native trace also proves separability: the irreducible characteristic polynomial has a nonzero coefficient of X^N-1, so its derivative is nonzero. See pathtower/DYADIC_AS_TOWER167.md for all field and filtration details.
Exact mathematical passage
R=2^{d-2}q_{d-1}
q_{d-j+1}Use in source · 5 passages
Search aliases: q_d-j+1
Use in source · Read passage · Paper_XI_Traditional_Final.tex, line 2028
[Restricted odd-prime arithmetic realization] Let p be an odd prime, F= F_p((t)), fix d≥1, and let 0<h_1<⋯<h_d, pnmid h_i, q_i=(p+1)h_i, N=p^d. There is an explicit maximal simple stratum in M_N(F) with totally ramified full-degree parameter field, a literal defining sequence of degrees p,p^2,…,p^d, and levels n_j=p^d-jq_d-j+1. Every native parameter and consecutive correction has trace zero over its recursive base field. The stratum admits a full simple character trivial on U_F^1. Its fixed can…
Exact mathematical passage
n_j=p^{d-j}q_{d-j+1}
q_iUse in source · 5 passages
Search aliases: q_i
Use in source · Read passage · Paper_XI_Traditional_Final.tex, line 2021
[Restricted odd-prime arithmetic realization] Let p be an odd prime, F= F_p((t)), fix d≥1, and let 0<h_1<⋯<h_d, pnmid h_i, q_i=(p+1)h_i, N=p^d. There is an explicit maximal simple stratum in M_N(F) with totally ramified full-degree parameter field, a literal defining sequence of degrees p,p^2,…,p^d, and levels n_j=p^d-jq_d-j+1. Every native parameter and consecutive correction has trace zero over its recursive base field. The stratum admits a full simple character trivial on U_F^1. Its fixed can…
Exact mathematical passage
0<h_1<\cdots<h_d,\qquad p\nmid h_i, \qquad q_i=(p+1)h_i,\qquad N=p^d.
q_{\rm src}Use in source · 3 passages
Search aliases: q_rm src rm
Use in source · Read passage · Paper_XI_Traditional_Final.tex, line 2154
The original low-rank maximal-type capacity obstruction remains true. The new pair is a common arithmetic assembly carrier; it is not renamed as the original maximal-simple-character pair. For an independently fixed native quadratic marking, a preassigned embedding must satisfy q_W(f(v))=q_rm src(v). At equal capacity, a free embedding requires equal Arf invariant. Spare capacity for d≥4 admits either source Arf type when the embedding is free. Retaining only the central {±1} would erase the dis…
Exact mathematical passage
q_W(f(v))=q_{\rm src}(v)
q_vUse in source · 6 passages
Search aliases: q_v
Use in source · Read passage · Paper_XI_Traditional_Final.tex, line 1549
[Translation of quadratic refinements] For v∈ V, put q_v(x)=q(x)+b(v,x). Then Arf(q_v)=Arf(q)+q(v).
Exact mathematical passage
q_v(x)=q(x)+b(v,x).
q_WUse in source · 4 passages
Search aliases: q_W
Use in source · Read passage · Paper_XI_Traditional_Final.tex, line 2154
The original low-rank maximal-type capacity obstruction remains true. The new pair is a common arithmetic assembly carrier; it is not renamed as the original maximal-simple-character pair. For an independently fixed native quadratic marking, a preassigned embedding must satisfy q_W(f(v))=q_rm src(v). At equal capacity, a free embedding requires equal Arf invariant. Spare capacity for d≥4 admits either source Arf type when the embedding is free. Retaining only the central {±1} would erase the dis…
Exact mathematical passage
q_W(f(v))=q_{\rm src}(v)
\mathcal RUse in source · 3 passages
Search aliases: R mathcal
Use in source · Read passage · Paper_XI_Traditional_Final.tex, line 1857
An arithmetic realization witness R(d_ρ,π) consists of actual arithmetic inducing data for an admissible supercuspidal π, together with explicit comparison and recovery maps carrying the prescribed assembled finite packet. The maps preserve its declared stabilizer action, all character values explicitly included in the supplied datum, central-extension and phase data, and compatible transports. Recovery returns that same π, with its fixed central character and noncompact completion. An enlarged …
Exact mathematical passage
\mathcal R(d_\rho,\pi)
R^mUse in source · 1 passage
Search aliases: R^m
Use in source · Read passage · rank2.tex, line 131
For if ku∈U_A m with k∈U_K 1 and u∈U_A (m+1), then k∈ K^×∩ U_A^m=U_K^m. The reverse inclusion is immediate. The complete period-two A-grade R^m/R^m+1 has F2-dimension2, and its field grade p_K^m/ p_K^m+1 has dimension1. Thus
Exact mathematical passage
R^m/R^{m+1}
R^{m+1}Use in source · 1 passage
Search aliases: R^m+1
Use in source · Read passage · rank2.tex, line 131
For if ku∈U_A m with k∈U_K 1 and u∈U_A (m+1), then k∈ K^×∩ U_A^m=U_K^m. The reverse inclusion is immediate. The complete period-two A-grade R^m/R^m+1 has F2-dimension2, and its field grade p_K^m/ p_K^m+1 has dimension1. Thus
Exact mathematical passage
R^m/R^{m+1}
r_iUse in source · 4 passages
Search aliases: r_i
Use in source · Read passage · Paper_XI_Traditional_Final.tex, line 658
[Tame tensor local factor] The local factor of the tame tensor product is the product of (1-a_i u^r_i)^-1 over precisely those refined packets whose inertia exponent vanishes.
Exact mathematical passage
(1-a_i u^{r_i})^{-1}
\operatorname{rank}Use in source · 1 passage
Search aliases: rank operatorname
Use in source · Read passage · odd_synthesis.tex, line 39
The same candidate has rank(π) · =N, · t(π) · =1, Sw(π) · =p^d-1(p+1)h_d, · dim_ F_p(J^1/H^1) · =N-1, dimη · =p^(N-1)/2, a(π^vee×π) · =p^2d-1+∑_i=1^d(p^2i-p^2i-2)h_i. These numerical identities do not identify a Weil representation, an endo-class, or the pair epsilon factors required by H5.
Exact mathematical passage
\begin{aligned} \operatorname{rank}(\pi)&=N, &t(\pi)&=1,\\ \operatorname{Sw}(\pi)&=p^{d-1}(p+1)h_d, & \dim_{\mathbb F_p}(J^1/H^1)&=N-1,\\ \dim\eta&=p^{(N-1)/2},\\ a(\pi^\vee\times\pi) &=p^{2d}-1+\sum_{i=1}^d(p^{2i}-p^{2i-2})h_i. \end{aligned}
\rfloorUse in source · 3 passages
Search aliases: rfloor
Use in source · Read passage · Paper_XI_Traditional_Final.tex, line 2117
ently each native correction δ satisfies v_F(trδ)+s≥1 (s=1), which makes its additive scalar correction trivial on U_F^1. The actual field and defining-sequence proof is [source reference]. Full beta-determinant transfer and character extension at each new maximal native field prove the full-character claim, including the odd-critical endpoint 2(lfloor r/2rfloor+1)-r=1; see [source reference]. The exact original quotient has dimension 2^d-2, which accommodates the required 2d source dimensions p…
Exact mathematical passage
2(\lfloor r/2\rfloor+1)-r=1
\rmUse in source · 5 passages
Search aliases: rm
Use in source · Read passage · Paper_XI_Traditional_Final.tex, line 2154
The original low-rank maximal-type capacity obstruction remains true. The new pair is a common arithmetic assembly carrier; it is not renamed as the original maximal-simple-character pair. For an independently fixed native quadratic marking, a preassigned embedding must satisfy q_W(f(v))=q_rm src(v). At equal capacity, a free embedding requires equal Arf invariant. Spare capacity for d≥4 admits either source Arf type when the embedding is free. Retaining only the central {±1} would erase the dis…
Exact mathematical passage
q_W(f(v))=q_{\rm src}(v)
\SpUse in source · 4 passages
Search aliases: Sp operatorname
Use in source · Read passage · Paper_XI_Traditional_Final.tex, line 1292
Let p be odd, let (V,b) be a finite symplectic F_p space of dimension 2r, and let g∈Sp(V) have no fixed vector. Fix one of the standard finite Weil normalizations used in the literature [source reference].
Exact mathematical passage
g\in\Sp(V)
\operatorname{Sp}Use in source · 2 passages
Search aliases: Sp operatorname
Use in source · Read passage · Paper_XI_Traditional_Final.tex, line 241
[Indecomposable monodromy blocks] Every Frobenius semisimple Weil Deligne representation is a direct sum of blocks Sp_m(τ), where τ is an irreducible Weil representation and the nilpotent operator is one Jordan chain of length m.
Exact mathematical passage
\operatorname{Sp}_m(\tau),
\operatorname{Sp}_mUse in source · 2 passages
Search aliases: Sp_m operatorname
Use in source · Read passage · Paper_XI_Traditional_Final.tex, line 241
[Indecomposable monodromy blocks] Every Frobenius semisimple Weil Deligne representation is a direct sum of blocks Sp_m(τ), where τ is an irreducible Weil representation and the nilpotent operator is one Jordan chain of length m.
Exact mathematical passage
\operatorname{Sp}_m(\tau),
\mathrm{src}Use in source · 4 passages
Search aliases: src mathrm
Use in source · Read passage · odd_synthesis.tex, line 123
Assume the one-vertex scalar-pushout contract: there are d labelled, actually supplied full character extensions with whole symplectic quotients V_i of dimension two over F_p. Their balanced C^×-central product has quotient V_src=⊕_iV_i and Heisenberg carrier dimension p^d=N. Initially take the declared stabilizer C={1}.
Exact mathematical passage
V_{\mathrm{src}}=\bigoplus_iV_i
\operatorname{St}Use in source · 1 passage
Search aliases: St operatorname
Use in source · Read passage · Paper_XI_Traditional_Final.tex, line 299
[Formal atomic reconstruction] Assume that every irreducible Weil representation τ has been assigned a supercuspidal representation σ_τ, compatibly with isomorphism and unramified twist. Then the rules Sp_m(τ) ⟼ St_m(σ_τ) and normalized parabolic induction determine a candidate correspondence on all Frobenius semisimple Weil Deligne parameters.
Exact mathematical passage
\operatorname{Sp}_m(\tau) \longmapsto \operatorname{St}_m(\sigma_\tau)
\operatorname{St}_mUse in source · 1 passage
Search aliases: St_m operatorname
Use in source · Read passage · Paper_XI_Traditional_Final.tex, line 299
[Formal atomic reconstruction] Assume that every irreducible Weil representation τ has been assigned a supercuspidal representation σ_τ, compatibly with isomorphism and unramified twist. Then the rules Sp_m(τ) ⟼ St_m(σ_τ) and normalized parabolic induction determine a candidate correspondence on all Frobenius semisimple Weil Deligne parameters.
Exact mathematical passage
\operatorname{Sp}_m(\tau) \longmapsto \operatorname{St}_m(\sigma_\tau)
\SwUse in source · 3 passages
Search aliases: Sw operatorname
Use in source · Paper_XI_Visual_Guide_Final.tex, line 198
An inertia-trivial companion cycle has a Weil-fixed vector exactly when a_i=1, and then its fixed space is one dimensional. Therefore dim(τ_ξ⊗τ_η)^W_K =#{i:c_i=0, a_i=1}. For an irreducible nontrivial tame atom of dimension n>1, wild inertia acts trivially and inertia has no fixed vectors. Hence Sw(τ)=0, Art(τ)=n. boundaryThese exact tame formulas do not establish the full wild pair-factor identities required by H5.
Exact mathematical passage
\Sw(\tau)=0,\qquad \operatorname{Art}(\tau)=n.
\operatorname{Sw}Use in source · 3 passages
Search aliases: Sw operatorname
Use in source · Read passage · Paper_XI_Traditional_Final.tex, line 1833
The dimension and orbit follow from [source reference]. The cycle polynomial is [source reference]. For rank greater than one, the conductor is [source reference]. In rank one, the Swan conductor is zero, and the Artin conductor is zero for trivial inertia and one otherwise, by the formula Art=dim V-dim V^I_K+Sw. The local polynomial is [source reference]. Tensor refinement is [source reference]. The Gauss sum is its defining finite sum; for a nontrivial multiplicative character its unit phase i…
Exact mathematical passage
\operatorname{Art}=\dim V-\dim V^{I_K}+\operatorname{Sw}
\mathrm{symp}Use in source · 2 passages
Search aliases: symp mathrm
Use in source · Read passage · Paper_XI_Traditional_Final.tex, line 987
[Radical free envelope] For an arbitrary scalar projective action of A, the quotient A^symp=A/rad b_c is the maximal quotient detected by noncommuting projective intertwiners. If A is an elementary abelian p group, then A^symp is a symplectic F_p space.
Exact mathematical passage
\boxed{ A^{\mathrm{symp}}=A/\rad b_c }
t^2Use in source · 1 passage
Search aliases: t^2
Use in source · Read passage · odd_synthesis.tex, line 220
The old fixed-field central argument genuinely failed. At p=3, (h_1,h_2)=(1,5), its naturally transferred full character takes the value ζ_3≠1 on 1+t^2, although the desired wild center is trivial. A shallow character choice cannot repair a forced value. The two-stage changing-field repair is retained in pathcentral/CENTRAL_REPAIR166.md; the trace-zero tower above resolves this central issue uniformly in every length.
Exact mathematical passage
1+t^2
T^iUse in source · 2 passages
Search aliases: T^i
Use in source · Read passage · dyadic_synthesis.tex, line 366
There is also a recovery from compact data explicitly recording that root. With A_0=Λ(ϖ_K) and X=c - Ind_J^Gη, define (Tf)(g)=A_0^-1f(ϖ_Kg). Then X/(T-1)X≃π. After imposing the fixed central character, N/(F^× J) is cyclic of order N, and the finite average N^-1∑_i=0^N-1T^i selects this same extension. These operators act on complex representation spaces, so their scalar division by N is valid. This is a different recovery from the Block159 projector onto a sector of η. The exact kernel and finit…
Exact mathematical passage
N^{-1}\sum_{i=0}^{N-1}T^i
T^mUse in source · 2 passages
Search aliases: T^m
Use in source · Read passage · Paper_XI_Traditional_Final.tex, line 1259
Let C=⟨ c⟩ have order m, and let T be a lift of the projective generator. Since c^m=1, the operator T^m is scalar. Choose λ∈ C^× with λ^mT^m=I. Replacing T by λ T gives an honest cyclic representation.
Exact mathematical passage
T^m
T^NUse in source · 1 passage
Search aliases: T^N
Use in source · Paper_XI_Visual_Guide_Final.tex, line 579
After imposing the already fixed central character, the relevant quotient is cyclic of order N. Then T^N=1 and the finite average e_Λ=frac1N∑_i=0^N-1T^i selects the same extension. The operator acts on a complex representation space, where division by N is valid. boundaryThis is a different recovery mechanism from a finite sector inside the original η. The root is retained, not reconstructed from the compact packet alone.
Exact mathematical passage
T^N=1
t^qUse in source · 1 passage
Search aliases: t^q
Use in source · Read passage · Paper_XI_Traditional_Final.tex, line 334
The first assertion is the Fourier idempotent calculation for a cyclic group. For the geometric Frobenius convention of [source reference], the relation is Φ^-1tΦ=t^q. Equivalently, Φ tΦ^-1=t^q^-1, with inverse taken modulo M since gcd(q,M)=1. Conjugating the Fourier sum and changing its summation index therefore gives Φ P_jΦ^-1=P_qj. The inverse Frobenius convention reverses the label direction while preserving the same orbits.
Exact mathematical passage
\Phi^{-1}t\Phi=t^q
t^{q^{-1}}Use in source · 1 passage
Search aliases: t^q^-1
Use in source · Read passage · Paper_XI_Traditional_Final.tex, line 334
The first assertion is the Fourier idempotent calculation for a cyclic group. For the geometric Frobenius convention of [source reference], the relation is Φ^-1tΦ=t^q. Equivalently, Φ tΦ^-1=t^q^-1, with inverse taken modulo M since gcd(q,M)=1. Conjugating the Fourier sum and changing its summation index therefore gives Φ P_jΦ^-1=P_qj. The inverse Frobenius convention reverses the label direction while preserving the same orbits.
Exact mathematical passage
\Phi t\Phi^{-1}=t^{q^{-1}}
t_LUse in source · 8 passages
Search aliases: t_L
Use in source · Read passage · dyadic_synthesis.tex, line 46
At a recursive coefficient field L of characteristic two, with residue field F_2 and uniformizer t_L, put E=L(α), α^2-α=t_L^-q, a=(q+1)/2, z=t_L^aα. The extension is separable and totally ramified of degree two; z is a uniformizer and z^2+t_L^az+t_L=0, Tr_E/L(α)=1. For constant-coefficient projection P_0 in the basis 1,z, the normalized trace corestriction and exact filtered right inverse are κ=α^-1, s=κpr, s(P_0)=t_L/z^2, Lift_E(C)=(z^2/t_L)CP_0. They satisfy s(Lift_E(C))=C, preserve the exact …
Exact mathematical passage
t_L
t_L^{-q}Use in source · 3 passages
Search aliases: t_L^-q
Use in source · Read passage · dyadic_synthesis.tex, line 46
At a recursive coefficient field L of characteristic two, with residue field F_2 and uniformizer t_L, put E=L(α), α^2-α=t_L^-q, a=(q+1)/2, z=t_L^aα. The extension is separable and totally ramified of degree two; z is a uniformizer and z^2+t_L^az+t_L=0, Tr_E/L(α)=1. For constant-coefficient projection P_0 in the basis 1,z, the normalized trace corestriction and exact filtered right inverse are κ=α^-1, s=κpr, s(P_0)=t_L/z^2, Lift_E(C)=(z^2/t_L)CP_0. They satisfy s(Lift_E(C))=C, preserve the exact …
Exact mathematical passage
E=L(\alpha),\qquad \alpha^2-\alpha=t_L^{-q},\qquad a=(q+1)/2,\qquad z=t_L^a\alpha.
t_L^{-q_d}Use in source · 1 passage
Search aliases: t_L^-q_d
Use in source · Read passage · odd_synthesis.tex, line 54
At a recursive base (L,t_L), introduce E=L(α), α^p-α=t_L^-q_d. This is a separable totally ramified extension of degree p, with v_E(α)=-q_d and Tr_E/Lα=0. Choose 1≤ b<p with q_db≡-1pmod p, set a=(q_db+1)/p, and put z=t_L^aα^b. Then z is an explicit uniformizer. Write its Eisenstein polynomial as f(X)=X^p+⋯+a_0.
Exact mathematical passage
E=L(\alpha),\qquad \alpha^p-\alpha=t_L^{-q_d}.
t_L^aUse in source · 6 passages
Search aliases: t_L^a
Use in source · Read passage · dyadic_synthesis.tex, line 46
At a recursive coefficient field L of characteristic two, with residue field F_2 and uniformizer t_L, put E=L(α), α^2-α=t_L^-q, a=(q+1)/2, z=t_L^aα. The extension is separable and totally ramified of degree two; z is a uniformizer and z^2+t_L^az+t_L=0, Tr_E/L(α)=1. For constant-coefficient projection P_0 in the basis 1,z, the normalized trace corestriction and exact filtered right inverse are κ=α^-1, s=κpr, s(P_0)=t_L/z^2, Lift_E(C)=(z^2/t_L)CP_0. They satisfy s(Lift_E(C))=C, preserve the exact …
Exact mathematical passage
E=L(\alpha),\qquad \alpha^2-\alpha=t_L^{-q},\qquad a=(q+1)/2,\qquad z=t_L^a\alpha.
T_nUse in source · 4 passages
Search aliases: T_n
Use in source · Read passage · Paper_XI_Traditional_Final.tex, line 455
Let G_n=GL_n( F_q), T_n≅ F_q^n^× with T_n an elliptic maximal torus.
Exact mathematical passage
G_n=\GL_n(\FF_q), \qquad T_n\cong\FF_{q^n}^{\times}
\operatorname{Teich}Use in source · 3 passages
Search aliases: Teich operatorname
Use in source · Read passage · odd_synthesis.tex, line 194
Indeed, for residue cardinality q, let B=ω(Teich(k^×)). The actual inertia hypothesis gives χ(C)^N⊂ B. Since N is a power of p and χ(C) has order prime to p, this implies χ(C)⊂ B. Choose k_0 inverse to N modulo q-1, and set δ(u)=ω(Teich(u))^k_0, widetilde C={(c,u):χ(c)=δ(u)}. The matrix Teich(u^k_0)I_N acts by χ(c). The kernel acts trivially on the entire sector induction and ambient compact induction, because these matrices are central. This preserves the fixed Λ, its canonical beta-extension f…
Exact mathematical passage
B=\omega(\operatorname{Teich}(k^\times))
\operatorname{Tr}Use in source · 4 passages
Search aliases: Tr operatorname
Use in source · Read passage · dyadic_synthesis.tex, line 50
At a recursive coefficient field L of characteristic two, with residue field F_2 and uniformizer t_L, put E=L(α), α^2-α=t_L^-q, a=(q+1)/2, z=t_L^aα. The extension is separable and totally ramified of degree two; z is a uniformizer and z^2+t_L^az+t_L=0, Tr_E/L(α)=1. For constant-coefficient projection P_0 in the basis 1,z, the normalized trace corestriction and exact filtered right inverse are κ=α^-1, s=κpr, s(P_0)=t_L/z^2, Lift_E(C)=(z^2/t_L)CP_0. They satisfy s(Lift_E(C))=C, preserve the exact …
Exact mathematical passage
z^2+t_L^az+t_L=0,\qquad \operatorname{Tr}_{E/L}(\alpha)=1.
\operatorname{tr}Use in source · 5 passages
Search aliases: tr operatorname
Use in source · Read passage · Paper_XI_Traditional_Final.tex, line 2109
The Artin--Schreier filtered lift has native trace one. Consequently each native correction δ satisfies v_F(trδ)+s≥1 (s=1), which makes its additive scalar correction trivial on U_F^1. The actual field and defining-sequence proof is [source reference]. Full beta-determinant transfer and character extension at each new maximal native field prove the full-character claim, including the odd-critical endpoint 2(lfloor r/2rfloor+1)-r=1; see [source reference]. The exact original quotient has dimensio…
Exact mathematical passage
v_F(\operatorname{tr}\delta)+s\ge1 \quad (s=1),
\trUse in source · 5 passages
Search aliases: tr operatorname
Use in source · Paper_XI_Visual_Guide_Final.tex, line 418
For its Eisenstein polynomial f and constant-coordinate projection P_0, the normalized corestriction uses κ=-f'(z)/z^p-1. Writing u_0=s(P_0), Lift_E(C)=u_0^-1CP_0, tr_LLift_E(C) =Tr_E/L(κ^-1tr_EC). If tr_EC=0, the lift has trace zero. Adding α I_m preserves trace zero. Induction therefore gives trace zero at every native stage and every native correction.
Exact mathematical passage
\operatorname{Lift}_E(C)=u_0^{-1}CP_0,\qquad \tr_L\operatorname{Lift}_E(C) =\Tr_{E/L}(\kappa^{-1}\tr_EC).
\operatorname{tr}_EUse in source · 2 passages
Search aliases: tr_E operatorname
Use in source · Read passage · dyadic_synthesis.tex, line 58
_E/L(α)=1. For constant-coefficient projection P_0 in the basis 1,z, the normalized trace corestriction and exact filtered right inverse are κ=α^-1, s=κpr, s(P_0)=t_L/z^2, Lift_E(C)=(z^2/t_L)CP_0. They satisfy s(Lift_E(C))=C, preserve the exact filtration valuation, and give the identity tr_LLift_E(C) =Tr_E/L (αtr_E C). The actual recursive matrix is B(L,t_L;h_1,…,h_d) =α I_2^d-1+ Lift_E(B(E,z;h_1,…,h_d-1)), with multiplication by α as the rank-two base case.
Exact mathematical passage
\operatorname{tr}_L\operatorname{Lift}_E(C) =\operatorname{Tr}_{E/L} \bigl(\alpha\operatorname{tr}_E C\bigr).
\tr_EUse in source · 3 passages
Search aliases: tr_E tr operatorname
Use in source · Paper_XI_Visual_Guide_Final.tex, line 418
For its Eisenstein polynomial f and constant-coordinate projection P_0, the normalized corestriction uses κ=-f'(z)/z^p-1. Writing u_0=s(P_0), Lift_E(C)=u_0^-1CP_0, tr_LLift_E(C) =Tr_E/L(κ^-1tr_EC). If tr_EC=0, the lift has trace zero. Adding α I_m preserves trace zero. Induction therefore gives trace zero at every native stage and every native correction.
Exact mathematical passage
\operatorname{Lift}_E(C)=u_0^{-1}CP_0,\qquad \tr_L\operatorname{Lift}_E(C) =\Tr_{E/L}(\kappa^{-1}\tr_EC).
\Tr_{E/L}Use in source · 4 passages
Search aliases: Tr_E/L Tr operatorname
Use in source · Paper_XI_Visual_Guide_Final.tex, line 415
lessonPART III / ACTUAL ARITHMETIC CONSTRUCTIONOdd towers retain native trace zeroBlock 166, recursive tower and full-character proofs For odd p, Tr_E/L(α)=0. Choose b with qb≡-1pmod p, let a=(qb+1)/p, and set z=t_L^aα^b. Then z is a uniformizer.
Exact mathematical passage
\Tr_{E/L}(\alpha)=0
\operatorname{Tr}_{E/L}Use in source · 4 passages
Search aliases: Tr_E/L operatorname
Use in source · Read passage · dyadic_synthesis.tex, line 50
At a recursive coefficient field L of characteristic two, with residue field F_2 and uniformizer t_L, put E=L(α), α^2-α=t_L^-q, a=(q+1)/2, z=t_L^aα. The extension is separable and totally ramified of degree two; z is a uniformizer and z^2+t_L^az+t_L=0, Tr_E/L(α)=1. For constant-coefficient projection P_0 in the basis 1,z, the normalized trace corestriction and exact filtered right inverse are κ=α^-1, s=κpr, s(P_0)=t_L/z^2, Lift_E(C)=(z^2/t_L)CP_0. They satisfy s(Lift_E(C))=C, preserve the exact …
Exact mathematical passage
z^2+t_L^az+t_L=0,\qquad \operatorname{Tr}_{E/L}(\alpha)=1.
\operatorname{tr}_LUse in source · 2 passages
Search aliases: tr_L operatorname
Use in source · Read passage · dyadic_synthesis.tex, line 58
_E/L(α)=1. For constant-coefficient projection P_0 in the basis 1,z, the normalized trace corestriction and exact filtered right inverse are κ=α^-1, s=κpr, s(P_0)=t_L/z^2, Lift_E(C)=(z^2/t_L)CP_0. They satisfy s(Lift_E(C))=C, preserve the exact filtration valuation, and give the identity tr_LLift_E(C) =Tr_E/L (αtr_E C). The actual recursive matrix is B(L,t_L;h_1,…,h_d) =α I_2^d-1+ Lift_E(B(E,z;h_1,…,h_d-1)), with multiplication by α as the rank-two base case.
Exact mathematical passage
\operatorname{tr}_L\operatorname{Lift}_E(C) =\operatorname{Tr}_{E/L} \bigl(\alpha\operatorname{tr}_E C\bigr).
\tr_LUse in source · 2 passages
Search aliases: tr_L tr operatorname
Use in source · Paper_XI_Visual_Guide_Final.tex, line 418
For its Eisenstein polynomial f and constant-coordinate projection P_0, the normalized corestriction uses κ=-f'(z)/z^p-1. Writing u_0=s(P_0), Lift_E(C)=u_0^-1CP_0, tr_LLift_E(C) =Tr_E/L(κ^-1tr_EC). If tr_EC=0, the lift has trace zero. Adding α I_m preserves trace zero. Induction therefore gives trace zero at every native stage and every native correction.
Exact mathematical passage
\operatorname{Lift}_E(C)=u_0^{-1}CP_0,\qquad \tr_L\operatorname{Lift}_E(C) =\Tr_{E/L}(\kappa^{-1}\tr_EC).
U^*Use in source · 3 passages
Search aliases: U^*
Use in source · Read passage · Paper_XI_Traditional_Final.tex, line 1416
[Hyperbolic phase formula] Let U be a finite dimensional F_p space, let a∈GL(U) have no eigenvalue one, and let g act on H(U)=U⊕ U^* by g=(a,(a^-1)^*). Then Φ_p(g,H(U)) = (fracdet ap).
Exact mathematical passage
H(U)=U\oplus U^*
U^{-1}Use in source · 1 passage
Search aliases: U^-1
Use in source · Read passage · Paper_XI_Traditional_Final.tex, line 1248
For each g, the operator Ω_θ(g)^-1UΩ_ρ(g)U^-1 commutes with the type side Heisenberg representation. It is therefore scalar. Since both stabilizer actions are honest representations, multiplication of group elements shows that the scalars form a character.
Exact mathematical passage
\Omega_\theta(g)^{-1}U\Omega_\rho(g)U^{-1}
u^{-1}Use in source · 1 passage
Search aliases: u^-1
Use in source · Read passage · Paper_XI_Traditional_Final.tex, line 653
For nontrivial inertia, the invariant space is zero. In the trivial case, Frobenius on the invariant packet is the companion matrix of X^r-a. Substituting X=u^-1 and multiplying by u^r gives 1-a u^r.
Exact mathematical passage
X=u^{-1}
U^1Use in source · 2 passages
Search aliases: U^1
Use in source · Read passage · odd_capacity.tex, line 191
The right side is the Heisenberg commutator pairing for the minimal simple stratum of gamma. Its quotient J^1(γ)/H^1(γ) is precisely U^m(A)/H^m(γ), because J^1(γ)=U^1(B_order)U^m(A), with U^1(B_order)⊂ H^1(γ). BK's nondegeneracy theorem for simple-character commutators therefore says that the form on S is nondegenerate.
Exact mathematical passage
J^1(\gamma)=U^1(B_{\mathrm{order}})U^m(A)
u^2Use in source · 2 passages
Search aliases: u^2
Use in source · Read passage · dyadic_synthesis.tex, line 106
For a later native step, first transfer the full beta-determinant unit character to multiplicity two. BK 3.6.13(ii) and 3.6.14 give, on scalar units, Θ_j(uI)=θ_j(u^2)=1. For odd critical level r, use m=lfloor r/2rfloor and D=H^m+1(β)=H^m+1(γ). The endpoint is valid because 2(m+1)-r=1: the additive-character cross term lies in the radical on which the matrix additive character is trivial. BK 3.3.18 supplies ϑ=Θ_j|_D ψ_β-γ. Integral trace makes it trivial on D∩ U_F^1.
Exact mathematical passage
\Theta_j(uI)=\theta_j(u^2)=1.
U^{2m}Use in source · 1 passage
Search aliases: U^2m
Use in source · Read passage · odd_capacity.tex, line 167
For u,v in U m(A), their group commutator is in U^2m(A)=U^N(A), which lies in U (m+1)(A). The terminal approximation satisfies
Exact mathematical passage
U^{2m}(A)=U^N(A)
U^aUse in source · 1 passage
Search aliases: U^a
Use in source · Read passage · odd_capacity.tex, line 73
Put P=rad(A), U^a(A)=1+P^a for atextgreater=1, and m=N/2.
Exact mathematical passage
U^a(A)=1+P^a
u^{k_0}Use in source · 1 passage
Search aliases: u^k_0
Use in source · Read passage · odd_synthesis.tex, line 201
y q, let B=ω(Teich(k^×)). The actual inertia hypothesis gives χ(C)^N⊂ B. Since N is a power of p and χ(C) has order prime to p, this implies χ(C)⊂ B. Choose k_0 inverse to N modulo q-1, and set δ(u)=ω(Teich(u))^k_0, widetilde C={(c,u):χ(c)=δ(u)}. The matrix Teich(u^k_0)I_N acts by χ(c). The kernel acts trivially on the entire sector induction and ambient compact induction, because these matrices are central. This preserves the fixed Λ, its canonical beta-extension factor, and all completion choi…
Exact mathematical passage
\operatorname{Teich}(u^{k_0})I_N
U^mUse in source · 3 passages
Search aliases: U^m
Use in source · Read passage · odd_capacity.tex, line 145
Here H m means H^1∩ U^m(A). Therefore the image S of U m(A) in W is canonically
Exact mathematical passage
H^1\cap U^m(A)
U^NUse in source · 1 passage
Search aliases: U^N
Use in source · Read passage · odd_capacity.tex, line 167
For u,v in U m(A), their group commutator is in U^2m(A)=U^N(A), which lies in U (m+1)(A). The terminal approximation satisfies
Exact mathematical passage
U^{2m}(A)=U^N(A)
u^pUse in source · 2 passages
Search aliases: u^p
Use in source · Read passage · odd_synthesis.tex, line 99
The full-character argument uses a specific order of operations. First construct a full character at native rank p^j, trivial on U_F^1. Transfer its full beta-determinant unit character to multiplicity p, using BK 3.6.13(ii) and 3.6.14. On a scalar u∈ U_F^1, this gives θ_j(u^p)=1. Next add the actual trace-zero correction, using BK 3.3.18. It remains trivial on the scalar intersection of its domain. Finally extend at the next maximal native parameter field: the forced deep character and the triv…
Exact mathematical passage
\theta_j(u^p)=1
u^rUse in source · 4 passages
Search aliases: u^r
Use in source · Read passage · Paper_XI_Traditional_Final.tex, line 642
[Local polynomial of one tame packet] Suppose an interaction packet has orbit length r and closing scalar a. Its local polynomial is det(I-uΦmid V^I_K) = 1-a u^r, · if the inertia character is trivial, 1, · otherwise.
Exact mathematical passage
\det\left(I-u\Phi\mid V^{I_K}\right) = \begin{cases} 1-a u^r,&\text{if the inertia character is trivial},\\ 1,&\text{otherwise}. \end{cases}
u^{r_i}Use in source · 2 passages
Search aliases: u^r_i
Use in source · Read passage · Paper_XI_Traditional_Final.tex, line 658
[Tame tensor local factor] The local factor of the tame tensor product is the product of (1-a_i u^r_i)^-1 over precisely those refined packets whose inertia exponent vanishes.
Exact mathematical passage
(1-a_i u^{r_i})^{-1}
u_0Use in source · 2 passages
Search aliases: u_0
Use in source · Read passage · odd_synthesis.tex, line 65
Let P_0 extract the constant coefficient in the basis 1,z,…,z^p-1. The normalized corestriction s=κpr, where κ=-f'(z)/z^p-1, satisfies u_0=s(P_0)=a_0/z^p∈ O_E^×, Lift_E(C)=u_0^-1CP_0, s(Lift_E(C))=C. The lift preserves the exact filtration valuation. Its crucial additional identity is tr_L(Lift_E(C)) =Tr_E/L (κ^-1tr_E(C)). Define B(L,t_L;h_1,…,h_d) =α I_p^d-1+ Lift_E(B(E,z;h_1,…,h_d-1)), with the degree-p multiplication matrix as the base case. Induction proves trace zero at every native rank. T…
Exact mathematical passage
u_0=s(P_0)=a_0/z^p\in\mathcal O_E^\times, \qquad \operatorname{Lift}_E(C)=u_0^{-1}CP_0, \qquad s(\operatorname{Lift}_E(C))=C.
u_0^{-1}Use in source · 2 passages
Search aliases: u_0^-1
Use in source · Read passage · odd_synthesis.tex, line 65
Let P_0 extract the constant coefficient in the basis 1,z,…,z^p-1. The normalized corestriction s=κpr, where κ=-f'(z)/z^p-1, satisfies u_0=s(P_0)=a_0/z^p∈ O_E^×, Lift_E(C)=u_0^-1CP_0, s(Lift_E(C))=C. The lift preserves the exact filtration valuation. Its crucial additional identity is tr_L(Lift_E(C)) =Tr_E/L (κ^-1tr_E(C)). Define B(L,t_L;h_1,…,h_d) =α I_p^d-1+ Lift_E(B(E,z;h_1,…,h_d-1)), with the degree-p multiplication matrix as the base case. Induction proves trace zero at every native rank. T…
Exact mathematical passage
u_0=s(P_0)=a_0/z^p\in\mathcal O_E^\times, \qquad \operatorname{Lift}_E(C)=u_0^{-1}CP_0, \qquad s(\operatorname{Lift}_E(C))=C.
u_aUse in source · 6 passages
Search aliases: u_a
Use in source · Read passage · Paper_XI_Traditional_Final.tex, line 922
Let A be a finite abelian group, and let c:A× A→ C^× be a normalized two cocycle. The twisted group algebra C^c[A] has basis (u_a)_a∈ A and multiplication u_au_b=c(a,b)u_a+b. Define the commutator bicharacter b_c(a,b)=c(a,b)c(b,a)^-1.
Exact mathematical passage
(u_a)_{a\in A}
u_{a+b}Use in source · 1 passage
Search aliases: u_a+b
Use in source · Read passage · Paper_XI_Traditional_Final.tex, line 923
Let A be a finite abelian group, and let c:A× A→ C^× be a normalized two cocycle. The twisted group algebra C^c[A] has basis (u_a)_a∈ A and multiplication u_au_b=c(a,b)u_a+b. Define the commutator bicharacter b_c(a,b)=c(a,b)c(b,a)^-1.
Exact mathematical passage
u_au_b=c(a,b)u_{a+b}.
u_a^{-1}Use in source · 1 passage
Search aliases: u_a^-1
Use in source · Read passage · Paper_XI_Traditional_Final.tex, line 979
Let z=∑_a∈ Az_au_a be central. Commuting with u_x gives z_ab_c(x,a)=z_a for every x. Nondegeneracy forces z_a=0 for every nonzero a. Thus the center is C. To prove semisimplicity directly, choose a linear projection P onto a submodule of a finite dimensional C^c[A]-module and form overline P=frac1abs A∑_a∈ Au_a P u_a^-1. Every summand maps into the submodule and is the identity there. Conjugation by any u_x permutes the summands: the cocycle scalars cancel between each operator and its inverse. …
Exact mathematical passage
\overline P=\frac{1}{\abs A}\sum_{a\in A}u_a P u_a^{-1}.
U_A^1Use in source · 2 passages
Search aliases: U_A^1
Use in source · Read passage · rank2.tex, line 97
K× normalizes A and every power of its radical, so U_K 1 normalizes U_A m. Therefore P is a subgroup and is compact open. It is contained in U_A 1 because mtextgreater=1 and U_K^1⊂ U_A^1.
Exact mathematical passage
U_K^1\subset U_A^1
U_A^jUse in source · 1 passage
Search aliases: U_A^j
Use in source · Read passage · rank4.tex, line 58
Use U_A^j=1+P_A^j and U_B^j=1+Q^j. BK3.1.7--9 give the actual groups
Exact mathematical passage
U_A^j=1+P_A^j
U_A^mUse in source · 4 passages
Search aliases: U_A^m
Use in source · Read passage · Paper_XI_Traditional_Final.tex, line 2123
At rank two, with q=2m+1, enlarge J=U_K^1U_A^m+1 to P=U_K^1U_A^m. The actual character ratio λ=θ^g/θ for g∉ J is nontrivial and sign-valued. With H_0=kerλ, the quotient is P/H_0≃ F_2^2, and Ind_J^Pθ is an irreducible two-dimensional Heisenberg carrier ([source reference]). At rank four, the actual outer quadratic centralizer gives P=J U_B^m, [P:J]=2, and P/H_0≃ F_2^4, with carrier Ind_J^Pη of dimension four ([source reference]). The finite scalar center can be larger than μ_2; no literal extrasp…
Exact mathematical passage
P=U_K^1U_A^m
U_A^{m+1}Use in source · 4 passages
Search aliases: U_A^m+1
Use in source · Read passage · Paper_XI_Traditional_Final.tex, line 2123
At rank two, with q=2m+1, enlarge J=U_K^1U_A^m+1 to P=U_K^1U_A^m. The actual character ratio λ=θ^g/θ for g∉ J is nontrivial and sign-valued. With H_0=kerλ, the quotient is P/H_0≃ F_2^2, and Ind_J^Pθ is an irreducible two-dimensional Heisenberg carrier ([source reference]). At rank four, the actual outer quadratic centralizer gives P=J U_B^m, [P:J]=2, and P/H_0≃ F_2^4, with carrier Ind_J^Pη of dimension four ([source reference]). The finite scalar center can be larger than μ_2; no literal extrasp…
Exact mathematical passage
J=U_K^1U_A^{m+1}
U_A^{n-r+m}Use in source · 1 passage
Search aliases: U_A^n-r+m
Use in source · Read passage · dyadic_synthesis.tex, line 343
At both low ranks, K^× normalizes the new P. In rank two this follows directly from normalization of A and its radical. In rank four, a K-uniformizer is approximated by the derived chain shift R times a unit of B, with relative error in P_A^n-r. Its conjugation error on U_B^m lies in U_A^n-r+m⊂ J, which proves normalization. The exact argument is pathlowrank/RANK4_INDUCTION_BY_STAGES167.md.
Exact mathematical passage
U_A^{n-r+m}\subset J
u_bUse in source · 1 passage
Search aliases: u_b
Use in source · Read passage · Paper_XI_Traditional_Final.tex, line 923
Let A be a finite abelian group, and let c:A× A→ C^× be a normalized two cocycle. The twisted group algebra C^c[A] has basis (u_a)_a∈ A and multiplication u_au_b=c(a,b)u_a+b. Define the commutator bicharacter b_c(a,b)=c(a,b)c(b,a)^-1.
Exact mathematical passage
u_au_b=c(a,b)u_{a+b}.
U_B^jUse in source · 1 passage
Search aliases: U_B^j
Use in source · Read passage · rank4.tex, line 58
Use U_A^j=1+P_A^j and U_B^j=1+Q^j. BK3.1.7--9 give the actual groups
Exact mathematical passage
U_B^j=1+Q^j
U_B^mUse in source · 5 passages
Search aliases: U_B^m
Use in source · Read passage · Paper_XI_Traditional_Final.tex, line 2129
two, with q=2m+1, enlarge J=U_K^1U_A^m+1 to P=U_K^1U_A^m. The actual character ratio λ=θ^g/θ for g∉ J is nontrivial and sign-valued. With H_0=kerλ, the quotient is P/H_0≃ F_2^2, and Ind_J^Pθ is an irreducible two-dimensional Heisenberg carrier ([source reference]). At rank four, the actual outer quadratic centralizer gives P=J U_B^m, [P:J]=2, and P/H_0≃ F_2^4, with carrier Ind_J^Pη of dimension four ([source reference]). The finite scalar center can be larger than μ_2; no literal extraspecial qu…
Exact mathematical passage
P=J U_B^m
U_B^{m+1}Use in source · 1 passage
Search aliases: U_B^m+1
Use in source · Read passage · rank4.tex, line 147
The quotient U_B^m/U_B^m+1 has dimension two. By Section2 the image of J cap U_B m on this grade is exactly its centralizing line, of dimension one. For the converse inclusion needed here, the B-approximation of a K-element of grade m differs from it by grade m+n-rtextgreater a; that error lies in U_A a subset J. Thus the whole centralizing line really occurs in the intersection, not only in an abstract associated graded space. Consequently
Exact mathematical passage
U_B^m/U_B^{m+1}
U_E^{(q+1)/2}Use in source · 1 passage
Search aliases: U_E^(q+1)/2
Use in source · Read passage · dyadic_synthesis.tex, line 99
At the minimal rank-two base, extend the forced character on U_E^(q+1)/2 to U_E^1, imposing triviality on U_F^1. The prescriptions agree on their intersection because the parameter trace is one and ψ_F(u-1)=1. This is an actual finite abelian character extension and gives the full minimal simple character.
Exact mathematical passage
U_E^{(q+1)/2}
U_E^1Use in source · 3 passages
Search aliases: U_E^1
Use in source · Read passage · dyadic_character.tex, line 269
Section4 constructs chi_E on U_E 1 extending phi and trivial on U_F 1. Define theta_1 on H^1=U_E^1 U_(A_1) a by chi_E on the first factor and psi_alpha on the second. The restrictions agree on the intersection. BK3.2.1 gives the actual full simple character, with theta_1textbar U_F 1=1.
Exact mathematical passage
H^1=U_E^1
U_F^sUse in source · 2 passages
Search aliases: U_F^s
Use in source · Read passage · dyadic_synthesis.tex, line 86
The replacement for trace zero is the depth inequality v_F(trδ)+s≥1. It makes the additive correction trivial on scalar U_F^s, because (u-1)trδ∈ t O_F. Our native corrections have trace one, so the inequality holds at s=1. This is the exact source-to-target transfer of the weighted-visibility principle; the Euler coefficient theorem is motivation, not an input to the arithmetic proof. In characteristic two, the weights are uniformizer depths, rather than integer powers of two in the coefficient …
Exact mathematical passage
U_F^s
U_K^1Use in source · 9 passages
Search aliases: U_K^1
Use in source · Read passage · Paper_XI_Traditional_Final.tex, line 2123
At rank two, with q=2m+1, enlarge J=U_K^1U_A^m+1 to P=U_K^1U_A^m. The actual character ratio λ=θ^g/θ for g∉ J is nontrivial and sign-valued. With H_0=kerλ, the quotient is P/H_0≃ F_2^2, and Ind_J^Pθ is an irreducible two-dimensional Heisenberg carrier ([source reference]). At rank four, the actual outer quadratic centralizer gives P=J U_B^m, [P:J]=2, and P/H_0≃ F_2^4, with carrier Ind_J^Pη of dimension four ([source reference]). The finite scalar center can be larger than μ_2; no literal extrasp…
Exact mathematical passage
J=U_K^1U_A^{m+1}
U_K^MUse in source · 2 passages
Search aliases: U_K^M
Use in source · Read passage · dyadic_character.tex, line 242
It is well defined by the intersection condition. Choose Mtextgreater=a with chitextbar U_K M=1. Then chi_0 is a character of a subgroup of the finite abelian group U_K^1/U_K^M. Extend it by adjoining generators and choosing the required complex roots, and pull it back. The resulting chi_K extends chi and is trivial on U_F 1.
Exact mathematical passage
U_K^1/U_K^M
U_K^mUse in source · 1 passage
Search aliases: U_K^m
Use in source · Read passage · rank2.tex, line 130
For if ku∈U_A m with k∈U_K 1 and u∈U_A (m+1), then k∈ K^×∩ U_A^m=U_K^m. The reverse inclusion is immediate. The complete period-two A-grade R^m/R^m+1 has F2-dimension2, and its field grade p_K^m/ p_K^m+1 has dimension1. Thus
Exact mathematical passage
k\in K^\times\cap U_A^m=U_K^m
U_rUse in source · 12 passages
Search aliases: U_r
Use in source · Read passage · Paper_XI_Traditional_Final.tex, line 1635
[Matrix holonomy criterion] Fix a root r and an isomorphism U_r:V_r⟶ W_r. The following conditions are equivalent.
Exact mathematical passage
U_r:V_r\longrightarrow W_r.
U_uUse in source · 7 passages
Search aliases: U_u
Use in source · Read passage · Paper_XI_Traditional_Final.tex, line 1642
item There are isomorphisms U_v:V_v→ W_v, with the declared U_r, such that U_vA_e=B_eU_u for every edge e:u→ v. item For one fundamental cycle basis based at r, U_rA_γ=B_γ U_r. item The same identity holds for every based loop γ.
Exact mathematical passage
U_vA_e=B_eU_u
U_vUse in source · 10 passages
Search aliases: U_v
Use in source · Read passage · Paper_XI_Traditional_Final.tex, line 1641
item There are isomorphisms U_v:V_v→ W_v, with the declared U_r, such that U_vA_e=B_eU_u for every edge e:u→ v. item For one fundamental cycle basis based at r, U_rA_γ=B_γ U_r. item The same identity holds for every based loop γ.
Exact mathematical passage
U_v:V_v\to W_v
u_xUse in source · 2 passages
Search aliases: u_x
Use in source · Read passage · Paper_XI_Traditional_Final.tex, line 974
Let z=∑_a∈ Az_au_a be central. Commuting with u_x gives z_ab_c(x,a)=z_a for every x. Nondegeneracy forces z_a=0 for every nonzero a. Thus the center is C. To prove semisimplicity directly, choose a linear projection P onto a submodule of a finite dimensional C^c[A]-module and form overline P=frac1abs A∑_a∈ Au_a P u_a^-1. Every summand maps into the submodule and is the identity there. Conjugation by any u_x permutes the summands: the cocycle scalars cancel between each operator and its inverse. …
Exact mathematical passage
u_x
V'Use in source · 2 passages
Search aliases: V'
Use in source · Read passage · Paper_XI_Traditional_Final.tex, line 1322
[Symplectic invariance] If ι:(V,b)⟶(V',b') is a symplectic isometry and g'=ι gι^-1, then Φ_p(g,V)=Φ_p(g',V').
Exact mathematical passage
\iota:(V,b)\longrightarrow(V',b')
V^*Use in source · 2 passages
Search aliases: V^*
Use in source · Read passage · Paper_XI_Traditional_Final.tex, line 1484
[Affine space of refinements] If one quadratic refinement exists, then the set of all quadratic refinements is an affine space over V^*.
Exact mathematical passage
V^*
V^{I_K}Use in source · 4 passages
Search aliases: V^I_K
Use in source · Read passage · Paper_XI_Traditional_Final.tex, line 638
Let V be one interaction packet. If its inertia character is nontrivial, then V^I_K=0. If the inertia character is trivial, Frobenius acts on the invariant packet by a companion cycle.
Exact mathematical passage
V^{I_K}=0
V_1Use in source · 2 passages
Search aliases: V_1
Use in source · Read passage · Paper_XI_Traditional_Final.tex, line 1337
[Orthogonal multiplicativity] Suppose V=V_1mathop⊥limits⋯mathop⊥limits V_s and g preserves every summand. If no g|_V_j has a fixed vector, then Φ_p(g,V) = ∏_j=1^s Φ_p(g|_V_j,V_j).
Exact mathematical passage
V=V_1\perpbig\cdots\perpbig V_s
V_2Use in source · 1 passage
Search aliases: V_2
Use in source · Read passage · Paper_XI_Traditional_Final.tex, line 1524
[Orthogonal dyadic multiplicativity] For an orthogonal sum (V,q)=(V_1,q_1)mathop⊥limits(V_2,q_2), we have Arf(q)=Arf(q_1)+Arf(q_2) and the normalized Gauss sign is the product of the two normalized signs.
Exact mathematical passage
(V,q)=(V_1,q_1)\perpbig(V_2,q_2),
v_EUse in source · 1 passage
Search aliases: v_E
Use in source · Read passage · odd_synthesis.tex, line 57
At a recursive base (L,t_L), introduce E=L(α), α^p-α=t_L^-q_d. This is a separable totally ramified extension of degree p, with v_E(α)=-q_d and Tr_E/Lα=0. Choose 1≤ b<p with q_db≡-1pmod p, set a=(q_db+1)/p, and put z=t_L^aα^b. Then z is an explicit uniformizer. Write its Eisenstein polynomial as f(X)=X^p+⋯+a_0.
Exact mathematical passage
v_E(\alpha)=-q_d
v_FUse in source · 5 passages
Search aliases: v_F
Use in source · Read passage · Paper_XI_Traditional_Final.tex, line 2109
The Artin--Schreier filtered lift has native trace one. Consequently each native correction δ satisfies v_F(trδ)+s≥1 (s=1), which makes its additive scalar correction trivial on U_F^1. The actual field and defining-sequence proof is [source reference]. Full beta-determinant transfer and character extension at each new maximal native field prove the full-character claim, including the odd-critical endpoint 2(lfloor r/2rfloor+1)-r=1; see [source reference]. The exact original quotient has dimensio…
Exact mathematical passage
v_F(\operatorname{tr}\delta)+s\ge1 \quad (s=1),
V_iUse in source · 2 passages
Search aliases: V_i
Use in source · Read passage · odd_synthesis.tex, line 121
Assume the one-vertex scalar-pushout contract: there are d labelled, actually supplied full character extensions with whole symplectic quotients V_i of dimension two over F_p. Their balanced C^×-central product has quotient V_src=⊕_iV_i and Heisenberg carrier dimension p^d=N. Initially take the declared stabilizer C={1}.
Exact mathematical passage
V_i
V_rUse in source · 2 passages
Search aliases: V_r
Use in source · Read passage · Paper_XI_Traditional_Final.tex, line 1635
[Matrix holonomy criterion] Fix a root r and an isomorphism U_r:V_r⟶ W_r. The following conditions are equivalent.
Exact mathematical passage
U_r:V_r\longrightarrow W_r.
V_{\rm src}Use in source · 1 passage
Search aliases: V_rm src rm
Use in source · Paper_XI_Visual_Guide_Final.tex, line 499
An orthogonal splitting W=V_rm src⊥ V_rm src^⊥ gives η_W≃η_rm src⊗η_⊥. A maximal-isotropic average on the complementary Heisenberg factor selects a one-dimensional representation line. Tensoring its projector with the identity gives a rank-N image-algebra projector. The retained induction/recovery theorem recovers the same Λ and then the same π. boundaryFor p>3, genuinely supplied plane data are required: a native minimal rank-p quotient has dimension p-1 and is not automatically a plane. The co…
Exact mathematical passage
W=V_{\rm src}\perp V_{\rm src}^{\perp}
V_{\rm src}^{\perp}Use in source · 1 passage
Search aliases: V_rm src^⊥ rm perp
Use in source · Paper_XI_Visual_Guide_Final.tex, line 499
An orthogonal splitting W=V_rm src⊥ V_rm src^⊥ gives η_W≃η_rm src⊗η_⊥. A maximal-isotropic average on the complementary Heisenberg factor selects a one-dimensional representation line. Tensoring its projector with the identity gives a rank-N image-algebra projector. The retained induction/recovery theorem recovers the same Λ and then the same π. boundaryFor p>3, genuinely supplied plane data are required: a native minimal rank-p quotient has dimension p-1 and is not automatically a plane. The co…
Exact mathematical passage
W=V_{\rm src}\perp V_{\rm src}^{\perp}
V_{\mathrm{src}}Use in source · 3 passages
Search aliases: V_src mathrm
Use in source · Read passage · odd_synthesis.tex, line 123
Assume the one-vertex scalar-pushout contract: there are d labelled, actually supplied full character extensions with whole symplectic quotients V_i of dimension two over F_p. Their balanced C^×-central product has quotient V_src=⊕_iV_i and Heisenberg carrier dimension p^d=N. Initially take the declared stabilizer C={1}.
Exact mathematical passage
V_{\mathrm{src}}=\bigoplus_iV_i
V_{\mathrm{src}}^\perpUse in source · 1 passage
Search aliases: V_src^⊥ mathrm perp
Use in source · Read passage · odd_synthesis.tex, line 153
Write W=V_src⊥ V_src^⊥. Finite Stone--von Neumann gives η_W≃η_src⊗η_⊥. Choose an isotropic eigenspace giving a line in the complementary factor. Its actual Heisenberg image-algebra projector e has rank N. The retained induction/idempotent theorem recovers the same inducing representation Λ from eΛ, and then the same π by compact induction. No candidate is changed at recovery.
Exact mathematical passage
W=V_{\mathrm{src}}\perp V_{\mathrm{src}}^\perp
V_uUse in source · 2 passages
Search aliases: V_u
Use in source · Read passage · Paper_XI_Traditional_Final.tex, line 1612
Let Γ be a connected finite graph. At every vertex v, place finite dimensional complex vector spaces V_v and W_v of the same dimension. For every oriented edge e:u→ v, let A_e:V_u⟶ V_v, B_e:W_u⟶ W_v be invertible. The reversed edge carries the inverse map.
Exact mathematical passage
A_e:V_u\longrightarrow V_v, \qquad B_e:W_u\longrightarrow W_v
V_vUse in source · 4 passages
Search aliases: V_v
Use in source · Read passage · Paper_XI_Traditional_Final.tex, line 1611
Let Γ be a connected finite graph. At every vertex v, place finite dimensional complex vector spaces V_v and W_v of the same dimension. For every oriented edge e:u→ v, let A_e:V_u⟶ V_v, B_e:W_u⟶ W_v be invertible. The reversed edge carries the inverse map.
Exact mathematical passage
V_v
wUse in source · 4 passages
Search aliases: w
Use in source · Read passage · Paper_XI_Traditional_Final.tex, line 235
Let W_K be the Weil group of K, let I_K be inertia, let P_K be wild inertia, and choose a geometric Frobenius lift Φ. A Weil Deligne representation is a pair (r,N) with r(w)Nr(w)^-1=absw_K N.
Exact mathematical passage
r(w)Nr(w)^{-1}=\abs{w}_K N.
W_{K_f}Use in source · 8 passages
Search aliases: W_K_f
Use in source · Read passage · Paper_XI_Traditional_Final.tex, line 338
[Tame orbit induction] Let τ be an irreducible complex representation of W_K with trivial wild inertia. Then the nonzero tame character packets form one transitive Frobenius orbit of length f. If K_f/K is the unramified extension of degree f, there is a character ξ of W_K_f such that τ≅Ind_W_K_f^W_Kξ.
Exact mathematical passage
W_{K_f}
W_rUse in source · 2 passages
Search aliases: W_r
Use in source · Read passage · Paper_XI_Traditional_Final.tex, line 1635
[Matrix holonomy criterion] Fix a root r and an isomorphism U_r:V_r⟶ W_r. The following conditions are equivalent.
Exact mathematical passage
U_r:V_r\longrightarrow W_r.
W_uUse in source · 2 passages
Search aliases: W_u
Use in source · Read passage · Paper_XI_Traditional_Final.tex, line 1612
Let Γ be a connected finite graph. At every vertex v, place finite dimensional complex vector spaces V_v and W_v of the same dimension. For every oriented edge e:u→ v, let A_e:V_u⟶ V_v, B_e:W_u⟶ W_v be invertible. The reversed edge carries the inverse map.
Exact mathematical passage
A_e:V_u\longrightarrow V_v, \qquad B_e:W_u\longrightarrow W_v
W_vUse in source · 4 passages
Search aliases: W_v
Use in source · Read passage · Paper_XI_Traditional_Final.tex, line 1611
Let Γ be a connected finite graph. At every vertex v, place finite dimensional complex vector spaces V_v and W_v of the same dimension. For every oriented edge e:u→ v, let A_e:V_u⟶ V_v, B_e:W_u⟶ W_v be invertible. The reversed edge carries the inverse map.
Exact mathematical passage
W_v
\widetilde CUse in source · 3 passages
Search aliases: widetilde C widetilde
Use in source · Read passage · odd_synthesis.tex, line 187
Then a finite cover widetilde C↠ C acts by actual central matrices in the unchanged candidate. Its represented operators descend to exactly χ on C. Hence the attachment and recovery above are equivariant for that action. A direct matrix homomorphism from C itself is not asserted and can fail.
Exact mathematical passage
\widetilde C\twoheadrightarrow C
\widetilde LUse in source · 4 passages
Search aliases: widetilde L widetilde
Use in source · Read passage · Paper_XI_Traditional_Final.tex, line 870
Let L⊆ V be maximal isotropic. Choose a subgroup lift widetilde L⊆ H(V) on which the central coordinate is zero.
Exact mathematical passage
\widetilde L\subseteq\frakH(V)
x'Use in source · 1 passage
Search aliases: x'
Use in source · Paper_XI_Visual_Guide_Final.tex, line 327
lessonPART II / FINITE WILD COMPARISONTwo dyadic refinements can have opposite signsOriginal Section 13 plate10_quadratic_planesThe entries are exact values in F_2; both polar forms are xy'+x'y. A quadratic refinement satisfies q(x+y)+q(x)+q(y)=b(x,y). On a plane, q_0(x,y)=xy, q_1(x,y)=xy+x+y. For q_0 there are three zeros and one one, so its Gauss sum is 2. For q_1 there is one zero and three ones, so its sum is -2. The alternating forms agree but the Arf invariants differ. ∑_x∈ V(-1)^q(x)=(-1)…
Exact mathematical passage
xy'+x'y
X^*Use in source · 3 passages
Search aliases: X^*
Use in source · Read passage · Paper_XI_Traditional_Final.tex, line 790
Let p be odd. Let X be an r dimensional vector space over F_p, and put V=X⊕ X^*. Define the alternating form b((x,ξ),(y,η)) = η(x)-ξ(y). Let Z= F_p, and define the Heisenberg group H(V)=V× Z with multiplication (v,z)(w,t) = (v+w,z+t+frac12 b(v,w)). Fix the additive character ψ(z)= e^2π iz/p.
Exact mathematical passage
V=X\oplus X^*.
x^2Use in source · 1 passage
Search aliases: x^2
Use in source · Read passage · rank4.tex, line 166
Its leading grade is outside the centralizing line, so u represents the nontrivial coset of P/J. Since u^2=1+x^2 in characteristic two, we have u 2 in U_B (2m) subset U_B (m+1) subset H.
Exact mathematical passage
u^2=1+x^2
X^nUse in source · 3 passages
Search aliases: X^n
Use in source · Read passage · Paper_XI_Traditional_Final.tex, line 529
[Frobenius cycle polynomial] The Frobenius transfer matrix has characteristic polynomial X^n-a and determinant detτ(Φ)=(-1)^n-1a.
Exact mathematical passage
X^n-a
X^{N-1}Use in source · 1 passage
Search aliases: X^N-1
Use in source · Read passage · dyadic_synthesis.tex, line 80
At an inductive field step d≥2, set n=2^d-1q_d, R=2^d-2q_d-1. Then n>2R and -2R>-n-q_1. BK Remark 2.2.7 gives the maximal field generated by the actual displayed matrix. Literal truncation and the filtered lift give the defining sequence. Nonzero native trace also proves separability: the irreducible characteristic polynomial has a nonzero coefficient of X^N-1, so its derivative is nonzero. See pathtower/DYADIC_AS_TOWER167.md for all field and filtration details.
Exact mathematical passage
X^{N-1}
X^pUse in source · 2 passages
Search aliases: X^p
Use in source · Read passage · odd_synthesis.tex, line 60
recursive base (L,t_L), introduce E=L(α), α^p-α=t_L^-q_d. This is a separable totally ramified extension of degree p, with v_E(α)=-q_d and Tr_E/Lα=0. Choose 1≤ b<p with q_db≡-1pmod p, set a=(q_db+1)/p, and put z=t_L^aα^b. Then z is an explicit uniformizer. Write its Eisenstein polynomial as f(X)=X^p+⋯+a_0.
Exact mathematical passage
f(X)=X^p+\cdots+a_0
X^{p-1}Use in source · 1 passage
Search aliases: X^p-1
Use in source · Read passage · odd_tower.tex, line 59
Then v_E(z)=pa-qb=1, so z is a uniformizer and E=L(z). Its monic minimal polynomial f(X)=X^p+a_p-1X^p-1+⋯+a_0 is Eisenstein. It is an explicit polynomial: compute the product over the p conjugates
Exact mathematical passage
f(X)=X^p+a_{p-1}X^{p-1}+\cdots+a_0
X^rUse in source · 1 passage
Search aliases: X^r
Use in source · Read passage · Paper_XI_Traditional_Final.tex, line 653
For nontrivial inertia, the invariant space is zero. In the trivial case, Frobenius on the invariant packet is the companion matrix of X^r-a. Substituting X=u^-1 and multiplying by u^r gives 1-a u^r.
Exact mathematical passage
X^r-a
X^{r_i}Use in source · 1 passage
Search aliases: X^r_i
Use in source · Read passage · Paper_XI_Traditional_Final.tex, line 678
A nonzero fixed vector must lie in an inertia trivial packet, so c_i=0. On that packet, Frobenius is a companion cycle with characteristic polynomial X^r_i-a_i. Its fixed space is the eigenspace for eigenvalue one. This eigenspace is one dimensional exactly when a_i=1, and it is zero otherwise. Summing over the refined packets proves the formula.
Exact mathematical passage
X^{r_i}-a_i
y'Use in source · 1 passage
Search aliases: y'
Use in source · Paper_XI_Visual_Guide_Final.tex, line 327
lessonPART II / FINITE WILD COMPARISONTwo dyadic refinements can have opposite signsOriginal Section 13 plate10_quadratic_planesThe entries are exact values in F_2; both polar forms are xy'+x'y. A quadratic refinement satisfies q(x+y)+q(x)+q(y)=b(x,y). On a plane, q_0(x,y)=xy, q_1(x,y)=xy+x+y. For q_0 there are three zeros and one one, so its Gauss sum is 2. For q_1 there is one zero and three ones, so its sum is -2. The alternating forms agree but the Arf invariants differ. ∑_x∈ V(-1)^q(x)=(-1)…
Exact mathematical passage
xy'+x'y
y^{-1}Use in source · 1 passage
Search aliases: y^-1
Use in source · Read passage · Paper_XI_Traditional_Final.tex, line 734
Compute absG(χ,ψ)^2 = ∑_x,y∈ k^× χ(xy^-1)ψ(x-y). Write x=ty. The inner sum over y equals q-1 when t=1 and equals -1 otherwise. Since ∑_t∈ k^×χ(t)=0, the result is q.
Exact mathematical passage
\abs{G(\chi,\psi)}^2 = \sum_{x,y\in k^{\times}} \chi(xy^{-1})\psi(x-y).
ZUse in source · 5 passages
Search aliases: Z
Use in source · Read passage · Paper_XI_Traditional_Final.tex, line 799
Let p be odd. Let X be an r dimensional vector space over F_p, and put V=X⊕ X^*. Define the alternating form b((x,ξ),(y,η)) = η(x)-ξ(y). Let Z= F_p, and define the Heisenberg group H(V)=V× Z with multiplication (v,z)(w,t) = (v+w,z+t+frac12 b(v,w)). Fix the additive character ψ(z)= e^2π iz/p.
Exact mathematical passage
Z=\FF_p
z^2Use in source · 3 passages
Search aliases: z^2
Use in source · Read passage · dyadic_synthesis.tex, line 50
At a recursive coefficient field L of characteristic two, with residue field F_2 and uniformizer t_L, put E=L(α), α^2-α=t_L^-q, a=(q+1)/2, z=t_L^aα. The extension is separable and totally ramified of degree two; z is a uniformizer and z^2+t_L^az+t_L=0, Tr_E/L(α)=1. For constant-coefficient projection P_0 in the basis 1,z, the normalized trace corestriction and exact filtered right inverse are κ=α^-1, s=κpr, s(P_0)=t_L/z^2, Lift_E(C)=(z^2/t_L)CP_0. They satisfy s(Lift_E(C))=C, preserve the exact …
Exact mathematical passage
z^2+t_L^az+t_L=0,\qquad \operatorname{Tr}_{E/L}(\alpha)=1.
z^pUse in source · 1 passage
Search aliases: z^p
Use in source · Read passage · odd_synthesis.tex, line 65
Let P_0 extract the constant coefficient in the basis 1,z,…,z^p-1. The normalized corestriction s=κpr, where κ=-f'(z)/z^p-1, satisfies u_0=s(P_0)=a_0/z^p∈ O_E^×, Lift_E(C)=u_0^-1CP_0, s(Lift_E(C))=C. The lift preserves the exact filtration valuation. Its crucial additional identity is tr_L(Lift_E(C)) =Tr_E/L (κ^-1tr_E(C)). Define B(L,t_L;h_1,…,h_d) =α I_p^d-1+ Lift_E(B(E,z;h_1,…,h_d-1)), with the degree-p multiplication matrix as the base case. Induction proves trace zero at every native rank. T…
Exact mathematical passage
u_0=s(P_0)=a_0/z^p\in\mathcal O_E^\times, \qquad \operatorname{Lift}_E(C)=u_0^{-1}CP_0, \qquad s(\operatorname{Lift}_E(C))=C.
z^{p-1}Use in source · 3 passages
Search aliases: z^p-1
Use in source · Read passage · odd_synthesis.tex, line 63
Let P_0 extract the constant coefficient in the basis 1,z,…,z^p-1. The normalized corestriction s=κpr, where κ=-f'(z)/z^p-1, satisfies u_0=s(P_0)=a_0/z^p∈ O_E^×, Lift_E(C)=u_0^-1CP_0, s(Lift_E(C))=C. The lift preserves the exact filtration valuation. Its crucial additional identity is tr_L(Lift_E(C)) =Tr_E/L (κ^-1tr_E(C)). Define B(L,t_L;h_1,…,h_d) =α I_p^d-1+ Lift_E(B(E,z;h_1,…,h_d-1)), with the degree-p multiplication matrix as the base case. Induction proves trace zero at every native rank. T…
Exact mathematical passage
1,z,\ldots,z^{p-1}
z_aUse in source · 3 passages
Search aliases: z_a
Use in source · Read passage · Paper_XI_Traditional_Final.tex, line 971
Let z=∑_a∈ Az_au_a be central. Commuting with u_x gives z_ab_c(x,a)=z_a for every x. Nondegeneracy forces z_a=0 for every nonzero a. Thus the center is C. To prove semisimplicity directly, choose a linear projection P onto a submodule of a finite dimensional C^c[A]-module and form overline P=frac1abs A∑_a∈ Au_a P u_a^-1. Every summand maps into the submodule and is the identity there. Conjugation by any u_x permutes the summands: the cocycle scalars cancel between each operator and its inverse. …
Exact mathematical passage
z=\sum_{a\in A}z_au_a
\ZZUse in source · 3 passages
Search aliases: ZZ Z mathbb
Use in source · Read passage · Paper_XI_Traditional_Final.tex, line 317
Assume that P_K acts trivially. After passing to a finite quotient, tame inertia is cyclic. Let t generate a quotient of order M, and put ζ_M= e^2π i/M. For j∈ Z/M Z, define P_j = frac1M ∑_a=0^M-1 ζ_M^-jar(t)^a.
Exact mathematical passage
j\in\ZZ/M\ZZ
|^{-1}Use in source · 2 passages
Search aliases: |^-1
Use in source · Paper_XI_Visual_Guide_Final.tex, line 138
For a proper parabolic P=LU, averaging over its unipotent radical gives e_U=|U|^-1∑_u∈ Uu, e_Ue_θ=e_θ e_U=0. The average selects U-fixed vectors; the cuspidal block has none. boundaryThe existence and classification of κ_θ are classical inputs. The idempotent calculation explains its exact finite selector.
Exact mathematical passage
e_U=|U|^{-1}\sum_{u\in U}u,\qquad e_Ue_\theta=e_\theta e_U=0.
|^2Use in source · 1 passage
Search aliases: |^2
Use in source · Paper_XI_Visual_Guide_Final.tex, line 210
lessonPART I / FINITE TAME COORDINATESFinite Gauss phases are unit scalarsOriginal Section 6 For a nontrivial additive character ψ of F_q and a nontrivial multiplicative character χ of F_q^×, define G(χ,ψ)=∑_x∈ F_q^×χ(x)ψ(x), γ(χ,ψ)=q^-1/2G(χ,ψ). The exact identities are |G(χ,ψ)|^2=q, γ(χ,ψ)γ(χ^-1,ψ)=χ(-1). The magnitude calculation substitutes x=ty in the double sum. The inner sum is q-1 for t=1 and -1 otherwise; multiplicative-character orthogonality supplies the result.
Exact mathematical passage
|G(\chi,\psi)|^2=q,\qquad \gamma(\chi,\psi)\gamma(\chi^{-1},\psi)=\chi(-1).
|_DUse in source · 1 passage
Search aliases: |_D
Use in source · Read passage · dyadic_synthesis.tex, line 112
minant unit character to multiplicity two. BK 3.6.13(ii) and 3.6.14 give, on scalar units, Θ_j(uI)=θ_j(u^2)=1. For odd critical level r, use m=lfloor r/2rfloor and D=H^m+1(β)=H^m+1(γ). The endpoint is valid because 2(m+1)-r=1: the additive-character cross term lies in the radical on which the matrix additive character is trivial. BK 3.3.18 supplies ϑ=Θ_j|_D ψ_β-γ. Integral trace makes it trivial on D∩ U_F^1.
Exact mathematical passage
\vartheta=\Theta_j|_D\,\psi_{\beta-\gamma}
|_{H_0}Use in source · 1 passage
Search aliases: |_H_0
Use in source · Read passage · dyadic_synthesis.tex, line 301
[P,J]⊂ U_A^m+1 lies in the forced additive-character domain, so λ has values {±1}. Put H_0=kerλ. Then P/H_0≃ F_2^2, E=Ind_J^Pθ is irreducible of dimension 2. The actual represented generators are a diagonal sign and an off-diagonal matrix that anticommute. Thus their commutator form is nondegenerate, and θ|_H_0 is the full character of an actual new arithmetic Heisenberg pair. No abstract operators create that group action.
Exact mathematical passage
\theta|_{H_0}
|_MUse in source · 1 passage
Search aliases: |_M
Use in source · Read passage · Paper_XI_Traditional_Final.tex, line 1785
[Breakwise phase product] In odd residue characteristic, let g∈Sp(V) preserve every break packet and suppose its restriction to M=im(g-I) is fixed point free. Put g_s=g|_V_s and M_s=im(g_s-I). The moving restrictions are symplectic, and Φ_p(g|_M,M) = ∏_s∈ SΦ_p(g_s|_M_s,M_s). In residue characteristic two, the analogous normalized Gauss sign is the product of the signs determined by the breakwise quadratic refinements.
Exact mathematical passage
\boxed{ \Phi_p(g|_M,M) = \prod_{s\in S}\Phi_p(g_s|_{M_s},M_s). }
|_{M_s}Use in source · 1 passage
Search aliases: |_M_s
Use in source · Read passage · Paper_XI_Traditional_Final.tex, line 1785
[Breakwise phase product] In odd residue characteristic, let g∈Sp(V) preserve every break packet and suppose its restriction to M=im(g-I) is fixed point free. Put g_s=g|_V_s and M_s=im(g_s-I). The moving restrictions are symplectic, and Φ_p(g|_M,M) = ∏_s∈ SΦ_p(g_s|_M_s,M_s). In residue characteristic two, the analogous normalized Gauss sign is the product of the signs determined by the breakwise quadratic refinements.
Exact mathematical passage
\boxed{ \Phi_p(g|_M,M) = \prod_{s\in S}\Phi_p(g_s|_{M_s},M_s). }
|_NUse in source · 1 passage
Search aliases: |_N
Use in source · Paper_XI_Visual_Guide_Final.tex, line 266
lessonPART II / FINITE WILD COMPARISONClifford multiplicity is the correct action spaceOriginal Section 9 Supply finite groups Ntriangleleft G, elementary abelian quotient A=G/N, and irreducibles Π and ρ, with ρ fixed up to isomorphism by G and Π|_N isotypic. The multiplicity space is M_ρ=Hom_N(W,X), W⊗ M_ρ≃ X. For a section s:A→ G and intertwiners I_a of ρ, write Π(s(a))=I_a⊗ T_a. The operators T_a give a projective action on M_ρ. Section and intertwiner changes alter it only by scalar changes …
Exact mathematical passage
\Pi|_N
|_PUse in source · 2 passages
Search aliases: |_P
Use in source · Read passage · dyadic_synthesis.tex, line 349
Consequently M=K^× P is an actual open group compact modulo the center, N=K^× J is normal of index two in M, and Σ=Ind_ N^ MΛ, Σ|_P≃ E, π≃c - Ind_ M^GΣ. Restriction to P is irreducible, so Σ is irreducible. Induction in stages proves the last equality for the original Λ, including its fixed central character and noncompact root. This is a genuine enlarged inducing representation; it changes neither the candidate nor its original canonical beta-extension data.
Exact mathematical passage
\Sigma=\operatorname{Ind}_{\mathcal N}^{\mathcal M}\Lambda, \qquad \Sigma|_P\simeq E, \qquad \boxed{\pi\simeq\operatorname{c\!-\!Ind}_{\mathcal M}^G\Sigma.}
|_{U_F^1}Use in source · 1 passage
Search aliases: |_U_F^1
Use in source · Read passage · odd_synthesis.tex, line 32
For this actual stratum there exists a full simple character θ with θ|_U_F^1=1. Retaining its canonical beta extension, the residue character and noncompact extension can be selected so that the resulting irreducible admissible supercuspidal π has any prescribed smooth central quasicharacter ω trivial on U_F^1. These completion choices must be free or already compatible; incompatible immutably supplied choices are not replaced.
Exact mathematical passage
\theta|_{U_F^1}=1
|_{V_j}Use in source · 2 passages
Search aliases: |_V_j
Use in source · Read passage · Paper_XI_Traditional_Final.tex, line 1340
[Orthogonal multiplicativity] Suppose V=V_1mathop⊥limits⋯mathop⊥limits V_s and g preserves every summand. If no g|_V_j has a fixed vector, then Φ_p(g,V) = ∏_j=1^s Φ_p(g|_V_j,V_j).
Exact mathematical passage
g|_{V_j}
|_{V_s}Use in source · 1 passage
Search aliases: |_V_s
Use in source · Read passage · Paper_XI_Traditional_Final.tex, line 1784
[Breakwise phase product] In odd residue characteristic, let g∈Sp(V) preserve every break packet and suppose its restriction to M=im(g-I) is fixed point free. Put g_s=g|_V_s and M_s=im(g_s-I). The moving restrictions are symplectic, and Φ_p(g|_M,M) = ∏_s∈ SΦ_p(g_s|_M_s,M_s). In residue characteristic two, the analogous normalized Gauss sign is the product of the signs determined by the breakwise quadratic refinements.
Exact mathematical passage
g_s=g|_{V_s}
|_{W_{K_f}}Use in source · 1 passage
Search aliases: |_W_K_f
Use in source · Read passage · Paper_XI_Traditional_Final.tex, line 383
[Unramified twist] Let ν be an unramified character of W_K. Then τ_ξ⊗ν ≅ Ind_W_K_f^W_K ( ξ ν|_W_K_f ). The tame inertia orbit is unchanged. If the original Frobenius cycle has closing scalar a, the twisted cycle has closing scalar aν(Φ)^f.
Exact mathematical passage
\tau_\xi\otimes\nu \cong \Ind_{W_{K_f}}^{W_K} \left( \xi\,\nu|_{W_{K_f}} \right).
\pmUse in source · 5 passages
Search aliases: ± pm
Use in source · Read passage · Paper_XI_Traditional_Final.tex, line 2157
mon arithmetic assembly carrier; it is not renamed as the original maximal-simple-character pair. For an independently fixed native quadratic marking, a preassigned embedding must satisfy q_W(f(v))=q_rm src(v). At equal capacity, a free embedding requires equal Arf invariant. Spare capacity for d≥4 admits either source Arf type when the embedding is free. Retaining only the central {±1} would erase the distinction between whole marked finite extensions such as D_8 and Q_8.
Exact mathematical passage
\{\pm1\}
\alphaUse in source · 20 passages
Search aliases: α alpha
Use in source · Read passage · dyadic_synthesis.tex, line 46
At a recursive coefficient field L of characteristic two, with residue field F_2 and uniformizer t_L, put E=L(α), α^2-α=t_L^-q, a=(q+1)/2, z=t_L^aα. The extension is separable and totally ramified of degree two; z is a uniformizer and z^2+t_L^az+t_L=0, Tr_E/L(α)=1. For constant-coefficient projection P_0 in the basis 1,z, the normalized trace corestriction and exact filtered right inverse are κ=α^-1, s=κpr, s(P_0)=t_L/z^2, Lift_E(C)=(z^2/t_L)CP_0. They satisfy s(Lift_E(C))=C, preserve the exact …
Exact mathematical passage
E=L(\alpha),\qquad \alpha^2-\alpha=t_L^{-q},\qquad a=(q+1)/2,\qquad z=t_L^a\alpha.
\alpha^{-1}Use in source · 1 passage
Search aliases: α^-1 alpha
Use in source · Read passage · dyadic_synthesis.tex, line 53
er t_L, put E=L(α), α^2-α=t_L^-q, a=(q+1)/2, z=t_L^aα. The extension is separable and totally ramified of degree two; z is a uniformizer and z^2+t_L^az+t_L=0, Tr_E/L(α)=1. For constant-coefficient projection P_0 in the basis 1,z, the normalized trace corestriction and exact filtered right inverse are κ=α^-1, s=κpr, s(P_0)=t_L/z^2, Lift_E(C)=(z^2/t_L)CP_0. They satisfy s(Lift_E(C))=C, preserve the exact filtration valuation, and give the identity tr_LLift_E(C) =Tr_E/L (αtr_E C). The actual recurs…
Exact mathematical passage
\kappa=\alpha^{-1},\qquad s=\kappa\operatorname{pr},\qquad s(P_0)=t_L/z^2,\qquad \operatorname{Lift}_E(C)=(z^2/t_L)CP_0.
\alpha^2Use in source · 1 passage
Search aliases: α^2 alpha
Use in source · Read passage · dyadic_synthesis.tex, line 46
At a recursive coefficient field L of characteristic two, with residue field F_2 and uniformizer t_L, put E=L(α), α^2-α=t_L^-q, a=(q+1)/2, z=t_L^aα. The extension is separable and totally ramified of degree two; z is a uniformizer and z^2+t_L^az+t_L=0, Tr_E/L(α)=1. For constant-coefficient projection P_0 in the basis 1,z, the normalized trace corestriction and exact filtered right inverse are κ=α^-1, s=κpr, s(P_0)=t_L/z^2, Lift_E(C)=(z^2/t_L)CP_0. They satisfy s(Lift_E(C))=C, preserve the exact …
Exact mathematical passage
E=L(\alpha),\qquad \alpha^2-\alpha=t_L^{-q},\qquad a=(q+1)/2,\qquad z=t_L^a\alpha.
\alpha^bUse in source · 2 passages
Search aliases: α^b alpha
Use in source · Read passage · odd_synthesis.tex, line 59
At a recursive base (L,t_L), introduce E=L(α), α^p-α=t_L^-q_d. This is a separable totally ramified extension of degree p, with v_E(α)=-q_d and Tr_E/Lα=0. Choose 1≤ b<p with q_db≡-1pmod p, set a=(q_db+1)/p, and put z=t_L^aα^b. Then z is an explicit uniformizer. Write its Eisenstein polynomial as f(X)=X^p+⋯+a_0.
Exact mathematical passage
z=t_L^a\alpha^b
\alpha^pUse in source · 3 passages
Search aliases: α^p alpha
Use in source · Read passage · odd_synthesis.tex, line 54
At a recursive base (L,t_L), introduce E=L(α), α^p-α=t_L^-q_d. This is a separable totally ramified extension of degree p, with v_E(α)=-q_d and Tr_E/Lα=0. Choose 1≤ b<p with q_db≡-1pmod p, set a=(q_db+1)/p, and put z=t_L^aα^b. Then z is an explicit uniformizer. Write its Eisenstein polynomial as f(X)=X^p+⋯+a_0.
Exact mathematical passage
E=L(\alpha),\qquad \alpha^p-\alpha=t_L^{-q_d}.
\beta_iUse in source · 1 passage
Search aliases: β_i beta
Use in source · Read passage · odd_capacity.tex, line 133
For H, BK 3.1.9(ii) states that H^a(β_i)=H^a(β_i+1) as soon as
Exact mathematical passage
H^a(\beta_i)=H^a(\beta_{i+1})
\beta_{i+1}Use in source · 1 passage
Search aliases: β_i+1 beta
Use in source · Read passage · odd_capacity.tex, line 133
For H, BK 3.1.9(ii) states that H^a(β_i)=H^a(β_i+1) as soon as
Exact mathematical passage
H^a(\beta_i)=H^a(\beta_{i+1})
\GammaUse in source · 2 passages
Search aliases: Γ Gamma
Use in source · Read passage · Paper_XI_Traditional_Final.tex, line 1611
Let Γ be a connected finite graph. At every vertex v, place finite dimensional complex vector spaces V_v and W_v of the same dimension. For every oriented edge e:u→ v, let A_e:V_u⟶ V_v, B_e:W_u⟶ W_v be invertible. The reversed edge carries the inverse map.
Exact mathematical passage
\Gamma
\deltaUse in source · 13 passages
Search aliases: δ delta
Use in source · Read passage · Paper_XI_Traditional_Final.tex, line 1234
[Cyclic discrepancy character] Assume that the two projective stabilizer actions have been linearized to honest representations of the cyclic group C. Then there is a character δ:C→ C^× such that UΩ_ρ(g) = δ(g)Ω_θ(g)U for every g∈ C.
Exact mathematical passage
\delta:C\to\CC^{\times}
\delta_{jk}Use in source · 1 passage
Search aliases: δ_jk delta
Use in source · Paper_XI_Visual_Guide_Final.tex, line 92
lessonPART I / FINITE TAME COORDINATESFourier projectors isolate tame inertiaOriginal Section 3; geometric Frobenius convention Let t generate a finite tame quotient of order M, with (q,M)=1. For j∈ Z/M Z define P_j=frac1M∑_a=0^M-1ζ_M^-jar(t)^a, ζ_M=e^2π i/M. The finite Fourier identities give P_jP_k=δ_jkP_j and ∑_jP_j=I. Thus P_jV is the ζ_M^j character space for r(t).
Exact mathematical passage
P_jP_k=\delta_{jk}P_j
\zetaUse in source · 5 passages
Search aliases: ζ zeta
Use in source · Read passage · Paper_XI_Traditional_Final.tex, line 317
Assume that P_K acts trivially. After passing to a finite quotient, tame inertia is cyclic. Let t generate a quotient of order M, and put ζ_M= e^2π i/M. For j∈ Z/M Z, define P_j = frac1M ∑_a=0^M-1 ζ_M^-jar(t)^a.
Exact mathematical passage
\zeta_M=\e^{2\pi i/M}
\zeta_3Use in source · 1 passage
Search aliases: ζ_3 zeta
Use in source · Read passage · odd_synthesis.tex, line 220
The old fixed-field central argument genuinely failed. At p=3, (h_1,h_2)=(1,5), its naturally transferred full character takes the value ζ_3≠1 on 1+t^2, although the desired wild center is trivial. A shallow character choice cannot repair a forced value. The two-stage changing-field repair is retained in pathcentral/CENTRAL_REPAIR166.md; the trace-zero tower above resolves this central issue uniformly in every length.
Exact mathematical passage
\zeta_3\ne1
\zeta_MUse in source · 2 passages
Search aliases: ζ_M zeta
Use in source · Read passage · Paper_XI_Traditional_Final.tex, line 317
Assume that P_K acts trivially. After passing to a finite quotient, tame inertia is cyclic. Let t generate a quotient of order M, and put ζ_M= e^2π i/M. For j∈ Z/M Z, define P_j = frac1M ∑_a=0^M-1 ζ_M^-jar(t)^a.
Exact mathematical passage
\zeta_M=\e^{2\pi i/M}
\zeta_M^{-ja}Use in source · 2 passages
Search aliases: ζ_M^-ja zeta
Use in source · Read passage · Paper_XI_Traditional_Final.tex, line 318
Assume that P_K acts trivially. After passing to a finite quotient, tame inertia is cyclic. Let t generate a quotient of order M, and put ζ_M= e^2π i/M. For j∈ Z/M Z, define P_j = frac1M ∑_a=0^M-1 ζ_M^-jar(t)^a.
Exact mathematical passage
P_j = \frac1M \sum_{a=0}^{M-1} \zeta_M^{-ja}r(t)^a.
\zeta_M^jUse in source · 1 passage
Search aliases: ζ_M^j zeta
Use in source · Paper_XI_Visual_Guide_Final.tex, line 92
ME COORDINATESFourier projectors isolate tame inertiaOriginal Section 3; geometric Frobenius convention Let t generate a finite tame quotient of order M, with (q,M)=1. For j∈ Z/M Z define P_j=frac1M∑_a=0^M-1ζ_M^-jar(t)^a, ζ_M=e^2π i/M. The finite Fourier identities give P_jP_k=δ_jkP_j and ∑_jP_j=I. Thus P_jV is the ζ_M^j character space for r(t).
Exact mathematical passage
\zeta_M^j
\eta_{L,m}Use in source · 1 passage
Search aliases: η_L,m eta
Use in source · Read passage · Paper_XI_Traditional_Final.tex, line 580
[Greatest common divisor and least common multiple packets] There is a decomposition τ_ξ⊗τ_η ≅ ⊕_i=0^d-1 Ind_W_K_L^W_K ( ξ_L,nη_L,m^Φ^i ), where ξ_L,n and η_L,m are the norm lifts to K_L.
Exact mathematical passage
\eta_{L,m}
\eta_{\rm src}Use in source · 1 passage
Search aliases: η_rm src eta rm
Use in source · Paper_XI_Visual_Guide_Final.tex, line 500
An orthogonal splitting W=V_rm src⊥ V_rm src^⊥ gives η_W≃η_rm src⊗η_⊥. A maximal-isotropic average on the complementary Heisenberg factor selects a one-dimensional representation line. Tensoring its projector with the identity gives a rank-N image-algebra projector. The retained induction/recovery theorem recovers the same Λ and then the same π. boundaryFor p>3, genuinely supplied plane data are required: a native minimal rank-p quotient has dimension p-1 and is not automatically a plane. The co…
Exact mathematical passage
\eta_W\simeq\eta_{\rm src}\otimes\eta_{\perp}.
\eta_{\mathrm{src}}Use in source · 1 passage
Search aliases: η_src eta mathrm
Use in source · Read passage · odd_synthesis.tex, line 155
Write W=V_src⊥ V_src^⊥. Finite Stone--von Neumann gives η_W≃η_src⊗η_⊥. Choose an isotropic eigenspace giving a line in the complementary factor. Its actual Heisenberg image-algebra projector e has rank N. The retained induction/idempotent theorem recovers the same inducing representation Λ from eΛ, and then the same π by compact induction. No candidate is changed at recovery.
Exact mathematical passage
\eta_W\simeq\eta_{\mathrm{src}}\otimes\eta_\perp
\eta_WUse in source · 2 passages
Search aliases: η_W eta
Use in source · Read passage · odd_synthesis.tex, line 155
Write W=V_src⊥ V_src^⊥. Finite Stone--von Neumann gives η_W≃η_src⊗η_⊥. Choose an isotropic eigenspace giving a line in the complementary factor. Its actual Heisenberg image-algebra projector e has rank N. The retained induction/idempotent theorem recovers the same inducing representation Λ from eΛ, and then the same π by compact induction. No candidate is changed at recovery.
Exact mathematical passage
\eta_W\simeq\eta_{\mathrm{src}}\otimes\eta_\perp
\eta_\thetaUse in source · 4 passages
Search aliases: η_θ eta theta
Use in source · Read passage · Paper_XI_Traditional_Final.tex, line 1170
[Type side Heisenberg quotient] The pairing h_θ is nondegenerate and alternating. There is one irreducible representation η_θ of J_θ^1 containing θ, and its dimension is sqrtabsJ_θ^1/H_θ^1.
Exact mathematical passage
\eta_\theta
\eta_\perpUse in source · 1 passage
Search aliases: η_⊥ eta perp
Use in source · Read passage · odd_synthesis.tex, line 155
Write W=V_src⊥ V_src^⊥. Finite Stone--von Neumann gives η_W≃η_src⊗η_⊥. Choose an isotropic eigenspace giving a line in the complementary factor. Its actual Heisenberg image-algebra projector e has rank N. The retained induction/idempotent theorem recovers the same inducing representation Λ from eΛ, and then the same π by compact induction. No candidate is changed at recovery.
Exact mathematical passage
\eta_W\simeq\eta_{\mathrm{src}}\otimes\eta_\perp
\eta_{\perp}Use in source · 1 passage
Search aliases: η_⊥ eta perp
Use in source · Paper_XI_Visual_Guide_Final.tex, line 500
An orthogonal splitting W=V_rm src⊥ V_rm src^⊥ gives η_W≃η_rm src⊗η_⊥. A maximal-isotropic average on the complementary Heisenberg factor selects a one-dimensional representation line. Tensoring its projector with the identity gives a rank-N image-algebra projector. The retained induction/recovery theorem recovers the same Λ and then the same π. boundaryFor p>3, genuinely supplied plane data are required: a native minimal rank-p quotient has dimension p-1 and is not automatically a plane. The co…
Exact mathematical passage
\eta_W\simeq\eta_{\rm src}\otimes\eta_{\perp}.
\ThetaUse in source · 3 passages
Search aliases: Θ Theta
Use in source · Read passage · dyadic_synthesis.tex, line 106
For a later native step, first transfer the full beta-determinant unit character to multiplicity two. BK 3.6.13(ii) and 3.6.14 give, on scalar units, Θ_j(uI)=θ_j(u^2)=1. For odd critical level r, use m=lfloor r/2rfloor and D=H^m+1(β)=H^m+1(γ). The endpoint is valid because 2(m+1)-r=1: the additive-character cross term lies in the radical on which the matrix additive character is trivial. BK 3.3.18 supplies ϑ=Θ_j|_D ψ_β-γ. Integral trace makes it trivial on D∩ U_F^1.
Exact mathematical passage
\Theta_j(uI)=\theta_j(u^2)=1.
\varthetaUse in source · 1 passage
Search aliases: ϑ vartheta
Use in source · Read passage · dyadic_synthesis.tex, line 112
minant unit character to multiplicity two. BK 3.6.13(ii) and 3.6.14 give, on scalar units, Θ_j(uI)=θ_j(u^2)=1. For odd critical level r, use m=lfloor r/2rfloor and D=H^m+1(β)=H^m+1(γ). The endpoint is valid because 2(m+1)-r=1: the additive-character cross term lies in the radical on which the matrix additive character is trivial. BK 3.3.18 supplies ϑ=Θ_j|_D ψ_β-γ. Integral trace makes it trivial on D∩ U_F^1.
Exact mathematical passage
\vartheta=\Theta_j|_D\,\psi_{\beta-\gamma}
\theta^gUse in source · 3 passages
Search aliases: θ^g theta
Use in source · Read passage · Paper_XI_Traditional_Final.tex, line 2124
At rank two, with q=2m+1, enlarge J=U_K^1U_A^m+1 to P=U_K^1U_A^m. The actual character ratio λ=θ^g/θ for g∉ J is nontrivial and sign-valued. With H_0=kerλ, the quotient is P/H_0≃ F_2^2, and Ind_J^Pθ is an irreducible two-dimensional Heisenberg carrier ([source reference]). At rank four, the actual outer quadratic centralizer gives P=J U_B^m, [P:J]=2, and P/H_0≃ F_2^4, with carrier Ind_J^Pη of dimension four ([source reference]). The finite scalar center can be larger than μ_2; no literal extrasp…
Exact mathematical passage
\lambda=\theta^g/\theta
\theta^uUse in source · 2 passages
Search aliases: θ^u theta
Use in source · Read passage · dyadic_synthesis.tex, line 322
eEnd_E_0(F^4). The odd lower critical level is r=3h_1=2m+1; the half leading level is a=3h_2>r. Define P=J U_B^m, where J=J^1 is the actual compact type group. The filtered centralizer estimate for the actual final field, without assuming E_0⊂ K, proves that P is a compact group with J normal of index two. For a concrete new coset representative u, λ=θ^u/θ on the actual H=H^1 is nontrivial with values {±1}. Put H_0=kerλ⊂ H. The calculation proves P/H_0≃ F_2^4, E=Ind_J^Pη is irreducible of dimens…
Exact mathematical passage
\lambda=\theta^u/\theta
\theta_iUse in source · 4 passages
Search aliases: θ_i theta
Use in source · Read passage · odd_synthesis.tex, line 144
Choose a labelled symplectic embedding V_src→ S. For each original full character form P_i=( C^×× J_i^1)/ {(θ_i(h)^-1,h):h∈ H_i^1}. The balanced product and the corresponding host restriction have the same alternating commutator. The center-fixing extension isomorphism, obtained with matched Darboux lifts and complex roots, therefore preserves every supplied value θ_i(h), including the shallow values.
Exact mathematical passage
\mathcal P_i=(\mathbb C^\times\times J_i^1)/ \{(\theta_i(h)^{-1},h):h\in H_i^1\}.
\Theta_jUse in source · 3 passages
Search aliases: Θ_j Theta
Use in source · Read passage · dyadic_synthesis.tex, line 106
For a later native step, first transfer the full beta-determinant unit character to multiplicity two. BK 3.6.13(ii) and 3.6.14 give, on scalar units, Θ_j(uI)=θ_j(u^2)=1. For odd critical level r, use m=lfloor r/2rfloor and D=H^m+1(β)=H^m+1(γ). The endpoint is valid because 2(m+1)-r=1: the additive-character cross term lies in the radical on which the matrix additive character is trivial. BK 3.3.18 supplies ϑ=Θ_j|_D ψ_β-γ. Integral trace makes it trivial on D∩ U_F^1.
Exact mathematical passage
\Theta_j(uI)=\theta_j(u^2)=1.
\theta_jUse in source · 3 passages
Search aliases: θ_j theta
Use in source · Read passage · dyadic_synthesis.tex, line 106
For a later native step, first transfer the full beta-determinant unit character to multiplicity two. BK 3.6.13(ii) and 3.6.14 give, on scalar units, Θ_j(uI)=θ_j(u^2)=1. For odd critical level r, use m=lfloor r/2rfloor and D=H^m+1(β)=H^m+1(γ). The endpoint is valid because 2(m+1)-r=1: the additive-character cross term lies in the radical on which the matrix additive character is trivial. BK 3.3.18 supplies ϑ=Θ_j|_D ψ_β-γ. Integral trace makes it trivial on D∩ U_F^1.
Exact mathematical passage
\Theta_j(uI)=\theta_j(u^2)=1.
\iota^{-1}Use in source · 1 passage
Search aliases: ι^-1 iota
Use in source · Read passage · Paper_XI_Traditional_Final.tex, line 1325
[Symplectic invariance] If ι:(V,b)⟶(V',b') is a symplectic isometry and g'=ι gι^-1, then Φ_p(g,V)=Φ_p(g',V').
Exact mathematical passage
g'=\iota g\iota^{-1}
\iota_HUse in source · 4 passages
Search aliases: ι_H iota
Use in source · Read passage · Paper_XI_Traditional_Final.tex, line 1218
[Restricted Heisenberg bridge] A restricted bridge datum determines an isomorphism between the two Heisenberg representations, unique up to a nonzero scalar. More precisely, there is an invertible map U: H_ρ⟶ H_θ such that Uω_ρ(h) = ω_θ(ι_H(h))U for every Heisenberg element h, and every other such map is a scalar multiple of U.
Exact mathematical passage
U\omega_\rho(h) = \omega_\theta(\iota_H(h))U
\kappaUse in source · 15 passages
Search aliases: κ kappa
Use in source · Read passage · Paper_XI_Traditional_Final.tex, line 463
[Regular elliptic torus characters] A regular character θ:T_n→ C^× determines an irreducible cuspidal representation κ_θ of G_n. Frobenius conjugate characters determine isomorphic representations, and every irreducible cuspidal representation of G_n occurs in this way.
Exact mathematical passage
\kappa_\theta
\kappa^{-1}Use in source · 2 passages
Search aliases: κ^-1 kappa
Use in source · Read passage · odd_synthesis.tex, line 71
(P_0 extract the constant coefficient in the basis 1,z,…,z^p-1. The normalized corestriction s=κpr, where κ=-f'(z)/z^p-1, satisfies u_0=s(P_0)=a_0/z^p∈ O_E^×, Lift_E(C)=u_0^-1CP_0, s(Lift_E(C))=C. The lift preserves the exact filtration valuation. Its crucial additional identity is tr_L(Lift_E(C)) =Tr_E/L (κ^-1tr_E(C)). Define B(L,t_L;h_1,…,h_d) =α I_p^d-1+ Lift_E(B(E,z;h_1,…,h_d-1)), with the degree-p multiplication matrix as the base case. Induction proves trace zero at every native rank. This…
Exact mathematical passage
\operatorname{tr}_L(\operatorname{Lift}_E(C)) =\operatorname{Tr}_{E/L} \bigl(\kappa^{-1}\operatorname{tr}_E(C)\bigr).
\kappa_\thetaUse in source · 9 passages
Search aliases: κ_θ kappa theta
Use in source · Read passage · Paper_XI_Traditional_Final.tex, line 463
[Regular elliptic torus characters] A regular character θ:T_n→ C^× determines an irreducible cuspidal representation κ_θ of G_n. Frobenius conjugate characters determine isomorphic representations, and every irreducible cuspidal representation of G_n occurs in this way.
Exact mathematical passage
\kappa_\theta
\LambdaUse in source · 16 passages
Search aliases: Λ Lambda
Use in source · Read passage · Paper_XI_Traditional_Final.tex, line 2062
y). A labelled symplectic embedding gives a center-fixing full scalar-pushout isomorphism. Finite Stone--von Neumann decomposition separates the source factor from its orthogonal complement. Maximal-isotropic averaging on the complementary Heisenberg factor selects a one-dimensional representation line and hence a source-sized sector. The retained induction/idempotent recovery returns the same Λ and π, as proved in [source reference].
Exact mathematical passage
\Lambda
\lambdaUse in source · 21 passages
Search aliases: λ lambda
Use in source · Read passage · Paper_XI_Traditional_Final.tex, line 932
[Scalar choice invariance] Replacing u_a by λ_au_a, with λ_a∈ C^×, changes c by a coboundary and leaves b_c unchanged.
Exact mathematical passage
\lambda_au_a
\lambda^{-1}Use in source · 2 passages
Search aliases: λ^-1 lambda
Use in source · Read passage · rank2.tex, line 204
As g²∈J, conjugation by g² fixes the character theta. Thus λ^g=λ^-1=λ. It follows that
Exact mathematical passage
\lambda^g=\lambda^{-1}=\lambda
\lambda^gUse in source · 1 passage
Search aliases: λ^g lambda
Use in source · Read passage · rank2.tex, line 204
As g²∈J, conjugation by g² fixes the character theta. Thus λ^g=λ^-1=λ. It follows that
Exact mathematical passage
\lambda^g=\lambda^{-1}=\lambda
\lambda^mUse in source · 1 passage
Search aliases: λ^m lambda
Use in source · Read passage · Paper_XI_Traditional_Final.tex, line 1259
Let C=⟨ c⟩ have order m, and let T be a lift of the projective generator. Since c^m=1, the operator T^m is scalar. Choose λ∈ C^× with λ^mT^m=I. Replacing T by λ T gives an honest cyclic representation.
Exact mathematical passage
\lambda^mT^m=I
\lambda^uUse in source · 1 passage
Search aliases: λ^u lambda
Use in source · Read passage · rank4.tex, line 234
First, lambda is invariant under u. Indeed u 2 belongs to J and theta is J-invariant, so λ^u=λ^-1=λ by (3). Consequently H0 is normal in P, and theta0 is P-invariant.
Exact mathematical passage
\lambda^u=\lambda^{-1}=\lambda
\lambda_aUse in source · 5 passages
Search aliases: λ_a lambda
Use in source · Read passage · Paper_XI_Traditional_Final.tex, line 932
[Scalar choice invariance] Replacing u_a by λ_au_a, with λ_a∈ C^×, changes c by a coboundary and leaves b_c unchanged.
Exact mathematical passage
\lambda_au_a
\lambda_{a+b}^{-1}Use in source · 2 passages
Search aliases: λ_a+b^-1 lambda
Use in source · Read passage · Paper_XI_Traditional_Final.tex, line 937
The new cocycle is c'(a,b) = λ_aλ_bλ_a+b^-1c(a,b). The scalar coboundary factor is symmetric in a and b, so it cancels from c'(a,b)c'(b,a)^-1.
Exact mathematical passage
c'(a,b) = \lambda_a\lambda_b\lambda_{a+b}^{-1}c(a,b).
\lambda_bUse in source · 2 passages
Search aliases: λ_b lambda
Use in source · Read passage · Paper_XI_Traditional_Final.tex, line 937
The new cocycle is c'(a,b) = λ_aλ_bλ_a+b^-1c(a,b). The scalar coboundary factor is symmetric in a and b, so it cancels from c'(a,b)c'(b,a)^-1.
Exact mathematical passage
c'(a,b) = \lambda_a\lambda_b\lambda_{a+b}^{-1}c(a,b).
\muUse in source · 13 passages
Search aliases: μ mu
Use in source · Read passage · Paper_XI_Traditional_Final.tex, line 414
[Möbius orbit count] The number of regular Frobenius orbits of length n is c_n(q) = frac1n ∑_dmid n μ(d) (q^n/d-1).
Exact mathematical passage
\boxed{ c_n(q) = \frac1n \sum_{d\mid n} \mu(d) \bigl(q^{n/d}-1\bigr). }
\mu_2Use in source · 6 passages
Search aliases: μ_2 mu
Use in source · Read passage · Paper_XI_Traditional_Final.tex, line 2132
-valued. With H_0=kerλ, the quotient is P/H_0≃ F_2^2, and Ind_J^Pθ is an irreducible two-dimensional Heisenberg carrier ([source reference]). At rank four, the actual outer quadratic centralizer gives P=J U_B^m, [P:J]=2, and P/H_0≃ F_2^4, with carrier Ind_J^Pη of dimension four ([source reference]). The finite scalar center can be larger than μ_2; no literal extraspecial quotient is claimed.
Exact mathematical passage
\mu_2
\mu_pUse in source · 5 passages
Search aliases: μ_p mu
Use in source · Read passage · Paper_XI_Traditional_Final.tex, line 1083
[Intrinsic commutator pairing] The pairing b_ρ(a,b) = c(a,b)c(b,a)^-1 is independent of every choice in [source reference]. It is an alternating bicharacter b_ρ:A× A→μ_p.
Exact mathematical passage
b_\rho:A\times A\to\mu_p.
\nuUse in source · 10 passages
Search aliases: ν nu
Use in source · Read passage · Paper_XI_Traditional_Final.tex, line 382
[Unramified twist] Let ν be an unramified character of W_K. Then τ_ξ⊗ν ≅ Ind_W_K_f^W_K ( ξ ν|_W_K_f ). The tame inertia orbit is unchanged. If the original Frobenius cycle has closing scalar a, the twisted cycle has closing scalar aν(Φ)^f.
Exact mathematical passage
\nu
\nu_aUse in source · 2 passages
Search aliases: ν_a nu
Use in source · Paper_XI_Visual_Guide_Final.tex, line 616
lessonPART IV / REALIZATION AND COMPARISONNumerical matching does not construct realizationBlocks 162, 166--167, profile and capacity audits For compact J⊂GL_n(F), the homomorphism v_F∘det has finite image in Z, hence image zero. An unramified character ν_a(g)=a^v_F(det g) is therefore trivial on J. The compact packet restrictions of π and π⊗ν_a agree, while the central character changes by a^n on a scalar uniformizer. Choosing a^n≠1 gives a different atom.
Exact mathematical passage
\nu_a(g)=a^{v_F(\det g)}
\xi^{-1}Use in source · 3 passages
Search aliases: ξ^-1 xi
Use in source · Read passage · Paper_XI_Traditional_Final.tex, line 365
[Dual and self dual tame packets] For a tame atom τ_ξ=Ind_W_K_f^W_Kξ, we have τ_ξ^vee ≅ Ind_W_K_f^W_Kξ^-1. The atom is self dual exactly when ξ^-1=ξ^Φ^j for some j∈ Z/f Z.
Exact mathematical passage
\tau_\xi^\vee \cong \Ind_{W_{K_f}}^{W_K}\xi^{-1}.
\xi^{\Phi^j}Use in source · 1 passage
Search aliases: ξ^Φ^j xi Phi
Use in source · Read passage · Paper_XI_Traditional_Final.tex, line 371
[Dual and self dual tame packets] For a tame atom τ_ξ=Ind_W_K_f^W_Kξ, we have τ_ξ^vee ≅ Ind_W_K_f^W_Kξ^-1. The atom is self dual exactly when ξ^-1=ξ^Φ^j for some j∈ Z/f Z.
Exact mathematical passage
\xi^{-1}=\xi^{\Phi^j}
\varpiUse in source · 4 passages
Search aliases: ϖ varpi
Use in source · Read passage · dyadic_synthesis.tex, line 361
There is also a recovery from compact data explicitly recording that root. With A_0=Λ(ϖ_K) and X=c - Ind_J^Gη, define (Tf)(g)=A_0^-1f(ϖ_Kg). Then X/(T-1)X≃π. After imposing the fixed central character, N/(F^× J) is cyclic of order N, and the finite average N^-1∑_i=0^N-1T^i selects this same extension. These operators act on complex representation spaces, so their scalar division by N is valid. This is a different recovery from the Block159 projector onto a sector of η. The exact kernel and finit…
Exact mathematical passage
A_0=\Lambda(\varpi_K)
\pi_1Use in source · 1 passage
Search aliases: π_1 pi
Use in source · Read passage · Paper_XI_Traditional_Final.tex, line 1662
[Unbased gauge classification] There exists some family of vertex isomorphisms if and only if the two holonomy representations π_1(Γ,r)⟶GL(V_r), π_1(Γ,r)⟶GL(W_r) are conjugate.
Exact mathematical passage
\pi_1(\Gamma,r)\longrightarrow\GL(V_r), \qquad \pi_1(\Gamma,r)\longrightarrow\GL(W_r)
\varpi_KUse in source · 4 passages
Search aliases: ϖ_K varpi
Use in source · Read passage · dyadic_synthesis.tex, line 361
There is also a recovery from compact data explicitly recording that root. With A_0=Λ(ϖ_K) and X=c - Ind_J^Gη, define (Tf)(g)=A_0^-1f(ϖ_Kg). Then X/(T-1)X≃π. After imposing the fixed central character, N/(F^× J) is cyclic of order N, and the finite average N^-1∑_i=0^N-1T^i selects this same extension. These operators act on complex representation spaces, so their scalar division by N is valid. This is a different recovery from the Block159 projector onto a sector of η. The exact kernel and finit…
Exact mathematical passage
A_0=\Lambda(\varpi_K)
\SigmaUse in source · 9 passages
Search aliases: Σ Sigma
Use in source · Read passage · Paper_XI_Traditional_Final.tex, line 2137
At both ranks K^× normalizes P. Thus N=K^× J is normal of index two in M=K^× P, and Σ=Ind_ N^ MΛ, π≃cInd_ M^GL_N(F)Σ. Induction in stages preserves the full original Λ, its central character, and its noncompact root; see [source reference]. The full marked scalar-pushout comparison applies to these actual enlarged pairs as well as to the original higher-rank types ([source reference]). The full synthesis and scope are in [source reference].
Exact mathematical passage
\Sigma=\Ind_{\mathcal N}^{\mathcal M}\Lambda, \qquad \pi\simeq\cInd_{\mathcal M}^{\GL_N(F)}\Sigma.
\sigmaUse in source · 3 passages
Search aliases: σ sigma
Use in source · Read passage · Paper_XI_Traditional_Final.tex, line 298
[Formal atomic reconstruction] Assume that every irreducible Weil representation τ has been assigned a supercuspidal representation σ_τ, compatibly with isomorphism and unramified twist. Then the rules Sp_m(τ) ⟼ St_m(σ_τ) and normalized parabolic induction determine a candidate correspondence on all Frobenius semisimple Weil Deligne parameters.
Exact mathematical passage
\sigma_\tau
\sigma_\tauUse in source · 3 passages
Search aliases: σ_τ sigma tau
Use in source · Read passage · Paper_XI_Traditional_Final.tex, line 298
[Formal atomic reconstruction] Assume that every irreducible Weil representation τ has been assigned a supercuspidal representation σ_τ, compatibly with isomorphism and unramified twist. Then the rules Sp_m(τ) ⟼ St_m(σ_τ) and normalized parabolic induction determine a candidate correspondence on all Frobenius semisimple Weil Deligne parameters.
Exact mathematical passage
\sigma_\tau
\tau^{I_K}Use in source · 2 passages
Search aliases: τ^I_K tau
Use in source · Read passage · Paper_XI_Traditional_Final.tex, line 697
[Conductor of a nontrivial tame atom] Let τ be an irreducible tame Weil representation of dimension n>1. Then Swan(τ)=0, dimτ^I_K=0, Art(τ)=n.
Exact mathematical passage
\Swan(\tau)=0, \qquad \dim\tau^{I_K}=0, \qquad \Art(\tau)=n.
\tau_\etaUse in source · 6 passages
Search aliases: τ_η tau eta
Use in source · Read passage · Paper_XI_Traditional_Final.tex, line 555
Let τ_ξ=Ind_W_K_n^W_Kξ, τ_η=Ind_W_K_m^W_Kη, where K_n/K and K_m/K are unramified. Put d=gcd(n,m), L=lcm(n,m).
Exact mathematical passage
\tau_\xi=\Ind_{W_{K_n}}^{W_K}\xi, \qquad \tau_\eta=\Ind_{W_{K_m}}^{W_K}\eta,
\tau_\xiUse in source · 7 passages
Search aliases: τ_ξ tau xi
Use in source · Read passage · Paper_XI_Traditional_Final.tex, line 361
[Dual and self dual tame packets] For a tame atom τ_ξ=Ind_W_K_f^W_Kξ, we have τ_ξ^vee ≅ Ind_W_K_f^W_Kξ^-1. The atom is self dual exactly when ξ^-1=ξ^Φ^j for some j∈ Z/f Z.
Exact mathematical passage
\tau_\xi=\Ind_{W_{K_f}}^{W_K}\xi,
\tau_\xi^\veeUse in source · 2 passages
Search aliases: τ_ξ^vee tau xi vee
Use in source · Read passage · Paper_XI_Traditional_Final.tex, line 365
[Dual and self dual tame packets] For a tame atom τ_ξ=Ind_W_K_f^W_Kξ, we have τ_ξ^vee ≅ Ind_W_K_f^W_Kξ^-1. The atom is self dual exactly when ξ^-1=ξ^Φ^j for some j∈ Z/f Z.
Exact mathematical passage
\tau_\xi^\vee \cong \Ind_{W_{K_f}}^{W_K}\xi^{-1}.
\Phi^{-1}Use in source · 8 passages
Search aliases: Φ^-1 Phi
Use in source · Read passage · Paper_XI_Traditional_Final.tex, line 328
[Tame character projectors] The operators P_j are pairwise orthogonal idempotents whose sum is the identity. Frobenius permutes them by Φ P_jΦ^-1=P_qj.
Exact mathematical passage
\Phi P_j\Phi^{-1}=P_{qj}.
\Phi^fUse in source · 4 passages
Search aliases: Φ^f Phi
Use in source · Read passage · Paper_XI_Traditional_Final.tex, line 347
The nonzero tame packets are permuted by Frobenius. A union of Frobenius orbits is stable under W_K, so irreducibility forces one orbit. Choose one packet V_θ. Its stabilizer is W_K_f=⟨ I_K,Φ^f⟩. A nonzero proper stable subspace of V_θ would generate a nonzero proper W_K stable subspace after summing its Frobenius translates. Thus V_θ is irreducible for the stabilizer. Inertia acts on it by the scalar character θ, and the remaining action is generated by Φ^f. The stabilizer action on this packet…
Exact mathematical passage
W_{K_f}=\langle I_K,\Phi^f\rangle
\Phi^jUse in source · 1 passage
Search aliases: Φ^j Phi
Use in source · Read passage · Paper_XI_Traditional_Final.tex, line 371
[Dual and self dual tame packets] For a tame atom τ_ξ=Ind_W_K_f^W_Kξ, we have τ_ξ^vee ≅ Ind_W_K_f^W_Kξ^-1. The atom is self dual exactly when ξ^-1=ξ^Φ^j for some j∈ Z/f Z.
Exact mathematical passage
\xi^{-1}=\xi^{\Phi^j}
\chi^{-1}Use in source · 3 passages
Search aliases: χ^-1 chi
Use in source · Read passage · Paper_XI_Traditional_Final.tex, line 755
[Duality of Gauss phases] For nontrivial χ, γ(χ,ψ)γ(χ^-1,ψ) = χ(-1).
Exact mathematical passage
\boxed{ \gamma(\chi,\psi)\gamma(\chi^{-1},\psi) = \chi(-1). }
\chi_\thetaUse in source · 7 passages
Search aliases: χ_θ chi theta
Use in source · Read passage · Paper_XI_Traditional_Final.tex, line 469
[Primitive cuspidal idempotent] Let χ_θ be the irreducible character of κ_θ, and let d_θ=χ_θ(1). Then e_θ = fracd_θabsG_n ∑_g∈ G_n χ_θ(g^-1)g is the primitive central idempotent selecting the κ_θ block in C[G_n].
Exact mathematical passage
\chi_\theta
\psi_0Use in source · 4 passages
Search aliases: ψ_0 psi
Use in source · Read passage · Paper_XI_Traditional_Final.tex, line 1161
Let θ be a maximal simple character for GL_n(K), with compact open subgroups H_θ^1⊆ J_θ^1⊆ J_θ. The quotient V_θ=J_θ^1/H_θ^1 is regarded as a finite vector space over F_p, where p is the residue characteristic. Fix a nontrivial additive character ψ_0: F_p→μ_p⊂ C^×, where μ_p is the group of p-th roots of unity. The simple-character commutator takes values in μ_p. Since ψ_0 is injective, the equation ψ_0(h_θ(bar x,bar y)) = θ([x,y]) uniquely defines the F_p-valued commutator pairing h_θ. We use t…
Exact mathematical passage
\psi_0:\FF_p\to\mu_p\subset\CC^\times
\psi_FUse in source · 4 passages
Search aliases: ψ_F psi
Use in source · Read passage · dyadic_synthesis.tex, line 13
Let F= F_2((t)), fix an additive character ψ_F of conductor t O_F, and take increasing odd positive integers h_1<⋯<h_d. Put q_i=3h_i, N=2^d, n_j=2^d-jq_d-j+1. There is an explicit β∈ M_N(F) generating a separable totally ramified degree-N field that normalizes the complete lattice chain. It gives a maximal simple stratum with leading level n_1, final critical exponent -q_1, and successive approximation-field degrees 2,4,…,2^d, with critical levels n_2,…,n_d.
Exact mathematical passage
\psi_F
\psi_{\beta-\gamma}Use in source · 1 passage
Search aliases: ψ_β-γ psi beta gamma
Use in source · Read passage · dyadic_synthesis.tex, line 112
minant unit character to multiplicity two. BK 3.6.13(ii) and 3.6.14 give, on scalar units, Θ_j(uI)=θ_j(u^2)=1. For odd critical level r, use m=lfloor r/2rfloor and D=H^m+1(β)=H^m+1(γ). The endpoint is valid because 2(m+1)-r=1: the additive-character cross term lies in the radical on which the matrix additive character is trivial. BK 3.3.18 supplies ϑ=Θ_j|_D ψ_β-γ. Integral trace makes it trivial on D∩ U_F^1.
Exact mathematical passage
\vartheta=\Theta_j|_D\,\psi_{\beta-\gamma}
\OmegaUse in source · 2 passages
Search aliases: Ω Omega
Use in source · Read passage · Paper_XI_Traditional_Final.tex, line 1238
[Cyclic discrepancy character] Assume that the two projective stabilizer actions have been linearized to honest representations of the cyclic group C. Then there is a character δ:C→ C^× such that UΩ_ρ(g) = δ(g)Ω_θ(g)U for every g∈ C.
Exact mathematical passage
U\Omega_\rho(g) = \delta(g)\Omega_\theta(g)U
\omega_VUse in source · 2 passages
Search aliases: ω_V omega
Use in source · Read passage · Paper_XI_Traditional_Final.tex, line 1764
[Heisenberg tensor factorization] Fix compatible central characters. The irreducible Heisenberg representation of an orthogonal sum is the tensor product of the irreducible Heisenberg representations of its summands: ω_V ≅ ⊗_s∈ Sω_V_s.
Exact mathematical passage
\boxed{ \omega_V \cong \bigotimes_{s\in S}\omega_{V_s}. }
\omega_{V_s}Use in source · 2 passages
Search aliases: ω_V_s omega
Use in source · Read passage · Paper_XI_Traditional_Final.tex, line 1764
[Heisenberg tensor factorization] Fix compatible central characters. The irreducible Heisenberg representation of an orthogonal sum is the tensor product of the irreducible Heisenberg representations of its summands: ω_V ≅ ⊗_s∈ Sω_V_s.
Exact mathematical passage
\boxed{ \omega_V \cong \bigotimes_{s\in S}\omega_{V_s}. }
\Omega_\thetaUse in source · 2 passages
Search aliases: Ω_θ Omega theta
Use in source · Read passage · Paper_XI_Traditional_Final.tex, line 1238
[Cyclic discrepancy character] Assume that the two projective stabilizer actions have been linearized to honest representations of the cyclic group C. Then there is a character δ:C→ C^× such that UΩ_ρ(g) = δ(g)Ω_θ(g)U for every g∈ C.
Exact mathematical passage
U\Omega_\rho(g) = \delta(g)\Omega_\theta(g)U
\omega_\thetaUse in source · 2 passages
Search aliases: ω_θ omega theta
Use in source · Read passage · Paper_XI_Traditional_Final.tex, line 1218
[Restricted Heisenberg bridge] A restricted bridge datum determines an isomorphism between the two Heisenberg representations, unique up to a nonzero scalar. More precisely, there is an invertible map U: H_ρ⟶ H_θ such that Uω_ρ(h) = ω_θ(ι_H(h))U for every Heisenberg element h, and every other such map is a scalar multiple of U.
Exact mathematical passage
U\omega_\rho(h) = \omega_\theta(\iota_H(h))U
\Omega_\rhoUse in source · 2 passages
Search aliases: Ω_ρ Omega rho
Use in source · Read passage · Paper_XI_Traditional_Final.tex, line 1238
[Cyclic discrepancy character] Assume that the two projective stabilizer actions have been linearized to honest representations of the cyclic group C. Then there is a character δ:C→ C^× such that UΩ_ρ(g) = δ(g)Ω_θ(g)U for every g∈ C.
Exact mathematical passage
U\Omega_\rho(g) = \delta(g)\Omega_\theta(g)U
\omega_\rhoUse in source · 2 passages
Search aliases: ω_ρ omega rho
Use in source · Read passage · Paper_XI_Traditional_Final.tex, line 1218
[Restricted Heisenberg bridge] A restricted bridge datum determines an isomorphism between the two Heisenberg representations, unique up to a nonzero scalar. More precisely, there is an invertible map U: H_ρ⟶ H_θ such that Uω_ρ(h) = ω_θ(ι_H(h))U for every Heisenberg element h, and every other such map is a scalar multiple of U.
Exact mathematical passage
U\omega_\rho(h) = \omega_\theta(\iota_H(h))U
\ellUse in source · 2 passages
Search aliases: ℓ ell
Use in source · Read passage · Paper_XI_Traditional_Final.tex, line 874
[Isotropic averaging projector] The operator P_L = frac1abs L ∑_ℓ∈widetilde L ρ(ℓ) is an orthogonal projector. In the Schrödinger model with L=X, it has rank one and selects the constant functions.
Exact mathematical passage
P_L = \frac1{\abs L} \sum_{\ell\in\widetilde L} \rho(\ell)
\twoheadrightarrowUse in source · 2 passages
Search aliases: ↠ twoheadrightarrow
Use in source · Read passage · odd_synthesis.tex, line 187
Then a finite cover widetilde C↠ C acts by actual central matrices in the unchanged candidate. Its represented operators descend to exactly χ on C. Hence the attachment and recovery above are equivariant for that action. A direct matrix homomorphism from C itself is not asserted and can fail.
Exact mathematical passage
\widetilde C\twoheadrightarrow C
\mapstoUse in source · 2 passages
Search aliases: ↦ mapsto
Use in source · Read passage · Paper_XI_Traditional_Final.tex, line 312
[source reference] is a reconstruction theorem. It does not produce the atomic assignment τ↦σ_τ, and it does not prove the required local factor identities.
Exact mathematical passage
\tau\mapsto\sigma_\tau
\RightarrowUse in source · 1 passage
Search aliases: ⇒ Rightarrow
Use in source · Paper_XI_Visual_Guide_Final.tex, line 639
There is a logical difference between ∀π [R(d,π)⇒H5(π)] and ∃π [R(d,π)landH5(π)]. The first statement can be vacuous. The second includes an actual witness. Choosing the established reference atom only supplies a profile-matching object; it leaves its recovery of d unproved. takeawayH5 remains a conditional premise about the same independently constructed candidate. It is not renamed as an unconditional factor-comparison theorem.
Exact mathematical passage
\forall\pi\,[R(d,\pi)\Rightarrow\mathrm{H5}(\pi)] \quad\text{and}\quad \exists\pi\,[R(d,\pi)\land\mathrm{H5}(\pi)].
\forallUse in source · 1 passage
Search aliases: ∀ forall
Use in source · Paper_XI_Visual_Guide_Final.tex, line 639
There is a logical difference between ∀π [R(d,π)⇒H5(π)] and ∃π [R(d,π)landH5(π)]. The first statement can be vacuous. The second includes an actual witness. Choosing the established reference atom only supplies a profile-matching object; it leaves its recovery of d unproved. takeawayH5 remains a conditional premise about the same independently constructed candidate. It is not renamed as an unconditional factor-comparison theorem.
Exact mathematical passage
\forall\pi\,[R(d,\pi)\Rightarrow\mathrm{H5}(\pi)] \quad\text{and}\quad \exists\pi\,[R(d,\pi)\land\mathrm{H5}(\pi)].
\existsUse in source · 1 passage
Search aliases: ∃ exists
Use in source · Paper_XI_Visual_Guide_Final.tex, line 639
There is a logical difference between ∀π [R(d,π)⇒H5(π)] and ∃π [R(d,π)landH5(π)]. The first statement can be vacuous. The second includes an actual witness. Choosing the established reference atom only supplies a profile-matching object; it leaves its recovery of d unproved. takeawayH5 remains a conditional premise about the same independently constructed candidate. It is not renamed as an unconditional factor-comparison theorem.
Exact mathematical passage
\forall\pi\,[R(d,\pi)\Rightarrow\mathrm{H5}(\pi)] \quad\text{and}\quad \exists\pi\,[R(d,\pi)\land\mathrm{H5}(\pi)].
\notinUse in source · 1 passage
Search aliases: ∉ notin
Use in source · Read passage · Paper_XI_Traditional_Final.tex, line 2124
At rank two, with q=2m+1, enlarge J=U_K^1U_A^m+1 to P=U_K^1U_A^m. The actual character ratio λ=θ^g/θ for g∉ J is nontrivial and sign-valued. With H_0=kerλ, the quotient is P/H_0≃ F_2^2, and Ind_J^Pθ is an irreducible two-dimensional Heisenberg carrier ([source reference]). At rank four, the actual outer quadratic centralizer gives P=J U_B^m, [P:J]=2, and P/H_0≃ F_2^4, with carrier Ind_J^Pη of dimension four ([source reference]). The finite scalar center can be larger than μ_2; no literal extrasp…
Exact mathematical passage
g\notin J
\prodUse in source · 5 passages
Search aliases: ∏ prod
Use in source · Read passage · Paper_XI_Traditional_Final.tex, line 1341
[Orthogonal multiplicativity] Suppose V=V_1mathop⊥limits⋯mathop⊥limits V_s and g preserves every summand. If no g|_V_j has a fixed vector, then Φ_p(g,V) = ∏_j=1^s Φ_p(g|_V_j,V_j).
Exact mathematical passage
\boxed{ \Phi_p(g,V) = \prod_{j=1}^{s} \Phi_p(g|_{V_j},V_j). }
\prod_jUse in source · 1 passage
Search aliases: ∏_j prod
Use in source · Read passage · Paper_XI_Traditional_Final.tex, line 1353
The half dimensions add, and det(g-Imid V) = ∏_jdet(g-Imid V_j). The quadratic character is multiplicative.
Exact mathematical passage
\det(g-I\mid V) = \prod_j\det(g-I\mid V_j).
\prod_{j=1}^{s}Use in source · 1 passage
Search aliases: ∏_j=1^s prod
Use in source · Read passage · Paper_XI_Traditional_Final.tex, line 1341
[Orthogonal multiplicativity] Suppose V=V_1mathop⊥limits⋯mathop⊥limits V_s and g preserves every summand. If no g|_V_j has a fixed vector, then Φ_p(g,V) = ∏_j=1^s Φ_p(g|_V_j,V_j).
Exact mathematical passage
\boxed{ \Phi_p(g,V) = \prod_{j=1}^{s} \Phi_p(g|_{V_j},V_j). }
\prod_sUse in source · 2 passages
Search aliases: ∏_s prod
Use in source · Read passage · Paper_XI_Traditional_Final.tex, line 1775
The Heisenberg group of an orthogonal sum is the central product of the Heisenberg groups of the summands. The tensor product representation has the declared central character and has dimension ∏_ssqrtabsV_s = sqrtabs V. It is irreducible by the finite Stone von Neumann theorem, so it is the required representation.
Exact mathematical passage
\prod_s\sqrt{\abs{V_s}} = \sqrt{\abs V}.
\prod_{s\in S}Use in source · 1 passage
Search aliases: ∏_s∈ S prod in
Use in source · Read passage · Paper_XI_Traditional_Final.tex, line 1785
[Breakwise phase product] In odd residue characteristic, let g∈Sp(V) preserve every break packet and suppose its restriction to M=im(g-I) is fixed point free. Put g_s=g|_V_s and M_s=im(g_s-I). The moving restrictions are symplectic, and Φ_p(g|_M,M) = ∏_s∈ SΦ_p(g_s|_M_s,M_s). In residue characteristic two, the analogous normalized Gauss sign is the product of the signs determined by the breakwise quadratic refinements.
Exact mathematical passage
\boxed{ \Phi_p(g|_M,M) = \prod_{s\in S}\Phi_p(g_s|_{M_s},M_s). }
\sum_{a=0}^{M-1}Use in source · 2 passages
Search aliases: ∑_a=0^M-1 sum
Use in source · Read passage · Paper_XI_Traditional_Final.tex, line 318
Assume that P_K acts trivially. After passing to a finite quotient, tame inertia is cyclic. Let t generate a quotient of order M, and put ζ_M= e^2π i/M. For j∈ Z/M Z, define P_j = frac1M ∑_a=0^M-1 ζ_M^-jar(t)^a.
Exact mathematical passage
P_j = \frac1M \sum_{a=0}^{M-1} \zeta_M^{-ja}r(t)^a.
\sum_{a\in A}Use in source · 2 passages
Search aliases: ∑_a∈ A sum in
Use in source · Read passage · Paper_XI_Traditional_Final.tex, line 971
Let z=∑_a∈ Az_au_a be central. Commuting with u_x gives z_ab_c(x,a)=z_a for every x. Nondegeneracy forces z_a=0 for every nonzero a. Thus the center is C. To prove semisimplicity directly, choose a linear projection P onto a submodule of a finite dimensional C^c[A]-module and form overline P=frac1abs A∑_a∈ Au_a P u_a^-1. Every summand maps into the submodule and is the identity there. Conjugation by any u_x permutes the summands: the cocycle scalars cancel between each operator and its inverse. …
Exact mathematical passage
z=\sum_{a\in A}z_au_a
\sum_{d\mid n}Use in source · 2 passages
Search aliases: ∑_dmid n sum mid
Use in source · Read passage · Paper_XI_Traditional_Final.tex, line 414
[Möbius orbit count] The number of regular Frobenius orbits of length n is c_n(q) = frac1n ∑_dmid n μ(d) (q^n/d-1).
Exact mathematical passage
\boxed{ c_n(q) = \frac1n \sum_{d\mid n} \mu(d) \bigl(q^{n/d}-1\bigr). }
\sum_{g\in G_n}Use in source · 3 passages
Search aliases: ∑_g∈ G_n sum in
Use in source · Read passage · Paper_XI_Traditional_Final.tex, line 470
[Primitive cuspidal idempotent] Let χ_θ be the irreducible character of κ_θ, and let d_θ=χ_θ(1). Then e_θ = fracd_θabsG_n ∑_g∈ G_n χ_θ(g^-1)g is the primitive central idempotent selecting the κ_θ block in C[G_n].
Exact mathematical passage
e_\theta = \frac{d_\theta}{\abs{G_n}} \sum_{g\in G_n} \chi_\theta(g^{-1})g
\sum_{i=0}^{N-1}Use in source · 2 passages
Search aliases: ∑_i=0^N-1 sum
Use in source · Read passage · dyadic_synthesis.tex, line 366
There is also a recovery from compact data explicitly recording that root. With A_0=Λ(ϖ_K) and X=c - Ind_J^Gη, define (Tf)(g)=A_0^-1f(ϖ_Kg). Then X/(T-1)X≃π. After imposing the fixed central character, N/(F^× J) is cyclic of order N, and the finite average N^-1∑_i=0^N-1T^i selects this same extension. These operators act on complex representation spaces, so their scalar division by N is valid. This is a different recovery from the Block159 projector onto a sector of η. The exact kernel and finit…
Exact mathematical passage
N^{-1}\sum_{i=0}^{N-1}T^i
\sum_jUse in source · 1 passage
Search aliases: ∑_j sum
Use in source · Paper_XI_Visual_Guide_Final.tex, line 92
lessonPART I / FINITE TAME COORDINATESFourier projectors isolate tame inertiaOriginal Section 3; geometric Frobenius convention Let t generate a finite tame quotient of order M, with (q,M)=1. For j∈ Z/M Z define P_j=frac1M∑_a=0^M-1ζ_M^-jar(t)^a, ζ_M=e^2π i/M. The finite Fourier identities give P_jP_k=δ_jkP_j and ∑_jP_j=I. Thus P_jV is the ζ_M^j character space for r(t).
Exact mathematical passage
\sum_jP_j=I
\sum_{j=1}^{m-1}Use in source · 2 passages
Search aliases: ∑_j=1^m-1 sum
Use in source · Read passage · Paper_XI_Traditional_Final.tex, line 263
[Exact block reconstruction] Let P_j be the projector onto V_j. Then N=∑_j=1^m-1P_j-1NP_j, N^m=0. Every power N^r is the sum of the packet paths of length r along the chain.
Exact mathematical passage
N=\sum_{j=1}^{m-1}P_{j-1}NP_j, \qquad N^m=0.
\sum_{t\in k^{\times}}Use in source · 1 passage
Search aliases: ∑_t∈ k^× sum in times
Use in source · Read passage · Paper_XI_Traditional_Final.tex, line 740
Compute absG(χ,ψ)^2 = ∑_x,y∈ k^× χ(xy^-1)ψ(x-y). Write x=ty. The inner sum over y equals q-1 when t=1 and equals -1 otherwise. Since ∑_t∈ k^×χ(t)=0, the result is q.
Exact mathematical passage
\sum_{t\in k^{\times}}\chi(t)=0
\sum_{u\in U}Use in source · 2 passages
Search aliases: ∑_u∈ U sum in
Use in source · Read passage · Paper_XI_Traditional_Final.tex, line 492
[Parabolic annihilation] Let P=LU be a proper parabolic subgroup of G_n, and put e_U=frac1abs U∑_u∈ Uu. If κ_θ is cuspidal, then e_Ue_θ=e_θ e_U=0.
Exact mathematical passage
e_U=\frac1{\abs U}\sum_{u\in U}u.
\sum_{u\in X}Use in source · 1 passage
Search aliases: ∑_u∈ X sum in
Use in source · Read passage · Paper_XI_Traditional_Final.tex, line 856
If the translation component x is nonzero, the operator permutes the standard basis without a fixed point, so its trace is zero. If x=0 but ξ≠0, the trace is ψ(z)∑_u∈ Xψ(ξ(u))=0. When x=ξ=0, the operator is the scalar ψ(z).
Exact mathematical passage
\psi(z)\sum_{u\in X}\psi(\xi(u))=0.
\sum_xUse in source · 2 passages
Search aliases: ∑_x sum
Use in source · Read passage · Paper_XI_Traditional_Final.tex, line 1566
Using the quadratic identity, q_v(x)=q(x+v)+q(v). Therefore ∑_x(-1)^q_v(x) = (-1)^q(v) ∑_x(-1)^q(x). Compare the signs in [source reference].
Exact mathematical passage
\sum_x(-1)^{q_v(x)} = (-1)^{q(v)} \sum_x(-1)^{q(x)}.
\sum_{x,y\in k^{\times}}Use in source · 1 passage
Search aliases: ∑_x,y∈ k^× sum in times
Use in source · Read passage · Paper_XI_Traditional_Final.tex, line 734
Compute absG(χ,ψ)^2 = ∑_x,y∈ k^× χ(xy^-1)ψ(x-y). Write x=ty. The inner sum over y equals q-1 when t=1 and equals -1 otherwise. Since ∑_t∈ k^×χ(t)=0, the result is q.
Exact mathematical passage
\abs{G(\chi,\psi)}^2 = \sum_{x,y\in k^{\times}} \chi(xy^{-1})\psi(x-y).
\sum_{x\in k^{\times}}Use in source · 1 passage
Search aliases: ∑_x∈ k^× sum in times
Use in source · Read passage · Paper_XI_Traditional_Final.tex, line 716
Let k= F_q, let ψ:k→ C^× be a nontrivial additive character, and let χ:k^×→ C^× be a multiplicative character. Define G(χ,ψ) = ∑_x∈ k^×χ(x)ψ(x). The standard theory of Gauss and Jacobi sums will be used in the form recorded in [source reference].
Exact mathematical passage
G(\chi,\psi) = \sum_{x\in k^{\times}}\chi(x)\psi(x).
\sum_{x\in V}Use in source · 2 passages
Search aliases: ∑_x∈ V sum in
Use in source · Read passage · Paper_XI_Traditional_Final.tex, line 1509
[Dyadic Gauss sign] Let dim V=2r, and let q be nondegenerate. Then ∑_x∈ V(-1)^q(x) = (-1)^Arf(q)2^r.
Exact mathematical passage
\boxed{ \sum_{x\in V}(-1)^{q(x)} = (-1)^{\Arf(q)}2^r. }
\sum_{x\in X}Use in source · 2 passages
Search aliases: ∑_x∈ X sum in
Use in source · Read passage · Paper_XI_Traditional_Final.tex, line 886
The lifted isotropic subgroup is abelian, so group averaging is an idempotent. Every operator in the average is unitary, and the subgroup is closed under inversion, so the average is self adjoint. For L=X, the average is the translation average (P_Lf)(u) = frac1abs X ∑_x∈ Xf(u+x), which is the constant function with value equal to the mean of f. Its range is one dimensional.
Exact mathematical passage
(P_Lf)(u) = \frac1{\abs X} \sum_{x\in X}f(u+x),
\sum_{x\in\F_q^\times}Use in source · 1 passage
Search aliases: ∑_x∈F_q^× ∑_x∈ F_q^× sum in F times sum in mathbb times
Use in source · Paper_XI_Visual_Guide_Final.tex, line 205
lessonPART I / FINITE TAME COORDINATESFinite Gauss phases are unit scalarsOriginal Section 6 For a nontrivial additive character ψ of F_q and a nontrivial multiplicative character χ of F_q^×, define G(χ,ψ)=∑_x∈ F_q^×χ(x)ψ(x), γ(χ,ψ)=q^-1/2G(χ,ψ). The exact identities are |G(χ,ψ)|^2=q, γ(χ,ψ)γ(χ^-1,ψ)=χ(-1). The magnitude calculation substitutes x=ty in the double sum. The inner sum is q-1 for t=1 and -1 otherwise; multiplicative-character orthogonality supplies the result.
Exact mathematical passage
G(\chi,\psi)=\sum_{x\in\F_q^\times}\chi(x)\psi(x),\qquad \gamma(\chi,\psi)=q^{-1/2}G(\chi,\psi).
\sum_{\ell\in\widetilde L}Use in source · 2 passages
Search aliases: ∑_ℓ∈widetilde L sum ell in widetilde
Use in source · Read passage · Paper_XI_Traditional_Final.tex, line 874
[Isotropic averaging projector] The operator P_L = frac1abs L ∑_ℓ∈widetilde L ρ(ℓ) is an orthogonal projector. In the Schrödinger model with L=X, it has rank one and selects the constant functions.
Exact mathematical passage
P_L = \frac1{\abs L} \sum_{\ell\in\widetilde L} \rho(\ell)
\setminusUse in source · 1 passage
Search aliases: ∖ setminus
Use in source · Read passage · dyadic_synthesis.tex, line 292
Write the odd minimal level as q=2m+1, with m≥1, and let J=H^1=J^1=U_K^1U_A^m+1. Define P=U_K^1U_A^m. This is an actual compact open group, with Jtriangleleft P, [P:J]=2 and P∩(K^× J)=J. For g∈ P∖ J, the actual intertwining theorem makes λ=θ^g/θ nontrivial. The commutator [P,J]⊂ U_A^m+1 lies in the forced additive-character domain, so λ has values {±1}. Put H_0=kerλ. Then P/H_0≃ F_2^2, E=Ind_J^Pθ is irreducible of dimension 2. The actual represented generators are a diagonal sign and an off-diag…
Exact mathematical passage
g\in P\setminus J
\circUse in source · 1 passage
Search aliases: ∘ circ
Use in source · Paper_XI_Visual_Guide_Final.tex, line 615
lessonPART IV / REALIZATION AND COMPARISONNumerical matching does not construct realizationBlocks 162, 166--167, profile and capacity audits For compact J⊂GL_n(F), the homomorphism v_F∘det has finite image in Z, hence image zero. An unramified character ν_a(g)=a^v_F(det g) is therefore trivial on J. The compact packet restrictions of π and π⊗ν_a agree, while the central character changes by a^n on a scalar uniformizer. Choosing a^n≠1 gives a different atom.
Exact mathematical passage
v_F\circ\det
\capUse in source · 5 passages
Search aliases: ∩ cap
Use in source · Read passage · dyadic_synthesis.tex, line 113
units, Θ_j(uI)=θ_j(u^2)=1. For odd critical level r, use m=lfloor r/2rfloor and D=H^m+1(β)=H^m+1(γ). The endpoint is valid because 2(m+1)-r=1: the additive-character cross term lies in the radical on which the matrix additive character is trivial. BK 3.3.18 supplies ϑ=Θ_j|_D ψ_β-γ. Integral trace makes it trivial on D∩ U_F^1.
Exact mathematical passage
D\cap U_F^1
\congUse in source · 9 passages
Search aliases: ≅ cong
Use in source · Read passage · Paper_XI_Traditional_Final.tex, line 339
[Tame orbit induction] Let τ be an irreducible complex representation of W_K with trivial wild inertia. Then the nonzero tame character packets form one transitive Frobenius orbit of length f. If K_f/K is the unramified extension of degree f, there is a character ξ of W_K_f such that τ≅Ind_W_K_f^W_Kξ.
Exact mathematical passage
\boxed{ \tau\cong\Ind_{W_{K_f}}^{W_K}\xi. }
\neUse in source · 5 passages
Search aliases: ≠ ne
Use in source · Read passage · Paper_XI_Traditional_Final.tex, line 842
[Character support] The character of the Schrödinger representation satisfies Trρ(v,z) = p^rψ(z), · v=0, 0, · v≠0.
Exact mathematical passage
\boxed{ \Tr\rho(v,z) = \begin{cases} p^r\psi(z),&v=0,\\ 0,&v\ne0. \end{cases} }
\equivUse in source · 2 passages
Search aliases: ≡ equiv
Use in source · Read passage · odd_synthesis.tex, line 58
At a recursive base (L,t_L), introduce E=L(α), α^p-α=t_L^-q_d. This is a separable totally ramified extension of degree p, with v_E(α)=-q_d and Tr_E/Lα=0. Choose 1≤ b<p with q_db≡-1pmod p, set a=(q_db+1)/p, and put z=t_L^aα^b. Then z is an explicit uniformizer. Write its Eisenstein polynomial as f(X)=X^p+⋯+a_0.
Exact mathematical passage
q_db\equiv-1\pmod p
\leUse in source · 6 passages
Search aliases: ≤ le
Use in source · Read passage · dyadic_character.tex, line 157
They are the compatible native truncations of (5), because the outer extension, uniformizer, and lift depend on the same retained largest h at each stage. For 1≤ j<d, in rank 2 (j+1), put
Exact mathematical passage
1\le j<d
\leqUse in source · 1 passage
Search aliases: ≤ leq
Use in source · Read passage · Paper_XI_Traditional_Final.tex, line 1727
[Quantitative finite gluing] With the notation of [source reference], normU_vA_e-B_eU_u ≤ normB_p_vnormD_enormA_p_u^-1. Thus small fundamental cycle defects give small non tree edge residuals whenever the tree transports are uniformly conditioned.
Exact mathematical passage
\norm{U_vA_e-B_eU_u} \leq \norm{B_{p_v}}\norm{D_e}\norm{A_{p_u}^{-1}}.
\geUse in source · 26 passages
Search aliases: ≥ ge
Use in source · Read passage · Paper_XI_Traditional_Final.tex, line 2020
[Restricted odd-prime arithmetic realization] Let p be an odd prime, F= F_p((t)), fix d≥1, and let 0<h_1<⋯<h_d, pnmid h_i, q_i=(p+1)h_i, N=p^d. There is an explicit maximal simple stratum in M_N(F) with totally ramified full-degree parameter field, a literal defining sequence of degrees p,p^2,…,p^d, and levels n_j=p^d-jq_d-j+1. Every native parameter and consecutive correction has trace zero over its recursive base field. The stratum admits a full simple character trivial on U_F^1. Its fixed can…
Exact mathematical passage
d\ge1
\subsetUse in source · 13 passages
Search aliases: ⊂ subset
Use in source · Read passage · Paper_XI_Traditional_Final.tex, line 1161
Let θ be a maximal simple character for GL_n(K), with compact open subgroups H_θ^1⊆ J_θ^1⊆ J_θ. The quotient V_θ=J_θ^1/H_θ^1 is regarded as a finite vector space over F_p, where p is the residue characteristic. Fix a nontrivial additive character ψ_0: F_p→μ_p⊂ C^×, where μ_p is the group of p-th roots of unity. The simple-character commutator takes values in μ_p. Since ψ_0 is injective, the equation ψ_0(h_θ(bar x,bar y)) = θ([x,y]) uniquely defines the F_p-valued commutator pairing h_θ. We use t…
Exact mathematical passage
\psi_0:\FF_p\to\mu_p\subset\CC^\times
\oplusUse in source · 6 passages
Search aliases: ⊕ oplus
Use in source · Read passage · Paper_XI_Traditional_Final.tex, line 790
Let p be odd. Let X be an r dimensional vector space over F_p, and put V=X⊕ X^*. Define the alternating form b((x,ξ),(y,η)) = η(x)-ξ(y). Let Z= F_p, and define the Heisenberg group H(V)=V× Z with multiplication (v,z)(w,t) = (v+w,z+t+frac12 b(v,w)). Fix the additive character ψ(z)= e^2π iz/p.
Exact mathematical passage
V=X\oplus X^*.
\bigoplus_iUse in source · 1 passage
Search aliases: ⊕_i bigoplus
Use in source · Read passage · odd_synthesis.tex, line 123
Assume the one-vertex scalar-pushout contract: there are d labelled, actually supplied full character extensions with whole symplectic quotients V_i of dimension two over F_p. Their balanced C^×-central product has quotient V_src=⊕_iV_i and Heisenberg carrier dimension p^d=N. Initially take the declared stabilizer C={1}.
Exact mathematical passage
V_{\mathrm{src}}=\bigoplus_iV_i
\bigotimesUse in source · 2 passages
Search aliases: ⊗ bigotimes
Use in source · Read passage · Paper_XI_Traditional_Final.tex, line 1764
[Heisenberg tensor factorization] Fix compatible central characters. The irreducible Heisenberg representation of an orthogonal sum is the tensor product of the irreducible Heisenberg representations of its summands: ω_V ≅ ⊗_s∈ Sω_V_s.
Exact mathematical passage
\boxed{ \omega_V \cong \bigotimes_{s\in S}\omega_{V_s}. }
\bigotimes_sUse in source · 1 passage
Search aliases: ⊗_s bigotimes
Use in source · Paper_XI_Visual_Guide_Final.tex, line 385
lessonPART II / FINITE WILD COMPARISONOrthogonal breaks become tensor factorsOriginal Section 15 plate13_break_tensorSymplectic direct sum and Heisenberg tensor product are compatible. For V=⊥_sV_s and compatible central characters, ω_V≃⊗_sω_V_s, ∏_ssqrt|V_s|=sqrt|V|. The groups combine by a central product. Tensoring the individual intertwiners gives the full finite bridge, once the stabilizer actions and phases agree. Transport-graph compatibility is then the matrix holonomy condition. boundar…
Exact mathematical passage
\omega_V\simeq\bigotimes_s\omega_{V_s},\qquad \prod_s\sqrt{|V_s|}=\sqrt{|V|}.
\bigotimes_{s\in S}Use in source · 1 passage
Search aliases: ⊗_s∈ S bigotimes in
Use in source · Read passage · Paper_XI_Traditional_Final.tex, line 1764
[Heisenberg tensor factorization] Fix compatible central characters. The irreducible Heisenberg representation of an orthogonal sum is the tensor product of the irreducible Heisenberg representations of its summands: ω_V ≅ ⊗_s∈ Sω_V_s.
Exact mathematical passage
\boxed{ \omega_V \cong \bigotimes_{s\in S}\omega_{V_s}. }
\perpUse in source · 8 passages
Search aliases: ⊥ perp
Use in source · Read passage · Paper_XI_Traditional_Final.tex, line 1796
The invariant orthogonal decomposition gives M=mathop⊥limits_s∈ SM_s. Symplectic invariance gives M^⊥=ker(g-I). Fixed point freeness on M therefore makes the restricted form nondegenerate on M and on each M_s. Each restriction of g_s-I to M_s is invertible, so [source reference] applies to the moving spaces. A zero moving space contributes one. In characteristic two, apply [source reference].
Exact mathematical passage
M=\mathop{\perp}\limits_{s\in S}M_s
\perp_sUse in source · 1 passage
Search aliases: ⊥_s perp
Use in source · Paper_XI_Visual_Guide_Final.tex, line 384
lessonPART II / FINITE WILD COMPARISONOrthogonal breaks become tensor factorsOriginal Section 15 plate13_break_tensorSymplectic direct sum and Heisenberg tensor product are compatible. For V=⊥_sV_s and compatible central characters, ω_V≃⊗_sω_V_s, ∏_ssqrt|V_s|=sqrt|V|. The groups combine by a central product. Tensoring the individual intertwiners gives the full finite bridge, once the stabilizer actions and phases agree. Transport-graph compatibility is then the matrix holonomy condition. boundar…
Exact mathematical passage
V=\perp_sV_s
\longmapstoUse in source · 2 passages
Search aliases: ⟼ longmapsto
Use in source · Read passage · Paper_XI_Traditional_Final.tex, line 299
[Formal atomic reconstruction] Assume that every irreducible Weil representation τ has been assigned a supercuspidal representation σ_τ, compatibly with isomorphism and unramified twist. Then the rules Sp_m(τ) ⟼ St_m(σ_τ) and normalized parabolic induction determine a candidate correspondence on all Frobenius semisimple Weil Deligne parameters.
Exact mathematical passage
\operatorname{Sp}_m(\tau) \longmapsto \operatorname{St}_m(\sigma_\tau)
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- papers/paper-xi/current/READING_CONCORDANCE.md · 0 mathematical passages
- papers/paper-xi/current/README.md · 0 mathematical passages
- papers/paper-xi/current/formalization/README.md · 0 mathematical passages
- papers/paper-xi/current/publication/source/lean_guide/README.md · 0 mathematical passages
- papers/paper-xi/current/publication/source/traditional/Paper_XI_Traditional_Final.tex · 696 mathematical passages
- papers/paper-xi/current/publication/source/traditional/appendices/dyadic_capacity.tex · 4 mathematical passages
- papers/paper-xi/current/publication/source/traditional/appendices/dyadic_character.tex · 7 mathematical passages
- papers/paper-xi/current/publication/source/traditional/appendices/dyadic_markings.tex · 0 mathematical passages
- papers/paper-xi/current/publication/source/traditional/appendices/dyadic_synthesis.tex · 169 mathematical passages
- papers/paper-xi/current/publication/source/traditional/appendices/dyadic_tower.tex · 1 mathematical passage
- papers/paper-xi/current/publication/source/traditional/appendices/lowrank_contract.tex · 0 mathematical passages
- papers/paper-xi/current/publication/source/traditional/appendices/odd_capacity.tex · 13 mathematical passages
- papers/paper-xi/current/publication/source/traditional/appendices/odd_character.tex · 4 mathematical passages
- papers/paper-xi/current/publication/source/traditional/appendices/odd_scalar.tex · 0 mathematical passages
- papers/paper-xi/current/publication/source/traditional/appendices/odd_synthesis.tex · 113 mathematical passages
- papers/paper-xi/current/publication/source/traditional/appendices/odd_tower.tex · 5 mathematical passages
- papers/paper-xi/current/publication/source/traditional/appendices/rank2.tex · 10 mathematical passages
- papers/paper-xi/current/publication/source/traditional/appendices/rank4.tex · 6 mathematical passages
- papers/paper-xi/current/publication/source/traditional/appendices/rank4_stages.tex · 1 mathematical passage
- papers/paper-xi/current/publication/source/traditional/appendices/stabilizer_contract.tex · 0 mathematical passages
- papers/paper-xi/current/publication/source/traditional/reader_route.tex · 1 mathematical passage
- papers/paper-xi/current/publication/source/visual_guide/Paper_XI_Visual_Guide_Final.tex · 387 mathematical passages
- papers/paper-xi/current/publication/source/visual_guide/reader_route.tex · 0 mathematical passages
PDF reading editions (44)
The text index does not extract PDF glyphs. Check the reading edition for notation in figures, tables, and PDF-only material.
- Paper_XI_Traditional_v1.1.0.pdf
- Paper_XI_Visual_Guide_v1.1.0.pdf
- Paper_XI_Lean_Guide_v1.1.0.pdf
- Paper_XI_Conjecture66_Progress_v0.4.0.pdf
- Paper_XI_Conjecture66_Visual_Atlas_v0.5.0.pdf
- 01_architecture.pdf
- 02_weil_deligne_chain.pdf
- 03_tame_frobenius_orbit.pdf
- 04_regular_orbit_counts.pdf
- 05_cuspidal_projector.pdf
- 06_gcd_lcm_refinement.pdf
- 07_gauss_phase_circle.pdf
- 08_heisenberg_geometry.pdf
- 09_commutator_envelope.pdf
- 10_one_break_envelope.pdf
- 11_restricted_bridge.pdf
- 12_odd_phase_blocks.pdf
- 13_dyadic_quadratic_refinement.pdf
- 14_matrix_holonomy.pdf
- 15_breakwise_decomposition.pdf
- 16_boundary_ladder.pdf
- 01_three_objects.pdf
- 02_monodromy.pdf
- 03_tame_orbit.pdf
- 04_orbit_counts.pdf
- 05_tensor_grid.pdf
- 06_cycle_scalar.pdf
- 07_heisenberg.pdf
- 08_radical.pdf
- 09_bridge.pdf
- 10_quadratic_planes.pdf
- 11_arf_counts.pdf
- 12_holonomy.pdf
- 13_break_tensor.pdf
- 14_recursive_parameter.pdf
- 15_degree_levels.pdf
- 16_scalar_depth.pdf
- 17_native_ambient_trace.pdf
- 18_capacity.pdf
- 21_marked_extension.pdf
- 19_enlarged_carrier.pdf
- 20_induction_stages.pdf
- 22_completion.pdf
- 23_conjecture_boundary.pdf