Notation by work
Strong Artin: source and evidence
All notation · Glossary · Source coverage
539 symbol spellings in 1410 mathematical passages across 8 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 539 entries are available below.
=Definition passage · 257 passages
Search aliases: =
Definition passage: Attached cyclic descent · body.tex, line 100
[Attached cyclic descent] Let E/F be cyclic of prime degree p, and let ρ be an irreducible finite image representation of G_F whose restriction τ=ρ|_G_E is irreducible. Suppose τ is automorphic with cuspidal representation Π of GL_n(Adele_E). An attached cyclic descent is a cuspidal descent π of Π such that rec_F_v(π_v) ≅ ρ|_W_F_v for every place v of F.
Exact mathematical passage
\tau=\rho|_{G_E}
ADefinition passage · 80 passages
Search aliases: A
Definition passage: Active coefficient degrees · body.tex, line 162
[Active coefficient degrees] Define S(A)={k∈{1,…,n}:e_k(A)≠0}.
Exact mathematical passage
S(A)=\{k\in\{1,\ldots,n\}:e_k(A)\neq0\}.
aDefinition passage · 33 passages
Search aliases: a
Definition passage: Extension spectrum polynomial · body.tex, line 388
[Extension spectrum polynomial] For a stable type τ, the multiplicity space M_τ(V)=Hom_H(τ,V) becomes a representation of G/H after comparing the action on V with widetildeτ. Write M_τ(V) ≅ ⊕_a∈ZZ/pZZ (ξ^a)^⊕ m_τ,a(V). Define P_τ,V(z) = ∑_a∈ZZ/pZZ m_τ,a(V)z^a in ZZ[z]/(z^p-1).
Exact mathematical passage
M_\tau(V) \cong \bigoplus_{a\in\ZZ/p\ZZ} (\xi^a)^{\oplus m_{\tau,a}(V)}.
a_iDefinition passage · 7 passages
Search aliases: a_i
Definition passage: Entire induction cone · body.tex, line 1529
[Entire induction cone] Fix finitely many known entire induced packets Θ_1,…,Θ_r in the character lattice of G. Define C_calE = { ∑_i=1^ra_iΘ_i: a_i∈ZZ_≥0 }. Its saturation is Sat(C_calE) = {x:some m>0 has mx∈ C_calE}.
Exact mathematical passage
C_{\calE} = \left\{ \sum_{i=1}^ra_i\Theta_i: a_i\in\ZZ_{\geq0} \right\}.
A_vDefinition passage · 7 passages
Search aliases: A_v
Definition passage: Local attachment set · body.tex, line 631
[Local attachment set] When the local quotient has order p, define calJ_v = {j∈ZZ/pZZ: A_v⊗ϵ_v^j≅ R_v}. At a split place, set calJ_v=ZZ/pZZ.
Exact mathematical passage
\calJ_v = \{j\in\ZZ/p\ZZ: A_v\otimes\epsilon_v^j\cong R_v\}.
\AdeleDefinition passage · 9 passages
Search aliases: Adele
Definition passage: Attached cyclic descent · body.tex, line 100
[Attached cyclic descent] Let E/F be cyclic of prime degree p, and let ρ be an irreducible finite image representation of G_F whose restriction τ=ρ|_G_E is irreducible. Suppose τ is automorphic with cuspidal representation Π of GL_n(Adele_E). An attached cyclic descent is a cuspidal descent π of Π such that rec_F_v(π_v) ≅ ρ|_W_F_v for every place v of F.
Exact mathematical passage
\GL_n(\Adele_E)
\Adele_EDefinition passage · 2 passages
Search aliases: Adele_E Adele
Definition passage: Attached cyclic descent · body.tex, line 100
[Attached cyclic descent] Let E/F be cyclic of prime degree p, and let ρ be an irreducible finite image representation of G_F whose restriction τ=ρ|_G_E is irreducible. Suppose τ is automorphic with cuspidal representation Π of GL_n(Adele_E). An attached cyclic descent is a cuspidal descent π of Π such that rec_F_v(π_v) ≅ ρ|_W_F_v for every place v of F.
Exact mathematical passage
\GL_n(\Adele_E)
\Adele_FDefinition passage · 5 passages
Search aliases: Adele_F Adele
Definition passage: Finite Satake cusp form · body.tex, line 779
[Finite Satake cusp form] A cuspidal representation of GL_n(Adele_F) is finite Satake when, outside one finite exceptional set, its normalized unramified Satake roots lie in one finite subgroup of CC^×. The exceptional set is fixed, not allowed to change separately for each coefficient or prime.
Exact mathematical passage
\GL_n(\Adele_F)
BDefinition passage · 31 passages
Search aliases: B
Definition passage: Universal bounded cyclotomic alphabet · body.tex, line 959
[Universal bounded cyclotomic alphabet] For positive integers n and B, let calQ_n,B be the finite set of monic degree n polynomials whose roots are roots of unity of order at most B.
Exact mathematical passage
B
CDefinition passage · 42 passages
Search aliases: C
Definition passage: Entire induction cone · body.tex, line 1529
[Entire induction cone] Fix finitely many known entire induced packets Θ_1,…,Θ_r in the character lattice of G. Define C_calE = { ∑_i=1^ra_iΘ_i: a_i∈ZZ_≥0 }. Its saturation is Sat(C_calE) = {x:some m>0 has mx∈ C_calE}.
Exact mathematical passage
C_{\calE} = \left\{ \sum_{i=1}^ra_i\Theta_i: a_i\in\ZZ_{\geq0} \right\}.
C_{\calE}Definition passage · 3 passages
Search aliases: C_calE calE
Definition passage: Entire induction cone · body.tex, line 1529
[Entire induction cone] Fix finitely many known entire induced packets Θ_1,…,Θ_r in the character lattice of G. Define C_calE = { ∑_i=1^ra_iΘ_i: a_i∈ZZ_≥0 }. Its saturation is Sat(C_calE) = {x:some m>0 has mx∈ C_calE}.
Exact mathematical passage
C_{\calE} = \left\{ \sum_{i=1}^ra_i\Theta_i: a_i\in\ZZ_{\geq0} \right\}.
\calADefinition passage · 10 passages
Search aliases: calA
Definition passage: Finite polynomial alphabet · body.tex, line 829
[Finite polynomial alphabet] The finite polynomial alphabet of the cyclic attachment problem is the union calA = calA_π ∪ calA_ρ, where calA_π is the set of normalized Hecke polynomials occurring for π outside the fixed exceptional set, and calA_ρ is the set of characteristic polynomials occurring for the finitely many twists ρ⊗ϵ^j, j∈ZZ/pZZ.
Exact mathematical passage
\calA = \calA_{\pi} \cup \calA_{\rho},
\calA_{\pi}Definition passage · 2 passages
Search aliases: calA_π calA pi
Definition passage: Finite polynomial alphabet · body.tex, line 829
[Finite polynomial alphabet] The finite polynomial alphabet of the cyclic attachment problem is the union calA = calA_π ∪ calA_ρ, where calA_π is the set of normalized Hecke polynomials occurring for π outside the fixed exceptional set, and calA_ρ is the set of characteristic polynomials occurring for the finitely many twists ρ⊗ϵ^j, j∈ZZ/pZZ.
Exact mathematical passage
\calA = \calA_{\pi} \cup \calA_{\rho},
\calA_{\rho}Definition passage · 2 passages
Search aliases: calA_ρ calA rho
Definition passage: Finite polynomial alphabet · body.tex, line 829
[Finite polynomial alphabet] The finite polynomial alphabet of the cyclic attachment problem is the union calA = calA_π ∪ calA_ρ, where calA_π is the set of normalized Hecke polynomials occurring for π outside the fixed exceptional set, and calA_ρ is the set of characteristic polynomials occurring for the finitely many twists ρ⊗ϵ^j, j∈ZZ/pZZ.
Exact mathematical passage
\calA = \calA_{\pi} \cup \calA_{\rho},
\calEDefinition passage · 5 passages
Search aliases: calE
Definition passage: Entire packet set · body.tex, line 1499
[Entire packet set] An entire packet set calE is a collection of characters of subgroups of G whose Artin L functions are known to be entire.
Exact mathematical passage
\calE
\calJDefinition passage · 7 passages
Search aliases: calJ
Definition passage: Alignment set · body.tex, line 456
[Alignment set] For V,W with the same restriction to H, define calJ_H(V,W) = {j∈ZZ/pZZ: P_τ,W=z^jP_τ,V for every stable τ}.
Exact mathematical passage
\boxed{ \calJ_H(V,W) = \{j\in\ZZ/p\ZZ: P_{\tau,W}=z^jP_{\tau,V} \text{ for every stable }\tau\}. }
\calJ_HDefinition passage · 2 passages
Search aliases: calJ_H calJ
Definition passage: Alignment set · body.tex, line 456
[Alignment set] For V,W with the same restriction to H, define calJ_H(V,W) = {j∈ZZ/pZZ: P_τ,W=z^jP_τ,V for every stable τ}.
Exact mathematical passage
\boxed{ \calJ_H(V,W) = \{j\in\ZZ/p\ZZ: P_{\tau,W}=z^jP_{\tau,V} \text{ for every stable }\tau\}. }
\calJ_vDefinition passage · 5 passages
Search aliases: calJ_v calJ
Definition passage: Local attachment set · body.tex, line 631
[Local attachment set] When the local quotient has order p, define calJ_v = {j∈ZZ/pZZ: A_v⊗ϵ_v^j≅ R_v}. At a split place, set calJ_v=ZZ/pZZ.
Exact mathematical passage
\calJ_v = \{j\in\ZZ/p\ZZ: A_v\otimes\epsilon_v^j\cong R_v\}.
\calQDefinition passage · 4 passages
Search aliases: calQ
Definition passage: Universal bounded cyclotomic alphabet · body.tex, line 959
[Universal bounded cyclotomic alphabet] For positive integers n and B, let calQ_n,B be the finite set of monic degree n polynomials whose roots are roots of unity of order at most B.
Exact mathematical passage
\calQ_{n,B}
\calQ_{n,B}Definition passage · 4 passages
Search aliases: calQ_n,B calQ
Definition passage: Universal bounded cyclotomic alphabet · body.tex, line 959
[Universal bounded cyclotomic alphabet] For positive integers n and B, let calQ_n,B be the finite set of monic degree n polynomials whose roots are roots of unity of order at most B.
Exact mathematical passage
\calQ_{n,B}
\CCDefinition passage · 14 passages
Search aliases: CC
Definition passage: Finite Satake cusp form · body.tex, line 779
[Finite Satake cusp form] A cuspidal representation of GL_n(Adele_F) is finite Satake when, outside one finite exceptional set, its normalized unramified Satake roots lie in one finite subgroup of CC^×. The exceptional set is fixed, not allowed to change separately for each coefficient or prime.
Exact mathematical passage
\CC^\times
\CC^\timesDefinition passage · 4 passages
Search aliases: CC^× CC times
Definition passage: Finite Satake cusp form · body.tex, line 779
[Finite Satake cusp form] A cuspidal representation of GL_n(Adele_F) is finite Satake when, outside one finite exceptional set, its normalized unramified Satake roots lie in one finite subgroup of CC^×. The exceptional set is fixed, not allowed to change separately for each coefficient or prime.
Exact mathematical passage
\CC^\times
EDefinition passage · 45 passages
Search aliases: E
Definition passage: Attached cyclic descent · body.tex, line 100
[Attached cyclic descent] Let E/F be cyclic of prime degree p, and let ρ be an irreducible finite image representation of G_F whose restriction τ=ρ|_G_E is irreducible. Suppose τ is automorphic with cuspidal representation Π of GL_n(Adele_E). An attached cyclic descent is a cuspidal descent π of Π such that rec_F_v(π_v) ≅ ρ|_W_F_v for every place v of F.
Exact mathematical passage
E/F
eDefinition passage · 34 passages
Search aliases: e
Definition passage: Active coefficient degrees · body.tex, line 162
[Active coefficient degrees] Define S(A)={k∈{1,…,n}:e_k(A)≠0}.
Exact mathematical passage
S(A)=\{k\in\{1,\ldots,n\}:e_k(A)\neq0\}.
e_kDefinition passage · 4 passages
Search aliases: e_k
Definition passage: Active coefficient degrees · body.tex, line 162
[Active coefficient degrees] Define S(A)={k∈{1,…,n}:e_k(A)≠0}.
Exact mathematical passage
S(A)=\{k\in\{1,\ldots,n\}:e_k(A)\neq0\}.
FDefinition passage · 69 passages
Search aliases: F
Definition passage: Attached cyclic descent · body.tex, line 100
[Attached cyclic descent] Let E/F be cyclic of prime degree p, and let ρ be an irreducible finite image representation of G_F whose restriction τ=ρ|_G_E is irreducible. Suppose τ is automorphic with cuspidal representation Π of GL_n(Adele_E). An attached cyclic descent is a cuspidal descent π of Π such that rec_F_v(π_v) ≅ ρ|_W_F_v for every place v of F.
Exact mathematical passage
E/F
F_vDefinition passage · 5 passages
Search aliases: F_v
Definition passage: Attached cyclic descent · body.tex, line 101
Attached cyclic descent] Let E/F be cyclic of prime degree p, and let ρ be an irreducible finite image representation of G_F whose restriction τ=ρ|_G_E is irreducible. Suppose τ is automorphic with cuspidal representation Π of GL_n(Adele_E). An attached cyclic descent is a cuspidal descent π of Π such that rec_F_v(π_v) ≅ ρ|_W_F_v for every place v of F.
Exact mathematical passage
\rec_{F_v}(\pi_v) \cong \rho|_{W_{F_v}}
GDefinition passage · 97 passages
Search aliases: G
Definition passage: Attached cyclic descent · body.tex, line 100
[Attached cyclic descent] Let E/F be cyclic of prime degree p, and let ρ be an irreducible finite image representation of G_F whose restriction τ=ρ|_G_E is irreducible. Suppose τ is automorphic with cuspidal representation Π of GL_n(Adele_E). An attached cyclic descent is a cuspidal descent π of Π such that rec_F_v(π_v) ≅ ρ|_W_F_v for every place v of F.
Exact mathematical passage
G_F
gDefinition passage · 66 passages
Search aliases: g
Definition passage: Stable and moving types · body.tex, line 361
[Stable and moving types] An irreducible representation τ∈Irr(H) is stable when τ^g≅τ for every g∈ G. Otherwise its orbit has size p and is called moving.
Exact mathematical passage
\tau^g\cong\tau
G_EDefinition passage · 10 passages
Search aliases: G_E
Definition passage: Attached cyclic descent · body.tex, line 100
[Attached cyclic descent] Let E/F be cyclic of prime degree p, and let ρ be an irreducible finite image representation of G_F whose restriction τ=ρ|_G_E is irreducible. Suppose τ is automorphic with cuspidal representation Π of GL_n(Adele_E). An attached cyclic descent is a cuspidal descent π of Π such that rec_F_v(π_v) ≅ ρ|_W_F_v for every place v of F.
Exact mathematical passage
\tau=\rho|_{G_E}
G_FDefinition passage · 17 passages
Search aliases: G_F
Definition passage: Attached cyclic descent · body.tex, line 100
[Attached cyclic descent] Let E/F be cyclic of prime degree p, and let ρ be an irreducible finite image representation of G_F whose restriction τ=ρ|_G_E is irreducible. Suppose τ is automorphic with cuspidal representation Π of GL_n(Adele_E). An attached cyclic descent is a cuspidal descent π of Π such that rec_F_v(π_v) ≅ ρ|_W_F_v for every place v of F.
Exact mathematical passage
G_F
\GLDefinition passage · 27 passages
Search aliases: GL
Definition passage: Attached cyclic descent · body.tex, line 100
[Attached cyclic descent] Let E/F be cyclic of prime degree p, and let ρ be an irreducible finite image representation of G_F whose restriction τ=ρ|_G_E is irreducible. Suppose τ is automorphic with cuspidal representation Π of GL_n(Adele_E). An attached cyclic descent is a cuspidal descent π of Π such that rec_F_v(π_v) ≅ ρ|_W_F_v for every place v of F.
Exact mathematical passage
\GL_n(\Adele_E)
\GL_nDefinition passage · 26 passages
Search aliases: GL_n GL
Definition passage: Attached cyclic descent · body.tex, line 100
[Attached cyclic descent] Let E/F be cyclic of prime degree p, and let ρ be an irreducible finite image representation of G_F whose restriction τ=ρ|_G_E is irreducible. Suppose τ is automorphic with cuspidal representation Π of GL_n(Adele_E). An attached cyclic descent is a cuspidal descent π of Π such that rec_F_v(π_v) ≅ ρ|_W_F_v for every place v of F.
Exact mathematical passage
\GL_n(\Adele_E)
HDefinition passage · 85 passages
Search aliases: H
Definition passage: Stable and moving types · body.tex, line 361
[Stable and moving types] An irreducible representation τ∈Irr(H) is stable when τ^g≅τ for every g∈ G. Otherwise its orbit has size p and is called moving.
Exact mathematical passage
\tau\in\Irr(H)
\HomDefinition passage · 2 passages
Search aliases: Hom
Definition passage: Extension spectrum polynomial · body.tex, line 384
[Extension spectrum polynomial] For a stable type τ, the multiplicity space M_τ(V)=Hom_H(τ,V) becomes a representation of G/H after comparing the action on V with widetildeτ. Write M_τ(V) ≅ ⊕_a∈ZZ/pZZ (ξ^a)^⊕ m_τ,a(V). Define P_τ,V(z) = ∑_a∈ZZ/pZZ m_τ,a(V)z^a in ZZ[z]/(z^p-1).
Exact mathematical passage
M_\tau(V)=\Hom_H(\tau,V)
\Hom_HDefinition passage · 1 passage
Search aliases: Hom_H Hom
Definition passage: Extension spectrum polynomial · body.tex, line 384
[Extension spectrum polynomial] For a stable type τ, the multiplicity space M_τ(V)=Hom_H(τ,V) becomes a representation of G/H after comparing the action on V with widetildeτ. Write M_τ(V) ≅ ⊕_a∈ZZ/pZZ (ξ^a)^⊕ m_τ,a(V). Define P_τ,V(z) = ∑_a∈ZZ/pZZ m_τ,a(V)z^a in ZZ[z]/(z^p-1).
Exact mathematical passage
M_\tau(V)=\Hom_H(\tau,V)
iDefinition passage · 39 passages
Search aliases: i
Definition passage: Frobenius transform of a polynomial · body.tex, line 1110
[Frobenius transform of a polynomial] If P(T)=∏_i=1^n(T-α_i), define P^[p](T) = ∏_i=1^n(T-α_i^p). This is a polynomial expression in the coefficients of P.
Exact mathematical passage
P(T)=\prod_{i=1}^{n}(T-\alpha_i),
\IrrDefinition passage · 1 passage
Search aliases: Irr
Definition passage: Stable and moving types · body.tex, line 361
[Stable and moving types] An irreducible representation τ∈Irr(H) is stable when τ^g≅τ for every g∈ G. Otherwise its orbit has size p and is called moving.
Exact mathematical passage
\tau\in\Irr(H)
jDefinition passage · 60 passages
Search aliases: j
Definition passage: Alignment set · body.tex, line 456
[Alignment set] For V,W with the same restriction to H, define calJ_H(V,W) = {j∈ZZ/pZZ: P_τ,W=z^jP_τ,V for every stable τ}.
Exact mathematical passage
\boxed{ \calJ_H(V,W) = \{j\in\ZZ/p\ZZ: P_{\tau,W}=z^jP_{\tau,V} \text{ for every stable }\tau\}. }
KDefinition passage · 30 passages
Search aliases: K
Definition passage: Separating coefficient prime · body.tex, line 853
[Separating coefficient prime] A finite place λ of K is separating for calA when reduction modulo λ is injective on calA.
Exact mathematical passage
K
kDefinition passage · 37 passages
Search aliases: k
Definition passage: Active coefficient degrees · body.tex, line 162
[Active coefficient degrees] Define S(A)={k∈{1,…,n}:e_k(A)≠0}.
Exact mathematical passage
S(A)=\{k\in\{1,\ldots,n\}:e_k(A)\neq0\}.
LDefinition passage · 31 passages
Search aliases: L
Definition passage: Entire packet set · body.tex, line 1499
[Entire packet set] An entire packet set calE is a collection of characters of subgroups of G whose Artin L functions are known to be entire.
Exact mathematical passage
L
MDefinition passage · 38 passages
Search aliases: M
Definition passage: Extension spectrum polynomial · body.tex, line 384
[Extension spectrum polynomial] For a stable type τ, the multiplicity space M_τ(V)=Hom_H(τ,V) becomes a representation of G/H after comparing the action on V with widetildeτ. Write M_τ(V) ≅ ⊕_a∈ZZ/pZZ (ξ^a)^⊕ m_τ,a(V). Define P_τ,V(z) = ∑_a∈ZZ/pZZ m_τ,a(V)z^a in ZZ[z]/(z^p-1).
Exact mathematical passage
M_\tau(V)=\Hom_H(\tau,V)
mDefinition passage · 30 passages
Search aliases: m
Definition passage: Extension spectrum polynomial · body.tex, line 388
[Extension spectrum polynomial] For a stable type τ, the multiplicity space M_τ(V)=Hom_H(τ,V) becomes a representation of G/H after comparing the action on V with widetildeτ. Write M_τ(V) ≅ ⊕_a∈ZZ/pZZ (ξ^a)^⊕ m_τ,a(V). Define P_τ,V(z) = ∑_a∈ZZ/pZZ m_τ,a(V)z^a in ZZ[z]/(z^p-1).
Exact mathematical passage
M_\tau(V) \cong \bigoplus_{a\in\ZZ/p\ZZ} (\xi^a)^{\oplus m_{\tau,a}(V)}.
M_\tauDefinition passage · 5 passages
Search aliases: M_τ tau
Definition passage: Extension spectrum polynomial · body.tex, line 384
[Extension spectrum polynomial] For a stable type τ, the multiplicity space M_τ(V)=Hom_H(τ,V) becomes a representation of G/H after comparing the action on V with widetildeτ. Write M_τ(V) ≅ ⊕_a∈ZZ/pZZ (ξ^a)^⊕ m_τ,a(V). Define P_τ,V(z) = ∑_a∈ZZ/pZZ m_τ,a(V)z^a in ZZ[z]/(z^p-1).
Exact mathematical passage
M_\tau(V)=\Hom_H(\tau,V)
m_{\tau,a}Definition passage · 3 passages
Search aliases: m_τ,a tau
Definition passage: Extension spectrum polynomial · body.tex, line 388
[Extension spectrum polynomial] For a stable type τ, the multiplicity space M_τ(V)=Hom_H(τ,V) becomes a representation of G/H after comparing the action on V with widetildeτ. Write M_τ(V) ≅ ⊕_a∈ZZ/pZZ (ξ^a)^⊕ m_τ,a(V). Define P_τ,V(z) = ∑_a∈ZZ/pZZ m_τ,a(V)z^a in ZZ[z]/(z^p-1).
Exact mathematical passage
M_\tau(V) \cong \bigoplus_{a\in\ZZ/p\ZZ} (\xi^a)^{\oplus m_{\tau,a}(V)}.
nDefinition passage · 114 passages
Search aliases: n
Definition passage: Attached cyclic descent · body.tex, line 100
[Attached cyclic descent] Let E/F be cyclic of prime degree p, and let ρ be an irreducible finite image representation of G_F whose restriction τ=ρ|_G_E is irreducible. Suppose τ is automorphic with cuspidal representation Π of GL_n(Adele_E). An attached cyclic descent is a cuspidal descent π of Π such that rec_F_v(π_v) ≅ ρ|_W_F_v for every place v of F.
Exact mathematical passage
\GL_n(\Adele_E)
\mathcal ODefinition passage · 1 passage
Search aliases: O mathcal
Definition passage: Pseudocharacter family gate · body.tex, line 1284
[Pseudocharacter family gate] A pseudocharacter family gate for a finite Satake descent π consists of a reduced p adic analytic space X, a closed point x valued in a finite extension of QQ_ℓ, and a continuous dimension n characteristic zero pseudorepresentation over O(X). It factors through the Galois group unramified outside one finite set, and its unramified characteristic polynomials specialize at x to the embedded Hecke polynomials of π. We use a point and coefficient setting in which charac…
Exact mathematical passage
\mathcal O(X)
\overline kDefinition passage · 1 passage
Search aliases: overline k overline
Definition passage: Residual realization of a descent · body.tex, line 998
[Residual realization of a descent] Let λ be a finite place of the coefficient field. A residual realization of π at λ is a continuous semisimple representation overline r_π,λ:G_F⟶GL_n(overline k_λ) whose characteristic polynomial at every unramified place outside a finite set is the reduction of the normalized Hecke polynomial of π.
Exact mathematical passage
\overline r_{\pi,\lambda}:G_F\longrightarrow\GL_n(\overline k_{\lambda})
\overline k_{\lambda}Definition passage · 1 passage
Search aliases: overline k_λ overline lambda
Definition passage: Residual realization of a descent · body.tex, line 998
[Residual realization of a descent] Let λ be a finite place of the coefficient field. A residual realization of π at λ is a continuous semisimple representation overline r_π,λ:G_F⟶GL_n(overline k_λ) whose characteristic polynomial at every unramified place outside a finite set is the reduction of the normalized Hecke polynomial of π.
Exact mathematical passage
\overline r_{\pi,\lambda}:G_F\longrightarrow\GL_n(\overline k_{\lambda})
\overline rDefinition passage · 10 passages
Search aliases: overline r overline
Definition passage: Residual realization of a descent · body.tex, line 998
[Residual realization of a descent] Let λ be a finite place of the coefficient field. A residual realization of π at λ is a continuous semisimple representation overline r_π,λ:G_F⟶GL_n(overline k_λ) whose characteristic polynomial at every unramified place outside a finite set is the reduction of the normalized Hecke polynomial of π.
Exact mathematical passage
\overline r_{\pi,\lambda}:G_F\longrightarrow\GL_n(\overline k_{\lambda})
\overline r_{\pi,\lambda}Definition passage · 5 passages
Search aliases: overline r_π,λ overline pi lambda
Definition passage: Residual realization of a descent · body.tex, line 998
[Residual realization of a descent] Let λ be a finite place of the coefficient field. A residual realization of π at λ is a continuous semisimple representation overline r_π,λ:G_F⟶GL_n(overline k_λ) whose characteristic polynomial at every unramified place outside a finite set is the reduction of the normalized Hecke polynomial of π.
Exact mathematical passage
\overline r_{\pi,\lambda}:G_F\longrightarrow\GL_n(\overline k_{\lambda})
PDefinition passage · 29 passages
Search aliases: P
Definition passage: Extension spectrum polynomial · body.tex, line 395
[Extension spectrum polynomial] For a stable type τ, the multiplicity space M_τ(V)=Hom_H(τ,V) becomes a representation of G/H after comparing the action on V with widetildeτ. Write M_τ(V) ≅ ⊕_a∈ZZ/pZZ (ξ^a)^⊕ m_τ,a(V). Define P_τ,V(z) = ∑_a∈ZZ/pZZ m_τ,a(V)z^a in ZZ[z]/(z^p-1).
Exact mathematical passage
\boxed{ P_{\tau,V}(z) = \sum_{a\in\ZZ/p\ZZ} m_{\tau,a}(V)z^a }
pDefinition passage · 162 passages
Search aliases: p
Definition passage: Attached cyclic descent · body.tex, line 100
[Attached cyclic descent] Let E/F be cyclic of prime degree p, and let ρ be an irreducible finite image representation of G_F whose restriction τ=ρ|_G_E is irreducible. Suppose τ is automorphic with cuspidal representation Π of GL_n(Adele_E). An attached cyclic descent is a cuspidal descent π of Π such that rec_F_v(π_v) ≅ ρ|_W_F_v for every place v of F.
Exact mathematical passage
p
P^{[p]}Definition passage · 1 passage
Search aliases: P^[p]
Definition passage: Frobenius transform of a polynomial · body.tex, line 1114
[Frobenius transform of a polynomial] If P(T)=∏_i=1^n(T-α_i), define P^[p](T) = ∏_i=1^n(T-α_i^p). This is a polynomial expression in the coefficients of P.
Exact mathematical passage
P^{[p]}(T) = \prod_{i=1}^{n}(T-\alpha_i^p).
P_{\tau,V}Definition passage · 8 passages
Search aliases: P_τ,V tau
Definition passage: Extension spectrum polynomial · body.tex, line 395
[Extension spectrum polynomial] For a stable type τ, the multiplicity space M_τ(V)=Hom_H(τ,V) becomes a representation of G/H after comparing the action on V with widetildeτ. Write M_τ(V) ≅ ⊕_a∈ZZ/pZZ (ξ^a)^⊕ m_τ,a(V). Define P_τ,V(z) = ∑_a∈ZZ/pZZ m_τ,a(V)z^a in ZZ[z]/(z^p-1).
Exact mathematical passage
\boxed{ P_{\tau,V}(z) = \sum_{a\in\ZZ/p\ZZ} m_{\tau,a}(V)z^a }
P_{\tau,W}Definition passage · 2 passages
Search aliases: P_τ,W tau
Definition passage: Alignment set · body.tex, line 456
[Alignment set] For V,W with the same restriction to H, define calJ_H(V,W) = {j∈ZZ/pZZ: P_τ,W=z^jP_τ,V for every stable τ}.
Exact mathematical passage
\boxed{ \calJ_H(V,W) = \{j\in\ZZ/p\ZZ: P_{\tau,W}=z^jP_{\tau,V} \text{ for every stable }\tau\}. }
\QQDefinition passage · 9 passages
Search aliases: QQ
Definition passage: Pseudocharacter family gate · body.tex, line 1284
[Pseudocharacter family gate] A pseudocharacter family gate for a finite Satake descent π consists of a reduced p adic analytic space X, a closed point x valued in a finite extension of QQ_ℓ, and a continuous dimension n characteristic zero pseudorepresentation over O(X). It factors through the Galois group unramified outside one finite set, and its unramified characteristic polynomials specialize at x to the embedded Hecke polynomials of π. We use a point and coefficient setting in which charac…
Exact mathematical passage
\QQ_\ell
\QQ_\ellDefinition passage · 4 passages
Search aliases: QQ_ℓ QQ ell
Definition passage: Pseudocharacter family gate · body.tex, line 1284
[Pseudocharacter family gate] A pseudocharacter family gate for a finite Satake descent π consists of a reduced p adic analytic space X, a closed point x valued in a finite extension of QQ_ℓ, and a continuous dimension n characteristic zero pseudorepresentation over O(X). It factors through the Galois group unramified outside one finite set, and its unramified characteristic polynomials specialize at x to the embedded Hecke polynomials of π. We use a point and coefficient setting in which charac…
Exact mathematical passage
\QQ_\ell
RDefinition passage · 29 passages
Search aliases: R
Definition passage: Local attachment set · body.tex, line 631
[Local attachment set] When the local quotient has order p, define calJ_v = {j∈ZZ/pZZ: A_v⊗ϵ_v^j≅ R_v}. At a split place, set calJ_v=ZZ/pZZ.
Exact mathematical passage
\calJ_v = \{j\in\ZZ/p\ZZ: A_v\otimes\epsilon_v^j\cong R_v\}.
rDefinition passage · 122 passages
Search aliases: r
Definition passage: Local spectral obstruction · body.tex, line 561
[Local spectral obstruction] The local spectral obstruction vanishes when, for every stable type τ, the zero Fourier modes of the two multiplicity spectra occur at the same indices and all ratios at the remaining indices satisfy widehat P_τ,W(r)/widehat P_τ,V(r)=ζ^jr for one common j. When widehat P_τ,V(r)=0, the condition requires widehat P_τ,W(r)=0, rather than omitting the comparison. Equivalently, all the Fourier identities in [source reference] hold. If no such j exists, the obstruction is …
Exact mathematical passage
\widehat P_{\tau,W}(r)/\widehat P_{\tau,V}(r)=\zeta^{jr}
R_vDefinition passage · 6 passages
Search aliases: R_v
Definition passage: Local attachment set · body.tex, line 631
[Local attachment set] When the local quotient has order p, define calJ_v = {j∈ZZ/pZZ: A_v⊗ϵ_v^j≅ R_v}. At a split place, set calJ_v=ZZ/pZZ.
Exact mathematical passage
\calJ_v = \{j\in\ZZ/p\ZZ: A_v\otimes\epsilon_v^j\cong R_v\}.
\recDefinition passage · 5 passages
Search aliases: rec
Definition passage: Attached cyclic descent · body.tex, line 101
Attached cyclic descent] Let E/F be cyclic of prime degree p, and let ρ be an irreducible finite image representation of G_F whose restriction τ=ρ|_G_E is irreducible. Suppose τ is automorphic with cuspidal representation Π of GL_n(Adele_E). An attached cyclic descent is a cuspidal descent π of Π such that rec_F_v(π_v) ≅ ρ|_W_F_v for every place v of F.
Exact mathematical passage
\rec_{F_v}(\pi_v) \cong \rho|_{W_{F_v}}
\rec_{F_v}Definition passage · 3 passages
Search aliases: rec_F_v rec
Definition passage: Attached cyclic descent · body.tex, line 101
Attached cyclic descent] Let E/F be cyclic of prime degree p, and let ρ be an irreducible finite image representation of G_F whose restriction τ=ρ|_G_E is irreducible. Suppose τ is automorphic with cuspidal representation Π of GL_n(Adele_E). An attached cyclic descent is a cuspidal descent π of Π such that rec_F_v(π_v) ≅ ρ|_W_F_v for every place v of F.
Exact mathematical passage
\rec_{F_v}(\pi_v) \cong \rho|_{W_{F_v}}
SDefinition passage · 18 passages
Search aliases: S
Definition passage: Active coefficient degrees · body.tex, line 162
[Active coefficient degrees] Define S(A)={k∈{1,…,n}:e_k(A)≠0}.
Exact mathematical passage
S(A)=\{k\in\{1,\ldots,n\}:e_k(A)\neq0\}.
\operatorname{Sat}Definition passage · 2 passages
Search aliases: Sat operatorname
Definition passage: Entire induction cone · body.tex, line 1538
[Entire induction cone] Fix finitely many known entire induced packets Θ_1,…,Θ_r in the character lattice of G. Define C_calE = { ∑_i=1^ra_iΘ_i: a_i∈ZZ_≥0 }. Its saturation is Sat(C_calE) = {x:some m>0 has mx∈ C_calE}.
Exact mathematical passage
\operatorname{Sat}(C_{\calE}) = \{x:\text{some }m>0\text{ has }mx\in C_{\calE}\}.
TDefinition passage · 46 passages
Search aliases: T
Definition passage: Frobenius transform of a polynomial · body.tex, line 1110
[Frobenius transform of a polynomial] If P(T)=∏_i=1^n(T-α_i), define P^[p](T) = ∏_i=1^n(T-α_i^p). This is a polynomial expression in the coefficients of P.
Exact mathematical passage
P(T)=\prod_{i=1}^{n}(T-\alpha_i),
VDefinition passage · 66 passages
Search aliases: V
Definition passage: Extension spectrum polynomial · body.tex, line 384
[Extension spectrum polynomial] For a stable type τ, the multiplicity space M_τ(V)=Hom_H(τ,V) becomes a representation of G/H after comparing the action on V with widetildeτ. Write M_τ(V) ≅ ⊕_a∈ZZ/pZZ (ξ^a)^⊕ m_τ,a(V). Define P_τ,V(z) = ∑_a∈ZZ/pZZ m_τ,a(V)z^a in ZZ[z]/(z^p-1).
Exact mathematical passage
M_\tau(V)=\Hom_H(\tau,V)
vDefinition passage · 71 passages
Search aliases: v
Definition passage: Attached cyclic descent · body.tex, line 101
Attached cyclic descent] Let E/F be cyclic of prime degree p, and let ρ be an irreducible finite image representation of G_F whose restriction τ=ρ|_G_E is irreducible. Suppose τ is automorphic with cuspidal representation Π of GL_n(Adele_E). An attached cyclic descent is a cuspidal descent π of Π such that rec_F_v(π_v) ≅ ρ|_W_F_v for every place v of F.
Exact mathematical passage
\rec_{F_v}(\pi_v) \cong \rho|_{W_{F_v}}
WDefinition passage · 35 passages
Search aliases: W
Definition passage: Attached cyclic descent · body.tex, line 101
Attached cyclic descent] Let E/F be cyclic of prime degree p, and let ρ be an irreducible finite image representation of G_F whose restriction τ=ρ|_G_E is irreducible. Suppose τ is automorphic with cuspidal representation Π of GL_n(Adele_E). An attached cyclic descent is a cuspidal descent π of Π such that rec_F_v(π_v) ≅ ρ|_W_F_v for every place v of F.
Exact mathematical passage
\rec_{F_v}(\pi_v) \cong \rho|_{W_{F_v}}
W_{F_v}Definition passage · 5 passages
Search aliases: W_F_v
Definition passage: Attached cyclic descent · body.tex, line 101
Attached cyclic descent] Let E/F be cyclic of prime degree p, and let ρ be an irreducible finite image representation of G_F whose restriction τ=ρ|_G_E is irreducible. Suppose τ is automorphic with cuspidal representation Π of GL_n(Adele_E). An attached cyclic descent is a cuspidal descent π of Π such that rec_F_v(π_v) ≅ ρ|_W_F_v for every place v of F.
Exact mathematical passage
\rec_{F_v}(\pi_v) \cong \rho|_{W_{F_v}}
\widehat PDefinition passage · 9 passages
Search aliases: widehat P widehat
Definition passage: Local spectral obstruction · body.tex, line 561
[Local spectral obstruction] The local spectral obstruction vanishes when, for every stable type τ, the zero Fourier modes of the two multiplicity spectra occur at the same indices and all ratios at the remaining indices satisfy widehat P_τ,W(r)/widehat P_τ,V(r)=ζ^jr for one common j. When widehat P_τ,V(r)=0, the condition requires widehat P_τ,W(r)=0, rather than omitting the comparison. Equivalently, all the Fourier identities in [source reference] hold. If no such j exists, the obstruction is …
Exact mathematical passage
\widehat P_{\tau,W}(r)/\widehat P_{\tau,V}(r)=\zeta^{jr}
\widehat P_{\tau,V}Definition passage · 7 passages
Search aliases: widehat P_τ,V widehat tau
Definition passage: Local spectral obstruction · body.tex, line 561
[Local spectral obstruction] The local spectral obstruction vanishes when, for every stable type τ, the zero Fourier modes of the two multiplicity spectra occur at the same indices and all ratios at the remaining indices satisfy widehat P_τ,W(r)/widehat P_τ,V(r)=ζ^jr for one common j. When widehat P_τ,V(r)=0, the condition requires widehat P_τ,W(r)=0, rather than omitting the comparison. Equivalently, all the Fourier identities in [source reference] hold. If no such j exists, the obstruction is …
Exact mathematical passage
\widehat P_{\tau,W}(r)/\widehat P_{\tau,V}(r)=\zeta^{jr}
\widehat P_{\tau,W}Definition passage · 4 passages
Search aliases: widehat P_τ,W widehat tau
Definition passage: Local spectral obstruction · body.tex, line 561
[Local spectral obstruction] The local spectral obstruction vanishes when, for every stable type τ, the zero Fourier modes of the two multiplicity spectra occur at the same indices and all ratios at the remaining indices satisfy widehat P_τ,W(r)/widehat P_τ,V(r)=ζ^jr for one common j. When widehat P_τ,V(r)=0, the condition requires widehat P_τ,W(r)=0, rather than omitting the comparison. Equivalently, all the Fourier identities in [source reference] hold. If no such j exists, the obstruction is …
Exact mathematical passage
\widehat P_{\tau,W}(r)/\widehat P_{\tau,V}(r)=\zeta^{jr}
\widetilde\tauDefinition passage · 3 passages
Search aliases: widetildeτ widetilde tau
Definition passage: Extension spectrum polynomial · body.tex, line 387
[Extension spectrum polynomial] For a stable type τ, the multiplicity space M_τ(V)=Hom_H(τ,V) becomes a representation of G/H after comparing the action on V with widetildeτ. Write M_τ(V) ≅ ⊕_a∈ZZ/pZZ (ξ^a)^⊕ m_τ,a(V). Define P_τ,V(z) = ∑_a∈ZZ/pZZ m_τ,a(V)z^a in ZZ[z]/(z^p-1).
Exact mathematical passage
\widetilde\tau
XDefinition passage · 28 passages
Search aliases: X
Definition passage: Pseudocharacter family gate · body.tex, line 1284
[Pseudocharacter family gate] A pseudocharacter family gate for a finite Satake descent π consists of a reduced p adic analytic space X, a closed point x valued in a finite extension of QQ_ℓ, and a continuous dimension n characteristic zero pseudorepresentation over O(X). It factors through the Galois group unramified outside one finite set, and its unramified characteristic polynomials specialize at x to the embedded Hecke polynomials of π. We use a point and coefficient setting in which charac…
Exact mathematical passage
X
xDefinition passage · 37 passages
Search aliases: x
Definition passage: Pseudocharacter family gate · body.tex, line 1284
[Pseudocharacter family gate] A pseudocharacter family gate for a finite Satake descent π consists of a reduced p adic analytic space X, a closed point x valued in a finite extension of QQ_ℓ, and a continuous dimension n characteristic zero pseudorepresentation over O(X). It factors through the Galois group unramified outside one finite set, and its unramified characteristic polynomials specialize at x to the embedded Hecke polynomials of π. We use a point and coefficient setting in which charac…
Exact mathematical passage
x
zDefinition passage · 24 passages
Search aliases: z
Definition passage: Extension spectrum polynomial · body.tex, line 395
[Extension spectrum polynomial] For a stable type τ, the multiplicity space M_τ(V)=Hom_H(τ,V) becomes a representation of G/H after comparing the action on V with widetildeτ. Write M_τ(V) ≅ ⊕_a∈ZZ/pZZ (ξ^a)^⊕ m_τ,a(V). Define P_τ,V(z) = ∑_a∈ZZ/pZZ m_τ,a(V)z^a in ZZ[z]/(z^p-1).
Exact mathematical passage
\boxed{ P_{\tau,V}(z) = \sum_{a\in\ZZ/p\ZZ} m_{\tau,a}(V)z^a }
z^aDefinition passage · 3 passages
Search aliases: z^a
Definition passage: Extension spectrum polynomial · body.tex, line 395
[Extension spectrum polynomial] For a stable type τ, the multiplicity space M_τ(V)=Hom_H(τ,V) becomes a representation of G/H after comparing the action on V with widetildeτ. Write M_τ(V) ≅ ⊕_a∈ZZ/pZZ (ξ^a)^⊕ m_τ,a(V). Define P_τ,V(z) = ∑_a∈ZZ/pZZ m_τ,a(V)z^a in ZZ[z]/(z^p-1).
Exact mathematical passage
\boxed{ P_{\tau,V}(z) = \sum_{a\in\ZZ/p\ZZ} m_{\tau,a}(V)z^a }
z^jDefinition passage · 4 passages
Search aliases: z^j
Definition passage: Alignment set · body.tex, line 456
[Alignment set] For V,W with the same restriction to H, define calJ_H(V,W) = {j∈ZZ/pZZ: P_τ,W=z^jP_τ,V for every stable τ}.
Exact mathematical passage
\boxed{ \calJ_H(V,W) = \{j\in\ZZ/p\ZZ: P_{\tau,W}=z^jP_{\tau,V} \text{ for every stable }\tau\}. }
z^pDefinition passage · 1 passage
Search aliases: z^p
Definition passage: Extension spectrum polynomial · body.tex, line 403
m polynomial] For a stable type τ, the multiplicity space M_τ(V)=Hom_H(τ,V) becomes a representation of G/H after comparing the action on V with widetildeτ. Write M_τ(V) ≅ ⊕_a∈ZZ/pZZ (ξ^a)^⊕ m_τ,a(V). Define P_τ,V(z) = ∑_a∈ZZ/pZZ m_τ,a(V)z^a in ZZ[z]/(z^p-1).
Exact mathematical passage
\ZZ[z]/(z^p-1)
\ZZDefinition passage · 21 passages
Search aliases: ZZ
Definition passage: Extension spectrum polynomial · body.tex, line 388
[Extension spectrum polynomial] For a stable type τ, the multiplicity space M_τ(V)=Hom_H(τ,V) becomes a representation of G/H after comparing the action on V with widetildeτ. Write M_τ(V) ≅ ⊕_a∈ZZ/pZZ (ξ^a)^⊕ m_τ,a(V). Define P_τ,V(z) = ∑_a∈ZZ/pZZ m_τ,a(V)z^a in ZZ[z]/(z^p-1).
Exact mathematical passage
M_\tau(V) \cong \bigoplus_{a\in\ZZ/p\ZZ} (\xi^a)^{\oplus m_{\tau,a}(V)}.
\ZZ_{\geq0}Definition passage · 1 passage
Search aliases: ZZ_≥0 ZZ geq
Definition passage: Entire induction cone · body.tex, line 1529
[Entire induction cone] Fix finitely many known entire induced packets Θ_1,…,Θ_r in the character lattice of G. Define C_calE = { ∑_i=1^ra_iΘ_i: a_i∈ZZ_≥0 }. Its saturation is Sat(C_calE) = {x:some m>0 has mx∈ C_calE}.
Exact mathematical passage
C_{\calE} = \left\{ \sum_{i=1}^ra_i\Theta_i: a_i\in\ZZ_{\geq0} \right\}.
|Definition passage · 14 passages
Search aliases: |
Definition passage: Attached cyclic descent · body.tex, line 100
[Attached cyclic descent] Let E/F be cyclic of prime degree p, and let ρ be an irreducible finite image representation of G_F whose restriction τ=ρ|_G_E is irreducible. Suppose τ is automorphic with cuspidal representation Π of GL_n(Adele_E). An attached cyclic descent is a cuspidal descent π of Π such that rec_F_v(π_v) ≅ ρ|_W_F_v for every place v of F.
Exact mathematical passage
\tau=\rho|_{G_E}
|_{G_E}Definition passage · 4 passages
Search aliases: |_G_E
Definition passage: Attached cyclic descent · body.tex, line 100
[Attached cyclic descent] Let E/F be cyclic of prime degree p, and let ρ be an irreducible finite image representation of G_F whose restriction τ=ρ|_G_E is irreducible. Suppose τ is automorphic with cuspidal representation Π of GL_n(Adele_E). An attached cyclic descent is a cuspidal descent π of Π such that rec_F_v(π_v) ≅ ρ|_W_F_v for every place v of F.
Exact mathematical passage
\tau=\rho|_{G_E}
|_{W_{F_v}}Definition passage · 3 passages
Search aliases: |_W_F_v
Definition passage: Attached cyclic descent · body.tex, line 101
Attached cyclic descent] Let E/F be cyclic of prime degree p, and let ρ be an irreducible finite image representation of G_F whose restriction τ=ρ|_G_E is irreducible. Suppose τ is automorphic with cuspidal representation Π of GL_n(Adele_E). An attached cyclic descent is a cuspidal descent π of Π such that rec_F_v(π_v) ≅ ρ|_W_F_v for every place v of F.
Exact mathematical passage
\rec_{F_v}(\pi_v) \cong \rho|_{W_{F_v}}
\timesDefinition passage · 8 passages
Search aliases: × times
Definition passage: Finite Satake cusp form · body.tex, line 779
[Finite Satake cusp form] A cuspidal representation of GL_n(Adele_F) is finite Satake when, outside one finite exceptional set, its normalized unramified Satake roots lie in one finite subgroup of CC^×. The exceptional set is fixed, not allowed to change separately for each coefficient or prime.
Exact mathematical passage
\CC^\times
\alphaDefinition passage · 9 passages
Search aliases: α alpha
Definition passage: Frobenius transform of a polynomial · body.tex, line 1110
[Frobenius transform of a polynomial] If P(T)=∏_i=1^n(T-α_i), define P^[p](T) = ∏_i=1^n(T-α_i^p). This is a polynomial expression in the coefficients of P.
Exact mathematical passage
P(T)=\prod_{i=1}^{n}(T-\alpha_i),
\alpha_iDefinition passage · 2 passages
Search aliases: α_i alpha
Definition passage: Frobenius transform of a polynomial · body.tex, line 1110
[Frobenius transform of a polynomial] If P(T)=∏_i=1^n(T-α_i), define P^[p](T) = ∏_i=1^n(T-α_i^p). This is a polynomial expression in the coefficients of P.
Exact mathematical passage
P(T)=\prod_{i=1}^{n}(T-\alpha_i),
\alpha_i^pDefinition passage · 1 passage
Search aliases: α_i^p alpha
Definition passage: Frobenius transform of a polynomial · body.tex, line 1114
[Frobenius transform of a polynomial] If P(T)=∏_i=1^n(T-α_i), define P^[p](T) = ∏_i=1^n(T-α_i^p). This is a polynomial expression in the coefficients of P.
Exact mathematical passage
P^{[p]}(T) = \prod_{i=1}^{n}(T-\alpha_i^p).
\epsilonDefinition passage · 18 passages
Search aliases: ϵ epsilon
Definition passage: Local attachment set · body.tex, line 631
[Local attachment set] When the local quotient has order p, define calJ_v = {j∈ZZ/pZZ: A_v⊗ϵ_v^j≅ R_v}. At a split place, set calJ_v=ZZ/pZZ.
Exact mathematical passage
\calJ_v = \{j\in\ZZ/p\ZZ: A_v\otimes\epsilon_v^j\cong R_v\}.
\epsilon^jDefinition passage · 3 passages
Search aliases: ϵ^j epsilon
Definition passage: Finite polynomial alphabet · body.tex, line 837
nite polynomial alphabet] The finite polynomial alphabet of the cyclic attachment problem is the union calA = calA_π ∪ calA_ρ, where calA_π is the set of normalized Hecke polynomials occurring for π outside the fixed exceptional set, and calA_ρ is the set of characteristic polynomials occurring for the finitely many twists ρ⊗ϵ^j, j∈ZZ/pZZ.
Exact mathematical passage
\rho\otimes\epsilon^j, \qquad j\in\ZZ/p\ZZ.
\epsilon_v^jDefinition passage · 2 passages
Search aliases: ϵ_v^j epsilon
Definition passage: Local attachment set · body.tex, line 631
[Local attachment set] When the local quotient has order p, define calJ_v = {j∈ZZ/pZZ: A_v⊗ϵ_v^j≅ R_v}. At a split place, set calJ_v=ZZ/pZZ.
Exact mathematical passage
\calJ_v = \{j\in\ZZ/p\ZZ: A_v\otimes\epsilon_v^j\cong R_v\}.
\zetaDefinition passage · 18 passages
Search aliases: ζ zeta
Definition passage: Local spectral obstruction · body.tex, line 561
[Local spectral obstruction] The local spectral obstruction vanishes when, for every stable type τ, the zero Fourier modes of the two multiplicity spectra occur at the same indices and all ratios at the remaining indices satisfy widehat P_τ,W(r)/widehat P_τ,V(r)=ζ^jr for one common j. When widehat P_τ,V(r)=0, the condition requires widehat P_τ,W(r)=0, rather than omitting the comparison. Equivalently, all the Fourier identities in [source reference] hold. If no such j exists, the obstruction is …
Exact mathematical passage
\widehat P_{\tau,W}(r)/\widehat P_{\tau,V}(r)=\zeta^{jr}
\zeta^{jr}Definition passage · 6 passages
Search aliases: ζ^jr zeta
Definition passage: Local spectral obstruction · body.tex, line 561
[Local spectral obstruction] The local spectral obstruction vanishes when, for every stable type τ, the zero Fourier modes of the two multiplicity spectra occur at the same indices and all ratios at the remaining indices satisfy widehat P_τ,W(r)/widehat P_τ,V(r)=ζ^jr for one common j. When widehat P_τ,V(r)=0, the condition requires widehat P_τ,W(r)=0, rather than omitting the comparison. Equivalently, all the Fourier identities in [source reference] hold. If no such j exists, the obstruction is …
Exact mathematical passage
\widehat P_{\tau,W}(r)/\widehat P_{\tau,V}(r)=\zeta^{jr}
\ThetaDefinition passage · 2 passages
Search aliases: Θ Theta
Definition passage: Entire induction cone · body.tex, line 1528
[Entire induction cone] Fix finitely many known entire induced packets Θ_1,…,Θ_r in the character lattice of G. Define C_calE = { ∑_i=1^ra_iΘ_i: a_i∈ZZ_≥0 }. Its saturation is Sat(C_calE) = {x:some m>0 has mx∈ C_calE}.
Exact mathematical passage
\Theta_1,\ldots,\Theta_r
\Theta_1Definition passage · 1 passage
Search aliases: Θ_1 Theta
Definition passage: Entire induction cone · body.tex, line 1528
[Entire induction cone] Fix finitely many known entire induced packets Θ_1,…,Θ_r in the character lattice of G. Define C_calE = { ∑_i=1^ra_iΘ_i: a_i∈ZZ_≥0 }. Its saturation is Sat(C_calE) = {x:some m>0 has mx∈ C_calE}.
Exact mathematical passage
\Theta_1,\ldots,\Theta_r
\Theta_iDefinition passage · 1 passage
Search aliases: Θ_i Theta
Definition passage: Entire induction cone · body.tex, line 1529
[Entire induction cone] Fix finitely many known entire induced packets Θ_1,…,Θ_r in the character lattice of G. Define C_calE = { ∑_i=1^ra_iΘ_i: a_i∈ZZ_≥0 }. Its saturation is Sat(C_calE) = {x:some m>0 has mx∈ C_calE}.
Exact mathematical passage
C_{\calE} = \left\{ \sum_{i=1}^ra_i\Theta_i: a_i\in\ZZ_{\geq0} \right\}.
\Theta_rDefinition passage · 1 passage
Search aliases: Θ_r Theta
Definition passage: Entire induction cone · body.tex, line 1528
[Entire induction cone] Fix finitely many known entire induced packets Θ_1,…,Θ_r in the character lattice of G. Define C_calE = { ∑_i=1^ra_iΘ_i: a_i∈ZZ_≥0 }. Its saturation is Sat(C_calE) = {x:some m>0 has mx∈ C_calE}.
Exact mathematical passage
\Theta_1,\ldots,\Theta_r
\lambdaDefinition passage · 33 passages
Search aliases: λ lambda
Definition passage: Separating coefficient prime · body.tex, line 853
[Separating coefficient prime] A finite place λ of K is separating for calA when reduction modulo λ is injective on calA.
Exact mathematical passage
\lambda
\xiDefinition passage · 15 passages
Search aliases: ξ xi
Definition passage: Extension spectrum polynomial · body.tex, line 388
[Extension spectrum polynomial] For a stable type τ, the multiplicity space M_τ(V)=Hom_H(τ,V) becomes a representation of G/H after comparing the action on V with widetildeτ. Write M_τ(V) ≅ ⊕_a∈ZZ/pZZ (ξ^a)^⊕ m_τ,a(V). Define P_τ,V(z) = ∑_a∈ZZ/pZZ m_τ,a(V)z^a in ZZ[z]/(z^p-1).
Exact mathematical passage
M_\tau(V) \cong \bigoplus_{a\in\ZZ/p\ZZ} (\xi^a)^{\oplus m_{\tau,a}(V)}.
\xi^aDefinition passage · 2 passages
Search aliases: ξ^a xi
Definition passage: Extension spectrum polynomial · body.tex, line 388
[Extension spectrum polynomial] For a stable type τ, the multiplicity space M_τ(V)=Hom_H(τ,V) becomes a representation of G/H after comparing the action on V with widetildeτ. Write M_τ(V) ≅ ⊕_a∈ZZ/pZZ (ξ^a)^⊕ m_τ,a(V). Define P_τ,V(z) = ∑_a∈ZZ/pZZ m_τ,a(V)z^a in ZZ[z]/(z^p-1).
Exact mathematical passage
M_\tau(V) \cong \bigoplus_{a\in\ZZ/p\ZZ} (\xi^a)^{\oplus m_{\tau,a}(V)}.
\PiDefinition passage · 21 passages
Search aliases: Π Pi
Definition passage: Attached cyclic descent · body.tex, line 100
[Attached cyclic descent] Let E/F be cyclic of prime degree p, and let ρ be an irreducible finite image representation of G_F whose restriction τ=ρ|_G_E is irreducible. Suppose τ is automorphic with cuspidal representation Π of GL_n(Adele_E). An attached cyclic descent is a cuspidal descent π of Π such that rec_F_v(π_v) ≅ ρ|_W_F_v for every place v of F.
Exact mathematical passage
\Pi
\piDefinition passage · 84 passages
Search aliases: π pi
Definition passage: Attached cyclic descent · body.tex, line 100
[Attached cyclic descent] Let E/F be cyclic of prime degree p, and let ρ be an irreducible finite image representation of G_F whose restriction τ=ρ|_G_E is irreducible. Suppose τ is automorphic with cuspidal representation Π of GL_n(Adele_E). An attached cyclic descent is a cuspidal descent π of Π such that rec_F_v(π_v) ≅ ρ|_W_F_v for every place v of F.
Exact mathematical passage
\pi
\pi_vDefinition passage · 8 passages
Search aliases: π_v pi
Definition passage: Attached cyclic descent · body.tex, line 101
Attached cyclic descent] Let E/F be cyclic of prime degree p, and let ρ be an irreducible finite image representation of G_F whose restriction τ=ρ|_G_E is irreducible. Suppose τ is automorphic with cuspidal representation Π of GL_n(Adele_E). An attached cyclic descent is a cuspidal descent π of Π such that rec_F_v(π_v) ≅ ρ|_W_F_v for every place v of F.
Exact mathematical passage
\rec_{F_v}(\pi_v) \cong \rho|_{W_{F_v}}
\rhoDefinition passage · 57 passages
Search aliases: ρ rho
Definition passage: Attached cyclic descent · body.tex, line 100
[Attached cyclic descent] Let E/F be cyclic of prime degree p, and let ρ be an irreducible finite image representation of G_F whose restriction τ=ρ|_G_E is irreducible. Suppose τ is automorphic with cuspidal representation Π of GL_n(Adele_E). An attached cyclic descent is a cuspidal descent π of Π such that rec_F_v(π_v) ≅ ρ|_W_F_v for every place v of F.
Exact mathematical passage
\rho
\tauDefinition passage · 69 passages
Search aliases: τ tau
Definition passage: Attached cyclic descent · body.tex, line 100
[Attached cyclic descent] Let E/F be cyclic of prime degree p, and let ρ be an irreducible finite image representation of G_F whose restriction τ=ρ|_G_E is irreducible. Suppose τ is automorphic with cuspidal representation Π of GL_n(Adele_E). An attached cyclic descent is a cuspidal descent π of Π such that rec_F_v(π_v) ≅ ρ|_W_F_v for every place v of F.
Exact mathematical passage
\tau=\rho|_{G_E}
\tau^gDefinition passage · 1 passage
Search aliases: τ^g tau
Definition passage: Stable and moving types · body.tex, line 361
[Stable and moving types] An irreducible representation τ∈Irr(H) is stable when τ^g≅τ for every g∈ G. Otherwise its orbit has size p and is called moving.
Exact mathematical passage
\tau^g\cong\tau
\ldotsDefinition passage · 9 passages
Search aliases: … ldots
Definition passage: Active coefficient degrees · body.tex, line 162
[Active coefficient degrees] Define S(A)={k∈{1,…,n}:e_k(A)≠0}.
Exact mathematical passage
S(A)=\{k\in\{1,\ldots,n\}:e_k(A)\neq0\}.
\ellDefinition passage · 27 passages
Search aliases: ℓ ell
Definition passage: Pseudocharacter family gate · body.tex, line 1284
[Pseudocharacter family gate] A pseudocharacter family gate for a finite Satake descent π consists of a reduced p adic analytic space X, a closed point x valued in a finite extension of QQ_ℓ, and a continuous dimension n characteristic zero pseudorepresentation over O(X). It factors through the Galois group unramified outside one finite set, and its unramified characteristic polynomials specialize at x to the embedded Hecke polynomials of π. We use a point and coefficient setting in which charac…
Exact mathematical passage
\QQ_\ell
\inDefinition passage · 58 passages
Search aliases: ∈ in
Definition passage: Active coefficient degrees · body.tex, line 162
[Active coefficient degrees] Define S(A)={k∈{1,…,n}:e_k(A)≠0}.
Exact mathematical passage
S(A)=\{k\in\{1,\ldots,n\}:e_k(A)\neq0\}.
\prodDefinition passage · 5 passages
Search aliases: ∏ prod
Definition passage: Frobenius transform of a polynomial · body.tex, line 1110
[Frobenius transform of a polynomial] If P(T)=∏_i=1^n(T-α_i), define P^[p](T) = ∏_i=1^n(T-α_i^p). This is a polynomial expression in the coefficients of P.
Exact mathematical passage
P(T)=\prod_{i=1}^{n}(T-\alpha_i),
\prod_{i=1}^{n}Definition passage · 2 passages
Search aliases: ∏_i=1^n prod
Definition passage: Frobenius transform of a polynomial · body.tex, line 1110
[Frobenius transform of a polynomial] If P(T)=∏_i=1^n(T-α_i), define P^[p](T) = ∏_i=1^n(T-α_i^p). This is a polynomial expression in the coefficients of P.
Exact mathematical passage
P(T)=\prod_{i=1}^{n}(T-\alpha_i),
\sumDefinition passage · 12 passages
Search aliases: ∑ sum
Definition passage: Extension spectrum polynomial · body.tex, line 395
[Extension spectrum polynomial] For a stable type τ, the multiplicity space M_τ(V)=Hom_H(τ,V) becomes a representation of G/H after comparing the action on V with widetildeτ. Write M_τ(V) ≅ ⊕_a∈ZZ/pZZ (ξ^a)^⊕ m_τ,a(V). Define P_τ,V(z) = ∑_a∈ZZ/pZZ m_τ,a(V)z^a in ZZ[z]/(z^p-1).
Exact mathematical passage
\boxed{ P_{\tau,V}(z) = \sum_{a\in\ZZ/p\ZZ} m_{\tau,a}(V)z^a }
\sum_{a\in\ZZ/p\ZZ}Definition passage · 4 passages
Search aliases: ∑_a∈ZZ/pZZ sum in ZZ ZZ
Definition passage: Extension spectrum polynomial · body.tex, line 395
[Extension spectrum polynomial] For a stable type τ, the multiplicity space M_τ(V)=Hom_H(τ,V) becomes a representation of G/H after comparing the action on V with widetildeτ. Write M_τ(V) ≅ ⊕_a∈ZZ/pZZ (ξ^a)^⊕ m_τ,a(V). Define P_τ,V(z) = ∑_a∈ZZ/pZZ m_τ,a(V)z^a in ZZ[z]/(z^p-1).
Exact mathematical passage
\boxed{ P_{\tau,V}(z) = \sum_{a\in\ZZ/p\ZZ} m_{\tau,a}(V)z^a }
\sum_{i=1}^rDefinition passage · 1 passage
Search aliases: ∑_i=1^r sum
Definition passage: Entire induction cone · body.tex, line 1529
[Entire induction cone] Fix finitely many known entire induced packets Θ_1,…,Θ_r in the character lattice of G. Define C_calE = { ∑_i=1^ra_iΘ_i: a_i∈ZZ_≥0 }. Its saturation is Sat(C_calE) = {x:some m>0 has mx∈ C_calE}.
Exact mathematical passage
C_{\calE} = \left\{ \sum_{i=1}^ra_i\Theta_i: a_i\in\ZZ_{\geq0} \right\}.
\cupDefinition passage · 2 passages
Search aliases: ∪ cup
Definition passage: Finite polynomial alphabet · body.tex, line 829
[Finite polynomial alphabet] The finite polynomial alphabet of the cyclic attachment problem is the union calA = calA_π ∪ calA_ρ, where calA_π is the set of normalized Hecke polynomials occurring for π outside the fixed exceptional set, and calA_ρ is the set of characteristic polynomials occurring for the finitely many twists ρ⊗ϵ^j, j∈ZZ/pZZ.
Exact mathematical passage
\calA = \calA_{\pi} \cup \calA_{\rho},
\congDefinition passage · 33 passages
Search aliases: ≅ cong
Definition passage: Attached cyclic descent · body.tex, line 101
Attached cyclic descent] Let E/F be cyclic of prime degree p, and let ρ be an irreducible finite image representation of G_F whose restriction τ=ρ|_G_E is irreducible. Suppose τ is automorphic with cuspidal representation Π of GL_n(Adele_E). An attached cyclic descent is a cuspidal descent π of Π such that rec_F_v(π_v) ≅ ρ|_W_F_v for every place v of F.
Exact mathematical passage
\rec_{F_v}(\pi_v) \cong \rho|_{W_{F_v}}
\neqDefinition passage · 6 passages
Search aliases: ≠ neq
Definition passage: Active coefficient degrees · body.tex, line 162
[Active coefficient degrees] Define S(A)={k∈{1,…,n}:e_k(A)≠0}.
Exact mathematical passage
S(A)=\{k\in\{1,\ldots,n\}:e_k(A)\neq0\}.
\geqDefinition passage · 1 passage
Search aliases: ≥ geq
Definition passage: Entire induction cone · body.tex, line 1529
[Entire induction cone] Fix finitely many known entire induced packets Θ_1,…,Θ_r in the character lattice of G. Define C_calE = { ∑_i=1^ra_iΘ_i: a_i∈ZZ_≥0 }. Its saturation is Sat(C_calE) = {x:some m>0 has mx∈ C_calE}.
Exact mathematical passage
C_{\calE} = \left\{ \sum_{i=1}^ra_i\Theta_i: a_i\in\ZZ_{\geq0} \right\}.
\bigoplusDefinition passage · 5 passages
Search aliases: ⊕ bigoplus
Definition passage: Extension spectrum polynomial · body.tex, line 388
[Extension spectrum polynomial] For a stable type τ, the multiplicity space M_τ(V)=Hom_H(τ,V) becomes a representation of G/H after comparing the action on V with widetildeτ. Write M_τ(V) ≅ ⊕_a∈ZZ/pZZ (ξ^a)^⊕ m_τ,a(V). Define P_τ,V(z) = ∑_a∈ZZ/pZZ m_τ,a(V)z^a in ZZ[z]/(z^p-1).
Exact mathematical passage
M_\tau(V) \cong \bigoplus_{a\in\ZZ/p\ZZ} (\xi^a)^{\oplus m_{\tau,a}(V)}.
\oplusDefinition passage · 3 passages
Search aliases: ⊕ oplus
Definition passage: Extension spectrum polynomial · body.tex, line 388
[Extension spectrum polynomial] For a stable type τ, the multiplicity space M_τ(V)=Hom_H(τ,V) becomes a representation of G/H after comparing the action on V with widetildeτ. Write M_τ(V) ≅ ⊕_a∈ZZ/pZZ (ξ^a)^⊕ m_τ,a(V). Define P_τ,V(z) = ∑_a∈ZZ/pZZ m_τ,a(V)z^a in ZZ[z]/(z^p-1).
Exact mathematical passage
M_\tau(V) \cong \bigoplus_{a\in\ZZ/p\ZZ} (\xi^a)^{\oplus m_{\tau,a}(V)}.
\bigoplus_{a\in\ZZ/p\ZZ}Definition passage · 1 passage
Search aliases: ⊕_a∈ZZ/pZZ bigoplus in ZZ ZZ
Definition passage: Extension spectrum polynomial · body.tex, line 388
[Extension spectrum polynomial] For a stable type τ, the multiplicity space M_τ(V)=Hom_H(τ,V) becomes a representation of G/H after comparing the action on V with widetildeτ. Write M_τ(V) ≅ ⊕_a∈ZZ/pZZ (ξ^a)^⊕ m_τ,a(V). Define P_τ,V(z) = ∑_a∈ZZ/pZZ m_τ,a(V)z^a in ZZ[z]/(z^p-1).
Exact mathematical passage
M_\tau(V) \cong \bigoplus_{a\in\ZZ/p\ZZ} (\xi^a)^{\oplus m_{\tau,a}(V)}.
\otimesDefinition passage · 34 passages
Search aliases: ⊗ otimes
Definition passage: Local attachment set · body.tex, line 631
[Local attachment set] When the local quotient has order p, define calJ_v = {j∈ZZ/pZZ: A_v⊗ϵ_v^j≅ R_v}. At a split place, set calJ_v=ZZ/pZZ.
Exact mathematical passage
\calJ_v = \{j\in\ZZ/p\ZZ: A_v\otimes\epsilon_v^j\cong R_v\}.
\longrightarrowDefinition passage · 6 passages
Search aliases: ⟶ longrightarrow
Definition passage: Residual realization of a descent · body.tex, line 998
[Residual realization of a descent] Let λ be a finite place of the coefficient field. A residual realization of π at λ is a continuous semisimple representation overline r_π,λ:G_F⟶GL_n(overline k_λ) whose characteristic polynomial at every unramified place outside a finite set is the reduction of the normalized Hecke polynomial of π.
Exact mathematical passage
\overline r_{\pi,\lambda}:G_F\longrightarrow\GL_n(\overline k_{\lambda})
+Introduction passage · 61 passages
Search aliases: +
Introduction passage · formal_appendix.tex, line 24
Let C_e=⟨γ⟩ be a finite cyclic group of order e, acting linearly on a module M. Write T for the action of γ and set N=1+T+⋯+T^e-1. The relation T^e=1 implies N(T-1)=(T-1)N=0. These identities explain why the image of T-1 lies in the kernel of N, but they do not identify the two spaces.
Exact mathematical passage
N=1+T+\cdots+T^{e-1}.
C_eIntroduction passage · 4 passages
Search aliases: C_e
Introduction passage · formal_appendix.tex, line 22
Let C_e=⟨γ⟩ be a finite cyclic group of order e, acting linearly on a module M. Write T for the action of γ and set N=1+T+⋯+T^e-1. The relation T^e=1 implies N(T-1)=(T-1)N=0. These identities explain why the image of T-1 lies in the kernel of N, but they do not identify the two spaces.
Exact mathematical passage
C_e=\langle\gamma\rangle
\detIntroduction passage · 14 passages
Search aliases: det
Introduction passage · body.tex, line 16
At a finite place where ρ is unramified, fix the Frobenius convention used by the normalized local Langlands correspondence. Write P_ρ,v(T)=det(TI_n-ρ(Frob_v)). For an unramified automorphic local component π_v, write P_π,v(T) for the monic polynomial of its normalized Satake conjugacy class. The convention is chosen so that an attached pair has the same polynomial. This keeps a possible geometric versus arithmetic Frobenius inversion from entering a comparison silently.
Exact mathematical passage
P_{\rho,v}(T)=\det\bigl(TI_n-\rho(\Frob_v)\bigr).
\FrobIntroduction passage · 6 passages
Search aliases: Frob
Introduction passage · body.tex, line 16
At a finite place where ρ is unramified, fix the Frobenius convention used by the normalized local Langlands correspondence. Write P_ρ,v(T)=det(TI_n-ρ(Frob_v)). For an unramified automorphic local component π_v, write P_π,v(T) for the monic polynomial of its normalized Satake conjugacy class. The convention is chosen so that an attached pair has the same polynomial. This keeps a possible geometric versus arithmetic Frobenius inversion from entering a comparison silently.
Exact mathematical passage
P_{\rho,v}(T)=\det\bigl(TI_n-\rho(\Frob_v)\bigr).
\Frob_vIntroduction passage · 5 passages
Search aliases: Frob_v Frob
Introduction passage · body.tex, line 16
At a finite place where ρ is unramified, fix the Frobenius convention used by the normalized local Langlands correspondence. Write P_ρ,v(T)=det(TI_n-ρ(Frob_v)). For an unramified automorphic local component π_v, write P_π,v(T) for the monic polynomial of its normalized Satake conjugacy class. The convention is chosen so that an attached pair has the same polynomial. This keeps a possible geometric versus arithmetic Frobenius inversion from entering a comparison silently.
Exact mathematical passage
P_{\rho,v}(T)=\det\bigl(TI_n-\rho(\Frob_v)\bigr).
IIntroduction passage · 25 passages
Search aliases: I
Introduction passage · body.tex, line 16
At a finite place where ρ is unramified, fix the Frobenius convention used by the normalized local Langlands correspondence. Write P_ρ,v(T)=det(TI_n-ρ(Frob_v)). For an unramified automorphic local component π_v, write P_π,v(T) for the monic polynomial of its normalized Satake conjugacy class. The convention is chosen so that an attached pair has the same polynomial. This keeps a possible geometric versus arithmetic Frobenius inversion from entering a comparison silently.
Exact mathematical passage
P_{\rho,v}(T)=\det\bigl(TI_n-\rho(\Frob_v)\bigr).
I_nIntroduction passage · 3 passages
Search aliases: I_n
Introduction passage · body.tex, line 16
At a finite place where ρ is unramified, fix the Frobenius convention used by the normalized local Langlands correspondence. Write P_ρ,v(T)=det(TI_n-ρ(Frob_v)). For an unramified automorphic local component π_v, write P_π,v(T) for the monic polynomial of its normalized Satake conjugacy class. The convention is chosen so that an attached pair has the same polynomial. This keeps a possible geometric versus arithmetic Frobenius inversion from entering a comparison silently.
Exact mathematical passage
P_{\rho,v}(T)=\det\bigl(TI_n-\rho(\Frob_v)\bigr).
NIntroduction passage · 30 passages
Search aliases: N
Introduction passage · formal_appendix.tex, line 24
Let C_e=⟨γ⟩ be a finite cyclic group of order e, acting linearly on a module M. Write T for the action of γ and set N=1+T+⋯+T^e-1. The relation T^e=1 implies N(T-1)=(T-1)N=0. These identities explain why the image of T-1 lies in the kernel of N, but they do not identify the two spaces.
Exact mathematical passage
N=1+T+\cdots+T^{e-1}.
P_{\pi,v}Introduction passage · 5 passages
Search aliases: P_π,v pi
Introduction passage · body.tex, line 19
At a finite place where ρ is unramified, fix the Frobenius convention used by the normalized local Langlands correspondence. Write P_ρ,v(T)=det(TI_n-ρ(Frob_v)). For an unramified automorphic local component π_v, write P_π,v(T) for the monic polynomial of its normalized Satake conjugacy class. The convention is chosen so that an attached pair has the same polynomial. This keeps a possible geometric versus arithmetic Frobenius inversion from entering a comparison silently.
Exact mathematical passage
P_{\pi,v}(T)
P_{\rho,v}Introduction passage · 3 passages
Search aliases: P_ρ,v rho
Introduction passage · body.tex, line 16
At a finite place where ρ is unramified, fix the Frobenius convention used by the normalized local Langlands correspondence. Write P_ρ,v(T)=det(TI_n-ρ(Frob_v)). For an unramified automorphic local component π_v, write P_π,v(T) for the monic polynomial of its normalized Satake conjugacy class. The convention is chosen so that an attached pair has the same polynomial. This keeps a possible geometric versus arithmetic Frobenius inversion from entering a comparison silently.
Exact mathematical passage
P_{\rho,v}(T)=\det\bigl(TI_n-\rho(\Frob_v)\bigr).
sIntroduction passage · 25 passages
Search aliases: s
Introduction passage · body.tex, line 1496
Let K/F be finite Galois with group G. For a complex character θ of a subgroup, write L(s,θ) for the corresponding Artin L function.
Exact mathematical passage
L(s,\theta)
T^eIntroduction passage · 1 passage
Search aliases: T^e
Introduction passage · formal_appendix.tex, line 27
Let C_e=⟨γ⟩ be a finite cyclic group of order e, acting linearly on a module M. Write T for the action of γ and set N=1+T+⋯+T^e-1. The relation T^e=1 implies N(T-1)=(T-1)N=0. These identities explain why the image of T-1 lies in the kernel of N, but they do not identify the two spaces.
Exact mathematical passage
T^e=1
T^{e-1}Introduction passage · 1 passage
Search aliases: T^e-1
Introduction passage · formal_appendix.tex, line 24
Let C_e=⟨γ⟩ be a finite cyclic group of order e, acting linearly on a module M. Write T for the action of γ and set N=1+T+⋯+T^e-1. The relation T^e=1 implies N(T-1)=(T-1)N=0. These identities explain why the image of T-1 lies in the kernel of N, but they do not identify the two spaces.
Exact mathematical passage
N=1+T+\cdots+T^{e-1}.
\gammaIntroduction passage · 18 passages
Search aliases: γ gamma
Introduction passage · formal_appendix.tex, line 22
Let C_e=⟨γ⟩ be a finite cyclic group of order e, acting linearly on a module M. Write T for the action of γ and set N=1+T+⋯+T^e-1. The relation T^e=1 implies N(T-1)=(T-1)N=0. These identities explain why the image of T-1 lies in the kernel of N, but they do not identify the two spaces.
Exact mathematical passage
C_e=\langle\gamma\rangle
\thetaIntroduction passage · 5 passages
Search aliases: θ theta
Introduction passage · body.tex, line 1496
Let K/F be finite Galois with group G. For a complex character θ of a subgroup, write L(s,θ) for the corresponding Artin L function.
Exact mathematical passage
\theta
\cdotsIntroduction passage · 1 passage
Search aliases: ⋯ cdots
Introduction passage · formal_appendix.tex, line 24
Let C_e=⟨γ⟩ be a finite cyclic group of order e, acting linearly on a module M. Write T for the action of γ and set N=1+T+⋯+T^e-1. The relation T^e=1 implies N(T-1)=(T-1)N=0. These identities explain why the image of T-1 lies in the kernel of N, but they do not identify the two spaces.
Exact mathematical passage
N=1+T+\cdots+T^{e-1}.
\langleIntroduction passage · 1 passage
Search aliases: ⟨ langle
Introduction passage · formal_appendix.tex, line 22
Let C_e=⟨γ⟩ be a finite cyclic group of order e, acting linearly on a module M. Write T for the action of γ and set N=1+T+⋯+T^e-1. The relation T^e=1 implies N(T-1)=(T-1)N=0. These identities explain why the image of T-1 lies in the kernel of N, but they do not identify the two spaces.
Exact mathematical passage
C_e=\langle\gamma\rangle
\rangleIntroduction passage · 1 passage
Search aliases: ⟩ rangle
Introduction passage · formal_appendix.tex, line 22
Let C_e=⟨γ⟩ be a finite cyclic group of order e, acting linearly on a module M. Write T for the action of γ and set N=1+T+⋯+T^e-1. The relation T^e=1 implies N(T-1)=(T-1)N=0. These identities explain why the image of T-1 lies in the kernel of N, but they do not identify the two spaces.
Exact mathematical passage
C_e=\langle\gamma\rangle
*Use in source · 1 passage
Search aliases: *
Use in source · formal_appendix.tex, line 254
Let J:V→ W be a linear map over a field k. The correction equation Ju+e=0 has a solution precisely when e∈imJ. A useful description of this image uses all the dual relations among the equations. Let S=ker(J^*:W^*→ V^*), C={w∈ W:ϕ(w)=0 for every ϕ∈ S}. Every Ju belongs to C. Conversely, if w does not belong to imJ, its nonzero class in W/imJ is detected by an algebraic linear functional. Pulling that functional back to W gives an element of S that does not annihilate w. Hence imJ=C. This algebrai…
Exact mathematical passage
S=\ker(J^*:W^*\to V^*),\qquad C=\{w\in W:\phi(w)=0\text{ for every }\phi\in S\}.
A^{-1}Use in source · 1 passage
Search aliases: A^-1
Use in source · body.tex, line 370
Choose g∈ G whose class generates G/H, so g^p∈ H. Stability gives an invertible intertwiner A with Aτ(h)A^-1=τ(ghg^-1) (h∈ H). Iterate this identity p times. The operator τ(g^p)^-1A^p commutes with τ(H), so Schur's lemma makes it a nonzero scalar cI. Also A commutes with τ(g^p), by the same intertwining identity at h=g^p. Choose b∈CC^× with b^p=c^-1 and put R=bA. Then Rτ(h)R^-1=τ(ghg^-1), R^p=τ(g^p). These are exactly the conjugation and closing relations for the coset presentation of G. Assigni…
Exact mathematical passage
A\tau(h)A^{-1}=\tau(ghg^{-1})\qquad(h\in H).
A^2Use in source · 4 passages
Search aliases: A^2
Use in source · formal_appendix.tex, line 397
For degree three, the linear term is D_A,3(h)=A^2h+AhA+hA^2, and the nonlinear remainder is R_A,3(h)=Ah^2+hAh+h^2A+h^3. The middle word hAh cannot be absorbed into Ah^2 in a matrix ring. The remainder estimate keeps all four words and bounds each by its norm, so it respects the ordered expansion. This also explains why the estimate is not an assertion of commutative binomial calculus in disguise. The scalar bound remains uniform because the norm of a finite ultrametric sum is controlled by its l…
Exact mathematical passage
D_{A,3}(h)=A^2h+AhA+hA^2,
A^dUse in source · 4 passages
Search aliases: A^d
Use in source · formal_appendix.tex, line 376
Let E be a complete normed ring with norm1=1, a submultiplicative norm, and the ultrametric inequality normx+y≤max(normx,normy). Assume norm A≤1. For a nonnegative integer d, define D_A,d(h) · =∑_i=0^d-1A^ihA^d-1-i, R_A,d(h) · =(A+h)^d-A^d-D_A,d(h). For d=0 the sum is empty and both maps are zero. The order of the factors in [source reference] matters. Replacing this operator by dA^d-1h requires A and h to commute. The replacement is invalid for general matrices.
Exact mathematical passage
D_{A,d}(h)&=\sum_{i=0}^{d-1}A^ihA^{d-1-i}, \label{eq:appendix-ordered-derivative}\\ R_{A,d}(h)&=(A+h)^d-A^d-D_{A,d}(h). \label{eq:appendix-ordered-remainder}
A^{d-1}Use in source · 1 passage
Search aliases: A^d-1
Use in source · formal_appendix.tex, line 384
R_A,d(h) · =(A+h)^d-A^d-D_A,d(h). For d=0 the sum is empty and both maps are zero. The order of the factors in [source reference] matters. Replacing this operator by dA^d-1h requires A and h to commute. The replacement is invalid for general matrices.
Exact mathematical passage
dA^{d-1}h
A^{d-1-i}Use in source · 1 passage
Search aliases: A^d-1-i
Use in source · formal_appendix.tex, line 376
Let E be a complete normed ring with norm1=1, a submultiplicative norm, and the ultrametric inequality normx+y≤max(normx,normy). Assume norm A≤1. For a nonnegative integer d, define D_A,d(h) · =∑_i=0^d-1A^ihA^d-1-i, R_A,d(h) · =(A+h)^d-A^d-D_A,d(h). For d=0 the sum is empty and both maps are zero. The order of the factors in [source reference] matters. Replacing this operator by dA^d-1h requires A and h to commute. The replacement is invalid for general matrices.
Exact mathematical passage
D_{A,d}(h)&=\sum_{i=0}^{d-1}A^ihA^{d-1-i}, \label{eq:appendix-ordered-derivative}\\ R_{A,d}(h)&=(A+h)^d-A^d-D_{A,d}(h). \label{eq:appendix-ordered-remainder}
A^iUse in source · 1 passage
Search aliases: A^i
Use in source · formal_appendix.tex, line 376
Let E be a complete normed ring with norm1=1, a submultiplicative norm, and the ultrametric inequality normx+y≤max(normx,normy). Assume norm A≤1. For a nonnegative integer d, define D_A,d(h) · =∑_i=0^d-1A^ihA^d-1-i, R_A,d(h) · =(A+h)^d-A^d-D_A,d(h). For d=0 the sum is empty and both maps are zero. The order of the factors in [source reference] matters. Replacing this operator by dA^d-1h requires A and h to commute. The replacement is invalid for general matrices.
Exact mathematical passage
D_{A,d}(h)&=\sum_{i=0}^{d-1}A^ihA^{d-1-i}, \label{eq:appendix-ordered-derivative}\\ R_{A,d}(h)&=(A+h)^d-A^d-D_{A,d}(h). \label{eq:appendix-ordered-remainder}
A^pUse in source · 1 passage
Search aliases: A^p
Use in source · body.tex, line 373
Choose g∈ G whose class generates G/H, so g^p∈ H. Stability gives an invertible intertwiner A with Aτ(h)A^-1=τ(ghg^-1) (h∈ H). Iterate this identity p times. The operator τ(g^p)^-1A^p commutes with τ(H), so Schur's lemma makes it a nonzero scalar cI. Also A commutes with τ(g^p), by the same intertwining identity at h=g^p. Choose b∈CC^× with b^p=c^-1 and put R=bA. Then Rτ(h)R^-1=τ(ghg^-1), R^p=τ(g^p). These are exactly the conjugation and closing relations for the coset presentation of G. Assigni…
Exact mathematical passage
\tau(g^p)^{-1}A^p
A_0Use in source · 2 passages
Search aliases: A_0
Use in source · formal_appendix.tex, line 642
There is a genuinely singular contrast. Let A_0=diag(1,-1) over Q_p. Then D_A_0,2(E_12)=0, although E_12≠0. Thus this derivative has no inverse over the fraction field, including when p is odd and 2 is a unit. The certificate theorem cannot cover this center. A universal claim that invertible centers always have invertible power derivatives would be false.
Exact mathematical passage
A_0=\operatorname{diag}(1,-1)
A_3Use in source · 3 passages
Search aliases: A_3
Use in source · body.tex, line 598
[A moving type and self twist] Let G=S_3, let H=A_3, and let V be the standard two dimensional representation. Its restriction is Res_HV=τ⊕τ^-1 for the two nontrivial characters of C_3. These characters form one moving orbit under G/H≅ C_2. Hence there is no stable extension spectrum, and V⊗sgn≅ V.
Exact mathematical passage
H=A_3
A_{\mathrm{bal}}Use in source · 1 passage
Search aliases: A_bal mathrm
Use in source · worked_appendix.tex, line 51
Now compare the balanced matrix A_bal=diag(1,ω,ω^2). Its polynomial is T^3-1. Multiplying all eigenvalues by ω merely permutes their multiset. Every lower coefficient vanishes, and the determinant is fixed. Its full characteristic polynomial is therefore blind to the label. This is an actual local rotational stabilizer. A claim that one characteristic polynomial always detects a quotient twist would fail on this example.
Exact mathematical passage
A_{\mathrm{bal}}=\operatorname{diag}(1,\omega,\omega^2).
\absUse in source · 8 passages
Search aliases: abs
Use in source · body.tex, line 900
[Finite trace image bound] Let k be algebraically closed of characteristic zero or characteristic ℓ>n. Let Γ⊆GL_n(k) be a subgroup, and assume its defining module k^n is semisimple. Suppose its trace set is finite, with t=#{Tr(γ):γ∈Γ}. Then Γ is finite and absΓ≤ t^dim_k A≤ t^n^2, A=span_kΓ. In particular this bounds a finite semisimple residual matrix group in characteristic greater than its rank.
Exact mathematical passage
\abs\Gamma\le t^{\dim_k A}\le t^{n^2}, \qquad A=\operatorname{span}_k\Gamma.
\Adele_\QQUse in source · 2 passages
Search aliases: Adele_QQ Adele QQ
Use in source · body.tex, line 1350
[Conditional construction of Kim and Yamauchi] Kim and Yamauchi's construction for GL_n(Adele_QQ) uses five declared inputs: a finite Hecke field, integrality with a common denominator, cuspidal Galois conjugates, residual Galois realizations for all sufficiently large coefficient primes, and a Rankin Selberg coefficient bound obtainable from the required exterior power functoriality. Their theorem is a conditional construction of Artin representations, not a theorem producing the residual reali…
Exact mathematical passage
\GL_n(\Adele_\QQ)
\AIUse in source · 1 passage
Search aliases: AI
Use in source · body.tex, line 326
ced object is torsor blind] Let Htriangleleft G with cyclic quotient character group X, and let V be an extension of an H representation. Then ⊕_χ∈ X V⊗χ ≅ Ind_H^GRes_HV. The left side is unchanged when V is replaced by another member of its extension torsor. Likewise, for cyclic base change, mathopboxplus_χ∈ X π⊗χ ≅ AI_E/F(BC_E/Fπ) inside the established automorphic induction formalism. Therefore comparison of the total induced objects cannot determine the cyclic attachment offset.
Exact mathematical passage
\boxed{ \mathop{\boxplus}_{\chi\in X} \pi\otimes\chi \cong \AI_{E/F}(\BC_{E/F}\pi) }
\AI_{E/F}Use in source · 1 passage
Search aliases: AI_E/F AI
Use in source · body.tex, line 326
ced object is torsor blind] Let Htriangleleft G with cyclic quotient character group X, and let V be an extension of an H representation. Then ⊕_χ∈ X V⊗χ ≅ Ind_H^GRes_HV. The left side is unchanged when V is replaced by another member of its extension torsor. Likewise, for cyclic base change, mathopboxplus_χ∈ X π⊗χ ≅ AI_E/F(BC_E/Fπ) inside the established automorphic induction formalism. Therefore comparison of the total induced objects cannot determine the cyclic attachment offset.
Exact mathematical passage
\boxed{ \mathop{\boxplus}_{\chi\in X} \pi\otimes\chi \cong \AI_{E/F}(\BC_{E/F}\pi) }
bUse in source · 33 passages
Search aliases: b
Use in source · body.tex, line 373
) whose class generates G/H, so g^p∈ H. Stability gives an invertible intertwiner A with Aτ(h)A^-1=τ(ghg^-1) (h∈ H). Iterate this identity p times. The operator τ(g^p)^-1A^p commutes with τ(H), so Schur's lemma makes it a nonzero scalar cI. Also A commutes with τ(g^p), by the same intertwining identity at h=g^p. Choose b∈CC^× with b^p=c^-1 and put R=bA. Then Rτ(h)R^-1=τ(ghg^-1), R^p=τ(g^p). These are exactly the conjugation and closing relations for the coset presentation of G. Assigning g↦ R an…
Exact mathematical passage
b\in\CC^\times
b^pUse in source · 1 passage
Search aliases: b^p
Use in source · body.tex, line 373
G/H, so g^p∈ H. Stability gives an invertible intertwiner A with Aτ(h)A^-1=τ(ghg^-1) (h∈ H). Iterate this identity p times. The operator τ(g^p)^-1A^p commutes with τ(H), so Schur's lemma makes it a nonzero scalar cI. Also A commutes with τ(g^p), by the same intertwining identity at h=g^p. Choose b∈CC^× with b^p=c^-1 and put R=bA. Then Rτ(h)R^-1=τ(ghg^-1), R^p=τ(g^p). These are exactly the conjugation and closing relations for the coset presentation of G. Assigning g↦ R and h↦τ(h) therefore defin…
Exact mathematical passage
b^p=c^{-1}
b_0Use in source · 4 passages
Search aliases: b_0
Use in source · formal_appendix.tex, line 209
Suppose an absolute two-cocycle has been cancelled by a continuous one-cochain b_0. Prescribed local lifts need not agree with the resulting global lift. Their remaining differences are local one-cocycles. Changing b_0 to another absolute primitive changes these differences by the restriction of a global one-cocycle. Conjugating the local lift at index i by a unit 1+x_i, with x_i∈ I, changes the local cocycle by a local coboundary. The conjugators x_i are independently chosen at each local index…
Exact mathematical passage
b_0
b_1Use in source · 2 passages
Search aliases: b_1
Use in source · formal_appendix.tex, line 235
To see the criterion, let a_i be the local mismatch after b_0. A match up to the allowed local conjugations exists exactly when a_i=res_i(z)+δ x_i for a global continuous one-cocycle z and kernel elements x_i. This is exactly the assertion that the image of (a_i)_i in [source reference] is zero. If b_1 is another absolute primitive, then b_1-b_0 is a global cocycle. Its restriction disappears in the cokernel, so the remaining class does not depend on which primitive was used.
Exact mathematical passage
b_1
B_xUse in source · 3 passages
Search aliases: B_x
Use in source · formal_appendix.tex, line 314
The graded correction argument retains an exact moving relation B_xF(x)=0 and a residual exactness statement imoverline J=keroverline B. If F(x)=π^r e, the moving identity gives π^r B_xe=0. A faithful leading layer allows this equality to imply overline B overline e=0. Equation [source reference] then supplies a residual correction overline u with overline J overline u=-overline e. Lift overline u to u and put x'=x+π^r u. The linear part of F(x') is divisible by π^r+1. For polynomial systems the…
Exact mathematical passage
B_xF(x)=0
\mathrm{bad}Use in source · 2 passages
Search aliases: bad mathrm
Use in source · formal_appendix.tex, line 625
For a negative control, take U_bad=1 · 10 0 · 1. The difference from A^2 is (0,8,0,0) in entry coordinates, and B sends it to (0,16,0,0). This is nonzero modulo 32, so no square root of U_bad lies in that ball. This is a failure for the specified prefix. It is not a claim that the target has no square root anywhere in a larger coefficient field.
Exact mathematical passage
U_{\mathrm{bad}}=\begin{pmatrix}1&10\\0&1\end{pmatrix}.
\mathrm{bal}Use in source · 1 passage
Search aliases: bal mathrm
Use in source · worked_appendix.tex, line 51
Now compare the balanced matrix A_bal=diag(1,ω,ω^2). Its polynomial is T^3-1. Multiplying all eigenvalues by ω merely permutes their multiset. Every lower coefficient vanishes, and the determinant is fixed. Its full characteristic polynomial is therefore blind to the label. This is an actual local rotational stabilizer. A claim that one characteristic polynomial always detects a quotient twist would fail on this example.
Exact mathematical passage
A_{\mathrm{bal}}=\operatorname{diag}(1,\omega,\omega^2).
\BCUse in source · 1 passage
Search aliases: BC
Use in source · body.tex, line 326
ced object is torsor blind] Let Htriangleleft G with cyclic quotient character group X, and let V be an extension of an H representation. Then ⊕_χ∈ X V⊗χ ≅ Ind_H^GRes_HV. The left side is unchanged when V is replaced by another member of its extension torsor. Likewise, for cyclic base change, mathopboxplus_χ∈ X π⊗χ ≅ AI_E/F(BC_E/Fπ) inside the established automorphic induction formalism. Therefore comparison of the total induced objects cannot determine the cyclic attachment offset.
Exact mathematical passage
\boxed{ \mathop{\boxplus}_{\chi\in X} \pi\otimes\chi \cong \AI_{E/F}(\BC_{E/F}\pi) }
\BC_{E/F}Use in source · 1 passage
Search aliases: BC_E/F BC
Use in source · body.tex, line 326
ced object is torsor blind] Let Htriangleleft G with cyclic quotient character group X, and let V be an extension of an H representation. Then ⊕_χ∈ X V⊗χ ≅ Ind_H^GRes_HV. The left side is unchanged when V is replaced by another member of its extension torsor. Likewise, for cyclic base change, mathopboxplus_χ∈ X π⊗χ ≅ AI_E/F(BC_E/Fπ) inside the established automorphic induction formalism. Therefore comparison of the total induced objects cannot determine the cyclic attachment offset.
Exact mathematical passage
\boxed{ \mathop{\boxplus}_{\chi\in X} \pi\otimes\chi \cong \AI_{E/F}(\BC_{E/F}\pi) }
\bigwedgeUse in source · 2 passages
Search aliases: bigwedge
Use in source · body.tex, line 157
The coefficient e_k(A) is the trace of bigwedge^k A. Scalar multiplication by ζ acts on bigwedge^kCC^n by ζ^k.
Exact mathematical passage
\bigwedge^k A
\bigwedge^kUse in source · 2 passages
Search aliases: bigwedge^k bigwedge
Use in source · body.tex, line 157
The coefficient e_k(A) is the trace of bigwedge^k A. Scalar multiplication by ζ acts on bigwedge^kCC^n by ζ^k.
Exact mathematical passage
\bigwedge^k A
\boxplusUse in source · 1 passage
Search aliases: boxplus
Use in source · body.tex, line 326
ced object is torsor blind] Let Htriangleleft G with cyclic quotient character group X, and let V be an extension of an H representation. Then ⊕_χ∈ X V⊗χ ≅ Ind_H^GRes_HV. The left side is unchanged when V is replaced by another member of its extension torsor. Likewise, for cyclic base change, mathopboxplus_χ∈ X π⊗χ ≅ AI_E/F(BC_E/Fπ) inside the established automorphic induction formalism. Therefore comparison of the total induced objects cannot determine the cyclic attachment offset.
Exact mathematical passage
\boxed{ \mathop{\boxplus}_{\chi\in X} \pi\otimes\chi \cong \AI_{E/F}(\BC_{E/F}\pi) }
cUse in source · 17 passages
Search aliases: c
Use in source · body.tex, line 373
Choose g∈ G whose class generates G/H, so g^p∈ H. Stability gives an invertible intertwiner A with Aτ(h)A^-1=τ(ghg^-1) (h∈ H). Iterate this identity p times. The operator τ(g^p)^-1A^p commutes with τ(H), so Schur's lemma makes it a nonzero scalar cI. Also A commutes with τ(g^p), by the same intertwining identity at h=g^p. Choose b∈CC^× with b^p=c^-1 and put R=bA. Then Rτ(h)R^-1=τ(ghg^-1), R^p=τ(g^p). These are exactly the conjugation and closing relations for the coset presentation of G. Assigni…
Exact mathematical passage
cI
c'Use in source · 1 passage
Search aliases: c'
Use in source · formal_appendix.tex, line 140
Change the chosen lifts to r'_g=(1+b(g))r_g, where b:G→ I and b(1)=0. Expanding without commuting the factors gives r'_gr'_h =(1+b(g)+g· b(h)+c(g,h))r_gh. To compare with r'_gh, multiply by 1-b(gh). The new defect is c'(g,h)=c(g,h)+g· b(h)-b(gh)+b(g). Thus changing the section adds a two-coboundary. The class [c]∈ H^2(G,I) does not depend on the section.
Exact mathematical passage
c'(g,h)=c(g,h)+g\cdot b(h)-b(gh)+b(g). \label{eq:appendix-defect-change}
c^{-1}Use in source · 1 passage
Search aliases: c^-1
Use in source · body.tex, line 373
G/H, so g^p∈ H. Stability gives an invertible intertwiner A with Aτ(h)A^-1=τ(ghg^-1) (h∈ H). Iterate this identity p times. The operator τ(g^p)^-1A^p commutes with τ(H), so Schur's lemma makes it a nonzero scalar cI. Also A commutes with τ(g^p), by the same intertwining identity at h=g^p. Choose b∈CC^× with b^p=c^-1 and put R=bA. Then Rτ(h)R^-1=τ(ghg^-1), R^p=τ(g^p). These are exactly the conjugation and closing relations for the coset presentation of G. Assigning g↦ R and h↦τ(h) therefore defin…
Exact mathematical passage
b^p=c^{-1}
C_2Use in source · 9 passages
Search aliases: C_2
Use in source · body.tex, line 579
[A stable cyclic type] Let G=C_3× C_2, H=C_3. Fix a nontrivial character τ of H. It is stable under G. After choosing the extension with trivial C_2 coordinate, the two one dimensional extensions have spectra P_τ,V_0(z)=1, P_τ,V_1(z)=z. The alignment set is the singleton containing the unique quotient twist carrying one extension to the other.
Exact mathematical passage
G=C_3\times C_2, \qquad H=C_3.
C_3Use in source · 7 passages
Search aliases: C_3
Use in source · body.tex, line 579
[A stable cyclic type] Let G=C_3× C_2, H=C_3. Fix a nontrivial character τ of H. It is stable under G. After choosing the extension with trivial C_2 coordinate, the two one dimensional extensions have spectra P_τ,V_0(z)=1, P_τ,V_1(z)=z. The alignment set is the singleton containing the unique quotient twist carrying one extension to the other.
Exact mathematical passage
G=C_3\times C_2, \qquad H=C_3.
C_3^2Use in source · 1 passage
Search aliases: C_3^2
Use in source · body.tex, line 45
The solvable two dimensional sectors treated by Langlands and Tunnell, further examples such as the solvable GO(4) range, and other established cases show that the conjecture is far from an empty expectation [source reference]. A recent preprint of Wang states the generalized octahedral GL_3 case with projective image C_3^2rtimesSL(2,3), together with a primitive solvable six dimensional consequence [source reference]. Its stated results have been checked against the submitted source. Its full p…
Exact mathematical passage
C_3^2\rtimes\mathrm{SL}(2,3)
C_6Use in source · 2 passages
Search aliases: C_6
Use in source · formal_appendix.tex, line 77
Let instead the generator of C_6 act by the same T. On the diagonal space, its norm is multiplication by 6=0 in F_3. On the off-diagonal space, the alternating six-term sum is also zero. Now N=0 on all of M, while the image of T-1 is still only the off-diagonal space. Therefore H^1(C_6,M) has dimension two. The generator matrix alone does not determine this cohomology without the order and coefficient characteristic.
Exact mathematical passage
C_6
C_pUse in source · 11 passages
Search aliases: C_p
Use in source · body.tex, line 49
The finite representation arguments have a parallel shape. If Htriangleleft G and G/H≅ C_p, an irreducible representation either restricts to a full orbit of p subgroup types or restricts irreducibly to one invariant type. In the first case induction reconstructs the representation. In the second case an extension must be selected. The second case is where two torsors meet. Each torsor exists, and each has p elements. There is no preferred origin in the statement that they are torsors under the …
Exact mathematical passage
G/H\cong C_p
\calPUse in source · 5 passages
Search aliases: calP
Use in source · body.tex, line 963
[One large residual realization gives an Artin representation] Let K be a number field, let S be a finite set of places of F, and let calP⊆OO_K[T] be a nonempty finite set of monic degree n polynomials. Assume all their roots are roots of unity. For v∉ S, choose P_v∈calP.
Exact mathematical passage
\calP\subseteq\OO_K[T]
\CC^nUse in source · 1 passage
Search aliases: CC^n CC
Use in source · body.tex, line 157
The coefficient e_k(A) is the trace of bigwedge^k A. Scalar multiplication by ζ acts on bigwedge^kCC^n by ζ^k.
Exact mathematical passage
\bigwedge^k\CC^n
\operatorname{coker}Use in source · 1 passage
Search aliases: coker operatorname
Use in source · formal_appendix.tex, line 218
After taking the local coboundary quotients, the remaining class belongs to coker( H^1_cont(G,I)⟶ ∏_i H^1_cont(G_i,I)). Here the local groups, their maps into G, and the precise continuous coefficient action must be supplied. The recovered source constructs the corresponding quotients directly in its square-zero data.
Exact mathematical passage
\operatorname{coker}\left( H^1_{\mathrm{cont}}(G,I)\longrightarrow \prod_i H^1_{\mathrm{cont}}(G_i,I)\right). \label{eq:appendix-gauge-cokernel}
\mathrm{cont}Use in source · 1 passage
Search aliases: cont mathrm
Use in source · formal_appendix.tex, line 218
After taking the local coboundary quotients, the remaining class belongs to coker( H^1_cont(G,I)⟶ ∏_i H^1_cont(G_i,I)). Here the local groups, their maps into G, and the precise continuous coefficient action must be supplied. The recovered source constructs the corresponding quotients directly in its square-zero data.
Exact mathematical passage
\operatorname{coker}\left( H^1_{\mathrm{cont}}(G,I)\longrightarrow \prod_i H^1_{\mathrm{cont}}(G_i,I)\right). \label{eq:appendix-gauge-cokernel}
DUse in source · 20 passages
Search aliases: D
Use in source · body.tex, line 974
Let t≥1 be the number of trace values in the source alphabet and put B=t^n^2. First fix a finite normal extension L/QQ containing K and the roots of unity of orders dividing D_B=lcm(1,2,…,B). Every polynomial in calQ_n,B, the bounded cyclotomic alphabet, has coefficients in L. Form the union of all Galois conjugates in L of calP∪calQ_n,B. This is a fixed finite set of integral polynomials. For each distinct pair choose one nonzero coefficient difference. Exclude the rational primes below the pri…
Exact mathematical passage
D_B=\operatorname{lcm}(1,2,\ldots,B).
dUse in source · 49 passages
Search aliases: d
Use in source · body.tex, line 723
Let E/F be a finite Galois extension of degree d. Suppose π base changes to an automorphic representation Π attached to a finite image Artin representation τ. Let M be the exponent of the finite image of τ. Fix a finite set S containing the exceptional comparison places and all ramification of the extension and the representations. The unramified conclusions in this section refer to places outside this fixed set. Weak attachment over the extension field suffices for these conclusions. If all loc…
Exact mathematical passage
d
D_{A,2}Use in source · 2 passages
Search aliases: D_A,2
Use in source · formal_appendix.tex, line 582
c · e, the ordered derivative is D_A,2(X)=AX+XA =2a+c · a+2b+e 2c · c+2e. Using the entry order (a,b,c,e), its matrix and a scaled inverse are J= 2 · 0 · 1 · 0
Exact mathematical passage
D_{A,2}(X)=AX+XA =\begin{pmatrix}2a+c&a+2b+e\\2c&c+2e\end{pmatrix}.
D_{A,3}Use in source · 1 passage
Search aliases: D_A,3
Use in source · formal_appendix.tex, line 397
For degree three, the linear term is D_A,3(h)=A^2h+AhA+hA^2, and the nonlinear remainder is R_A,3(h)=Ah^2+hAh+h^2A+h^3. The middle word hAh cannot be absorbed into Ah^2 in a matrix ring. The remainder estimate keeps all four words and bounds each by its norm, so it respects the ordered expansion. This also explains why the estimate is not an assertion of commutative binomial calculus in disguise. The scalar bound remains uniform because the norm of a finite ultrametric sum is controlled by its l…
Exact mathematical passage
D_{A,3}(h)=A^2h+AhA+hA^2,
D_{A,d}Use in source · 5 passages
Search aliases: D_A,d
Use in source · formal_appendix.tex, line 376
Let E be a complete normed ring with norm1=1, a submultiplicative norm, and the ultrametric inequality normx+y≤max(normx,normy). Assume norm A≤1. For a nonnegative integer d, define D_A,d(h) · =∑_i=0^d-1A^ihA^d-1-i, R_A,d(h) · =(A+h)^d-A^d-D_A,d(h). For d=0 the sum is empty and both maps are zero. The order of the factors in [source reference] matters. Replacing this operator by dA^d-1h requires A and h to commute. The replacement is invalid for general matrices.
Exact mathematical passage
D_{A,d}(h)&=\sum_{i=0}^{d-1}A^ihA^{d-1-i}, \label{eq:appendix-ordered-derivative}\\ R_{A,d}(h)&=(A+h)^d-A^d-D_{A,d}(h). \label{eq:appendix-ordered-remainder}
D_{A,d+1}Use in source · 1 passage
Search aliases: D_A,d+1
Use in source · formal_appendix.tex, line 388
The recursive identities are D_A,d+1(h) · =D_A,d(h)A+A^dh, R_A,d+1(h) · =R_A,d(h)(A+h)+D_A,d(h)h. They follow by multiplying (A+h)^d on the right by A+h and collecting the terms with zero, one, and at least two occurrences of h. This derivation does not rearrange any noncommuting factors.
Exact mathematical passage
D_{A,d+1}(h)&=D_{A,d}(h)A+A^dh,\\ R_{A,d+1}(h)&=R_{A,d}(h)(A+h)+D_{A,d}(h)h.
D_{A_0,2}Use in source · 1 passage
Search aliases: D_A_0,2
Use in source · formal_appendix.tex, line 643
There is a genuinely singular contrast. Let A_0=diag(1,-1) over Q_p. Then D_A_0,2(E_12)=0, although E_12≠0. Thus this derivative has no inverse over the fraction field, including when p is odd and 2 is a unit. The certificate theorem cannot cover this center. A universal claim that invertible centers always have invertible power derivatives would be false.
Exact mathematical passage
D_{A_0,2}(E_{12})=0
D_BUse in source · 1 passage
Search aliases: D_B
Use in source · body.tex, line 974
Let t≥1 be the number of trace values in the source alphabet and put B=t^n^2. First fix a finite normal extension L/QQ containing K and the roots of unity of orders dividing D_B=lcm(1,2,…,B). Every polynomial in calQ_n,B, the bounded cyclotomic alphabet, has coefficients in L. Form the union of all Galois conjugates in L of calP∪calQ_n,B. This is a fixed finite set of integral polynomials. For each distinct pair choose one nonzero coefficient difference. Exclude the rational primes below the pri…
Exact mathematical passage
D_B=\operatorname{lcm}(1,2,\ldots,B).
D_dUse in source · 4 passages
Search aliases: D_d
Use in source · formal_appendix.tex, line 429
Induct on d for the second estimate. The case d=0 is immediate. Write D_d=D_A,d and R_d=R_A,d. Subtracting the remainder recursions gives R_d+1(h)-R_d+1(k) = · (R_d(h)-R_d(k))(A+h)
Exact mathematical passage
D_d=D_{A,d}
d_iUse in source · 2 passages
Search aliases: d_i
Use in source · body.tex, line 909
oof Because products of group matrices are group matrices, A is a finite dimensional subalgebra of M_n(k). An A submodule is the same thing as a Γ submodule. The given module is semisimple and faithful for A. The Jacobson radical annihilates every simple constituent, hence annihilates this faithful module. It is therefore zero. The split semisimple algebra has the form A≅⊕_iM_d_i(k), k^n≅⊕_i(k^d_i)^⊕ m_i, ∑_i d_im_i=n. Faithfulness gives m_i≥1 for every block. The trace pairing B(x,y)=Tr_k^n(xy)…
Exact mathematical passage
A\cong\bigoplus_iM_{d_i}(k),\qquad k^n\cong\bigoplus_i(k^{d_i})^{\oplus m_i}, \qquad\sum_i d_im_i=n.
\det^aUse in source · 1 passage
Search aliases: det^a det
Use in source · body.tex, line 263
Every algebraic character of GL_n is an integral power of the determinant. Thus det∘ r=det^a for one integer a. Evaluate both sides at zI_n. The left side is z^wN, while the right side is z^an.
Exact mathematical passage
\det\circ r=\det^a
\operatorname{diag}Use in source · 6 passages
Search aliases: diag operatorname
Use in source · formal_appendix.tex, line 68
[The same generator action gives different cyclic cohomology] Over F_3, let M=Mat_2( F_3) and let T be conjugation by F=diag(1,-1). The diagonal entries of a matrix form the +1 eigenspace of T. The off-diagonal entries form its -1 eigenspace. Both spaces have dimension two.
Exact mathematical passage
F=\operatorname{diag}(1,-1)
\dimUse in source · 4 passages
Search aliases: dim
Use in source · body.tex, line 297
[Highly ramified epsilon twists are blind] Let V' = V⊗χ, where χ has order p, and assume pmiddim V. Then V and V' have the same determinant. For every sufficiently ramified auxiliary character η, ϵ(V⊗η,Ψ) = ϵ(V'⊗η,Ψ).
Exact mathematical passage
p\mid\dim V
\dim_kUse in source · 2 passages
Search aliases: dim_k dim
Use in source · body.tex, line 900
[Finite trace image bound] Let k be algebraically closed of characteristic zero or characteristic ℓ>n. Let Γ⊆GL_n(k) be a subgroup, and assume its defining module k^n is semisimple. Suppose its trace set is finite, with t=#{Tr(γ):γ∈Γ}. Then Γ is finite and absΓ≤ t^dim_k A≤ t^n^2, A=span_kΓ. In particular this bounds a finite semisimple residual matrix group in characteristic greater than its rank.
Exact mathematical passage
\abs\Gamma\le t^{\dim_k A}\le t^{n^2}, \qquad A=\operatorname{span}_k\Gamma.
e'Use in source · 3 passages
Search aliases: e'
Use in source · formal_appendix.tex, line 340
The normalization by π^r deserves care. Writing F(x)=π^r e need not determine e uniquely in a ring with torsion. If e' is another choice, then π^r(e-e')=0. The faithful leading layer used by the theorem ensures that e-e' has zero residue below the terminal cutoff. Thus the residue correction problem is independent of the chosen normalized error. The same hypothesis converts π^r B_xe=0 into the residue relation needed for exactness. These are two concrete reasons why the coefficient-ring conditio…
Exact mathematical passage
e'
e^{-1}Use in source · 1 passage
Search aliases: e^-1
Use in source · formal_appendix.tex, line 61
Suppose now that e is invertible in the coefficient ring. Define S=∑_j=0^e-1jT^j. A direct coefficient calculation gives (T-1)S=e-N. For x∈ker N, take y=e^-1Sx. Then (T-1)y=x, so every cocycle is a coboundary and H^1(C_e,M)=0. The averaging argument uses division by e at its decisive step. In residue characteristic dividing e, the same display remains an identity but no longer constructs y.
Exact mathematical passage
y=e^{-1}Sx
e_1Use in source · 1 passage
Search aliases: e_1
Use in source · worked_appendix.tex, line 39
Let ω be a primitive cube root of unity and put A=diag(1,1,ω). Expanding directly gives det(TI-A) =T^3-(2+ω)T^2+(1+2ω)T-ω. The three elementary symmetric coefficients are e_1=2+ω, e_2=1+2ω, e_3=ω. For scalar twist ω^jA, they become ω^j(2+ω), ω^2j(1+2ω), ω^3jω=ω. The determinant is constant for j=0,1,2, but the first coefficient distinguishes the three labels because 2+ω≠0. Thus rank divisibility causes blindness of the determinant, rather than blindness of every coefficient.
Exact mathematical passage
e_1=2+\omega,\qquad e_2=1+2\omega,\qquad e_3=\omega.
E_{12}Use in source · 2 passages
Search aliases: E_12
Use in source · formal_appendix.tex, line 643
There is a genuinely singular contrast. Let A_0=diag(1,-1) over Q_p. Then D_A_0,2(E_12)=0, although E_12≠0. Thus this derivative has no inverse over the fraction field, including when p is odd and 2 is a unit. The certificate theorem cannot cover this center. A universal claim that invertible centers always have invertible power derivatives would be false.
Exact mathematical passage
D_{A_0,2}(E_{12})=0
e_2Use in source · 1 passage
Search aliases: e_2
Use in source · worked_appendix.tex, line 39
Let ω be a primitive cube root of unity and put A=diag(1,1,ω). Expanding directly gives det(TI-A) =T^3-(2+ω)T^2+(1+2ω)T-ω. The three elementary symmetric coefficients are e_1=2+ω, e_2=1+2ω, e_3=ω. For scalar twist ω^jA, they become ω^j(2+ω), ω^2j(1+2ω), ω^3jω=ω. The determinant is constant for j=0,1,2, but the first coefficient distinguishes the three labels because 2+ω≠0. Thus rank divisibility causes blindness of the determinant, rather than blindness of every coefficient.
Exact mathematical passage
e_1=2+\omega,\qquad e_2=1+2\omega,\qquad e_3=\omega.
e_3Use in source · 1 passage
Search aliases: e_3
Use in source · worked_appendix.tex, line 39
Let ω be a primitive cube root of unity and put A=diag(1,1,ω). Expanding directly gives det(TI-A) =T^3-(2+ω)T^2+(1+2ω)T-ω. The three elementary symmetric coefficients are e_1=2+ω, e_2=1+2ω, e_3=ω. For scalar twist ω^jA, they become ω^j(2+ω), ω^2j(1+2ω), ω^3jω=ω. The determinant is constant for j=0,1,2, but the first coefficient distinguishes the three labels because 2+ω≠0. Thus rank divisibility causes blindness of the determinant, rather than blindness of every coefficient.
Exact mathematical passage
e_1=2+\omega,\qquad e_2=1+2\omega,\qquad e_3=\omega.
E_{ab}Use in source · 1 passage
Search aliases: E_ab
Use in source · body.tex, line 918
dule. It is therefore zero. The split semisimple algebra has the form A≅⊕_iM_d_i(k), k^n≅⊕_i(k^d_i)^⊕ m_i, ∑_i d_im_i=n. Faithfulness gives m_i≥1 for every block. The trace pairing B(x,y)=Tr_k^n(xy) restricts on the ith block to m_iTr(XY). Matrix trace pairing is nondegenerate, as is seen by pairing matrix units E_ab with E_ba. Each m_i is nonzero in k: this is automatic in characteristic zero, and follows from 1≤ m_i≤ n<ℓ in residual characteristic. Thus B is nondegenerate on A.
Exact mathematical passage
E_{ab}
E_{ba}Use in source · 1 passage
Search aliases: E_ba
Use in source · body.tex, line 918
efore zero. The split semisimple algebra has the form A≅⊕_iM_d_i(k), k^n≅⊕_i(k^d_i)^⊕ m_i, ∑_i d_im_i=n. Faithfulness gives m_i≥1 for every block. The trace pairing B(x,y)=Tr_k^n(xy) restricts on the ith block to m_iTr(XY). Matrix trace pairing is nondegenerate, as is seen by pairing matrix units E_ab with E_ba. Each m_i is nonzero in k: this is automatic in characteristic zero, and follows from 1≤ m_i≤ n<ℓ in residual characteristic. Thus B is nondegenerate on A.
Exact mathematical passage
E_{ba}
E_{ij}Use in source · 2 passages
Search aliases: E_ij
Use in source · formal_appendix.tex, line 537
There is a finite exact certificate for the inverse. For every matrix unit E_ij check QD(E_ij)=E_ij, DQ(E_ij)=E_ij. Linearity and the matrix-unit basis extend both identities to every matrix over Q_p. These equalities are exact in the coefficient field. They are not merely congruences modulo p. In particular, an operator singular over Q_p cannot satisfy them.
Exact mathematical passage
E_{ij}
E_wUse in source · 8 passages
Search aliases: E_w
Use in source · body.tex, line 623
For each place v of F, put R_v=ρ|_W_F_v, A_v=rec_F_v(π_0,v). Cyclic base change compatibility gives Res_W_E_wR_v ≅ Res_W_E_wA_v.
Exact mathematical passage
\Res_{W_{E_w}}R_v \cong \Res_{W_{E_w}}A_v.
\mathbb FUse in source · 7 passages
Search aliases: F mathbb
Use in source · bibliography.tex, line 12
bibitemKimYamauchi2016 H. H. Kim and T. Yamauchi, A uniform structure on subgroups of GL_n( F_q) and its application to a conditional construction of Artin representations of GL_n, Journal of the Ramanujan Mathematical Society 32(1) (2017), 75--99. Checked preprint https://arxiv.org/abs/1403.5535arXiv:1403.5535v3, 7 April 2016. Theorem 1.1 is conditional on its five inputs. The inherited citation key records the preprint year.
Exact mathematical passage
GL_n(\mathbb F_q)
fUse in source · 6 passages
Search aliases: f
Use in source · body.tex, line 735
Choose wmid v and let f=f(w/v). If the normalized Satake roots of π_v are α_1,…,α_n, unramified base change replaces the Satake conjugacy class by its fth power. Hence the roots of Π_w are α_i^f. Since Π_w is attached to a representation of exponent dividing M, (α_i^f)^M=1. Thus α_i∈μ_fM. Since fmid d, this group is contained in μ_dM.
Exact mathematical passage
f=f(w/v)
F^{-1}Use in source · 1 passage
Search aliases: F^-1
Use in source · formal_appendix.tex, line 86
In Frobenius coordinates the relevant action is often T(X)=FXF^-1. The images of T-1 and 1-T coincide, since multiplication of a source vector by -1 exchanges the signs. This is the bridge to the concrete quotient by Frobenius coboundaries. The recovered v0.18 source proves that bridge under the explicit zero-norm hypothesis. It does not claim that every Frobenius quotient is automatically an arithmetic local cohomology group. One must first identify the acting group and the coefficient module.
Exact mathematical passage
T(X)=FXF^{-1}
F^2Use in source · 1 passage
Search aliases: F^2
Use in source · formal_appendix.tex, line 606
Choose m=3, so the local root ball is A+8Mat_2( Z_2) and the finite target test is modulo 2^m+s=32. Consider F=1 · 1 8 · 1, U=F^2=9 · 2 16 · 9. Here AF=9 · 2
Exact mathematical passage
F=\begin{pmatrix}1&1\\8&1\end{pmatrix},\qquad U=F^2=\begin{pmatrix}9&2\\16&9\end{pmatrix}.
F^3Use in source · 1 passage
Search aliases: F^3
Use in source · formal_appendix.tex, line 691
Nor does a successful shallow node necessarily extend. For example, the scalar prefix F=4 modulo 9 satisfies F^3=1 there. Its lifts modulo 27 are 4,13,22, all with cube 10. That branch stops at the next level. Merely checking a finite number of successful levels gives no all-depth conclusion without another theorem.
Exact mathematical passage
F^3=1
F^dUse in source · 1 passage
Search aliases: F^d
Use in source · formal_appendix.tex, line 563
The root constructed over Q_p is integral. Its center is integral, and every correction entry has norm at most p^-m≤1. If d>0 and U is a unit, the root F is also a unit. Indeed, F commutes with U=F^d and hence with U^-1, and F^d-1U^-1 is a two-sided inverse of F. Thus the unit version has an actual output in GL_n( Z_p).
Exact mathematical passage
U=F^d
F^{d-1}Use in source · 1 passage
Search aliases: F^d-1
Use in source · formal_appendix.tex, line 564
The root constructed over Q_p is integral. Its center is integral, and every correction entry has norm at most p^-m≤1. If d>0 and U is a unit, the root F is also a unit. Indeed, F commutes with U=F^d and hence with U^-1, and F^d-1U^-1 is a two-sided inverse of F. Thus the unit version has an actual output in GL_n( Z_p).
Exact mathematical passage
F^{d-1}U^{-1}
\mathbb F_3Use in source · 6 passages
Search aliases: F_3 mathbb
Use in source · formal_appendix.tex, line 67
[The same generator action gives different cyclic cohomology] Over F_3, let M=Mat_2( F_3) and let T be conjugation by F=diag(1,-1). The diagonal entries of a matrix form the +1 eigenspace of T. The off-diagonal entries form its -1 eigenspace. Both spaces have dimension two.
Exact mathematical passage
\mathbb F_3
\mathbb F_qUse in source · 1 passage
Search aliases: F_q mathbb
Use in source · bibliography.tex, line 12
bibitemKimYamauchi2016 H. H. Kim and T. Yamauchi, A uniform structure on subgroups of GL_n( F_q) and its application to a conditional construction of Artin representations of GL_n, Journal of the Ramanujan Mathematical Society 32(1) (2017), 75--99. Checked preprint https://arxiv.org/abs/1403.5535arXiv:1403.5535v3, 7 April 2016. Theorem 1.1 is conditional on its five inputs. The inherited citation key records the preprint year.
Exact mathematical passage
GL_n(\mathbb F_q)
F_rUse in source · 2 passages
Search aliases: F_r
Use in source · formal_appendix.tex, line 671
This matters for simultaneous matrix prescriptions. At level r, a node can contain a shared matrix F_r together with one local conjugator z_i,r for each member of a finite family. The equations are z_i,rF_r^d_iz_i,r^-1=U_ipmod p^r, with the prescribed prefix of F_r and prescribed residual identity for each conjugator. These are a single joint predicate on a finite tuple. The arithmetic equations and prefixes are preserved by reduction. Compactness therefore produces compatible towers for the sha…
Exact mathematical passage
F_r
F_r^{d_i}Use in source · 1 passage
Search aliases: F_r^d_i
Use in source · formal_appendix.tex, line 673
This matters for simultaneous matrix prescriptions. At level r, a node can contain a shared matrix F_r together with one local conjugator z_i,r for each member of a finite family. The equations are z_i,rF_r^d_iz_i,r^-1=U_ipmod p^r, with the prescribed prefix of F_r and prescribed residual identity for each conjugator. These are a single joint predicate on a finite tuple. The arithmetic equations and prefixes are preserved by reduction. Compactness therefore produces compatible towers for the sha…
Exact mathematical passage
z_{i,r}F_r^{d_i}z_{i,r}^{-1}=U_i\pmod {p^r},
\operatorname{Frob}Use in source · 1 passage
Search aliases: Frob operatorname
Use in source · body.tex, line 1124
[Frobenius transform after reduction] Let P have integral coefficients at a place of residue characteristic p. Then overlineP^[p](T) = Frob(overline P(T)), where Frobenius raises every coefficient to its pth power. In particular, the map overline P⟼overlineP^[p] is injective over the residue field.
Exact mathematical passage
\boxed{ \overline{P^{[p]}}(T) = \operatorname{Frob}\bigl(\overline P(T)\bigr), }
\Frob_wUse in source · 1 passage
Search aliases: Frob_w Frob
Use in source · body.tex, line 1025
Restrict the residual realization to G_E. At every unramified place w of E above a good place v of F, cyclic base change identifies the characteristic polynomial of the restriction with the reduction of the Hecke polynomial of Π_w. Since Π is attached to τ, this is the characteristic polynomial of overlineτ(Frob_w). Chebotarev density and Brauer Nesbitt give ( overline r_π,λ|_G_E )^ss ≅ overlineτ. The right side is absolutely irreducible and has the full dimension n. Hence the restriction itself…
Exact mathematical passage
\overline\tau(\Frob_w)
g^{-1}Use in source · 2 passages
Search aliases: g^-1
Use in source · body.tex, line 370
Choose g∈ G whose class generates G/H, so g^p∈ H. Stability gives an invertible intertwiner A with Aτ(h)A^-1=τ(ghg^-1) (h∈ H). Iterate this identity p times. The operator τ(g^p)^-1A^p commutes with τ(H), so Schur's lemma makes it a nonzero scalar cI. Also A commutes with τ(g^p), by the same intertwining identity at h=g^p. Choose b∈CC^× with b^p=c^-1 and put R=bA. Then Rτ(h)R^-1=τ(ghg^-1), R^p=τ(g^p). These are exactly the conjugation and closing relations for the coset presentation of G. Assigni…
Exact mathematical passage
A\tau(h)A^{-1}=\tau(ghg^{-1})\qquad(h\in H).
g^pUse in source · 5 passages
Search aliases: g^p
Use in source · body.tex, line 369
Choose g∈ G whose class generates G/H, so g^p∈ H. Stability gives an invertible intertwiner A with Aτ(h)A^-1=τ(ghg^-1) (h∈ H). Iterate this identity p times. The operator τ(g^p)^-1A^p commutes with τ(H), so Schur's lemma makes it a nonzero scalar cI. Also A commutes with τ(g^p), by the same intertwining identity at h=g^p. Choose b∈CC^× with b^p=c^-1 and put R=bA. Then Rτ(h)R^-1=τ(ghg^-1), R^p=τ(g^p). These are exactly the conjugation and closing relations for the coset presentation of G. Assigni…
Exact mathematical passage
g^p\in H
g^rUse in source · 2 passages
Search aliases: g^r
Use in source · body.tex, line 503
[Fourier trace interpretation] Let g be a lift of a generator of G/H. For a stable type τ, widehat P_τ,V(r) = Tr(g^rmid M_τ(V)). If V is twisted by ξ^j, then widehat P_τ,V⊗ξ^j(r) = ζ^jrwidehat P_τ,V(r).
Exact mathematical passage
\boxed{ \widehat P_{\tau,V}(r) = \Tr\left(g^r\mid M_\tau(V)\right). }
g_1Use in source · 2 passages
Search aliases: g_1
Use in source · body.tex, line 869
Absolute irreducibility and Burnside's theorem give group elements g_1,…,g_n^2 whose matrices form a K basis of M_n(K). Vectorize these matrices in the fixed basis. Their determinant is nonzero. Exclude places at which that determinant is zero or has a denominator, together with denominators of the finitely many group matrices. At a remaining place the coordinate lattice is stable, and the reduced matrices span the full matrix algebra over the residue field. They continue to span it after extend…
Exact mathematical passage
g_1,\ldots,g_{n^2}
g_dUse in source · 1 passage
Search aliases: g_d
Use in source · body.tex, line 1387
The algebraic pseudocharacter realization theorem supplies a semisimple representation r with trace T. Its continuity deserves a separate argument. Let A be the span of its image. The trace pairing on A is nondegenerate, by the characteristic zero semisimple algebra argument in [source reference]. Choose a basis r(g_1),…,r(g_d) from the image. The functions g⟼Tr(r(g)r(g_i))=T(gg_i) are continuous. Nondegeneracy says that these finitely many trace functions determine all linear coordinates of r(g…
Exact mathematical passage
r(g_1),\ldots,r(g_d)
G_iUse in source · 1 passage
Search aliases: G_i
Use in source · formal_appendix.tex, line 218
After taking the local coboundary quotients, the remaining class belongs to coker( H^1_cont(G,I)⟶ ∏_i H^1_cont(G_i,I)). Here the local groups, their maps into G, and the precise continuous coefficient action must be supplied. The recovered source constructs the corresponding quotients directly in its square-zero data.
Exact mathematical passage
\operatorname{coker}\left( H^1_{\mathrm{cont}}(G,I)\longrightarrow \prod_i H^1_{\mathrm{cont}}(G_i,I)\right). \label{eq:appendix-gauge-cokernel}
g_iUse in source · 1 passage
Search aliases: g_i
Use in source · body.tex, line 1388
algebraic pseudocharacter realization theorem supplies a semisimple representation r with trace T. Its continuity deserves a separate argument. Let A be the span of its image. The trace pairing on A is nondegenerate, by the characteristic zero semisimple algebra argument in [source reference]. Choose a basis r(g_1),…,r(g_d) from the image. The functions g⟼Tr(r(g)r(g_i))=T(gg_i) are continuous. Nondegeneracy says that these finitely many trace functions determine all linear coordinates of r(g) in…
Exact mathematical passage
g\longmapsto\Tr(r(g)r(g_i))=T(gg_i)
g_{n^2}Use in source · 1 passage
Search aliases: g_n^2
Use in source · body.tex, line 869
Absolute irreducibility and Burnside's theorem give group elements g_1,…,g_n^2 whose matrices form a K basis of M_n(K). Vectorize these matrices in the fixed basis. Their determinant is nonzero. Exclude places at which that determinant is zero or has a denominator, together with denominators of the finitely many group matrices. At a remaining place the coordinate lattice is stable, and the reduced matrices span the full matrix algebra over the residue field. They continue to span it after extend…
Exact mathematical passage
g_1,\ldots,g_{n^2}
g_pUse in source · 3 passages
Search aliases: g_p
Use in source · body.tex, line 1088
For g∈ G, write its commuting primary decomposition as g=g_p g_s, where g_p has p power order and g_s has order prime to p. The image of g_s in C_p is trivial, so g_s∈ H. In residue characteristic p, the matrix of g_s is diagonalizable after extending the field, and the matrix of g_p is unipotent. They commute, so the latter preserves each eigenspace of the former. On an eigenspace with eigenvalue α, the product is α times a unipotent operator. Its characteristic polynomial has only the root α, …
Exact mathematical passage
g=g_p g_s
g_sUse in source · 7 passages
Search aliases: g_s
Use in source · body.tex, line 1088
For g∈ G, write its commuting primary decomposition as g=g_p g_s, where g_p has p power order and g_s has order prime to p. The image of g_s in C_p is trivial, so g_s∈ H. In residue characteristic p, the matrix of g_s is diagonalizable after extending the field, and the matrix of g_p is unipotent. They commute, so the latter preserves each eigenspace of the former. On an eigenspace with eigenvalue α, the product is α times a unipotent operator. Its characteristic polynomial has only the root α, …
Exact mathematical passage
g=g_p g_s
\GalUse in source · 2 passages
Search aliases: Gal
Use in source · body.tex, line 9
Let F be a number field, let G_F=Gal(overline F/F), and let ρ:G_F⟶GL_n(CC) be a continuous representation with finite image. Continuity is taken for the usual complex topology. Such a representation factors through the Galois group of a finite extension. Throughout the conjecture discussion, ρ is irreducible. The trivial one dimensional representation is excluded when discussing entireness of an Artin L function, since its L function is the Dedekind zeta function.
Exact mathematical passage
G_F=\Gal(\overline F/F)
\operatorname{GL}Use in source · 1 passage
Search aliases: GL operatorname
Use in source · formal_appendix.tex, line 565
The root constructed over Q_p is integral. Its center is integral, and every correction entry has norm at most p^-m≤1. If d>0 and U is a unit, the root F is also a unit. Indeed, F commutes with U=F^d and hence with U^-1, and F^d-1U^-1 is a two-sided inverse of F. Thus the unit version has an actual output in GL_n( Z_p).
Exact mathematical passage
\operatorname{GL}_n(\mathbb Z_p)
\GL_3Use in source · 1 passage
Search aliases: GL_3 GL
Use in source · body.tex, line 45
The solvable two dimensional sectors treated by Langlands and Tunnell, further examples such as the solvable GO(4) range, and other established cases show that the conjecture is far from an empty expectation [source reference]. A recent preprint of Wang states the generalized octahedral GL_3 case with projective image C_3^2rtimesSL(2,3), together with a primitive solvable six dimensional consequence [source reference]. Its stated results have been checked against the submitted source. Its full p…
Exact mathematical passage
\GL_3
\GL_NUse in source · 1 passage
Search aliases: GL_N GL
Use in source · body.tex, line 241
[Determinant character of an algebraic representation] Let r:GL_n(CC)⟶GL_N(CC) be an irreducible algebraic representation. Suppose the center acts with weight w, so r(zI_n)=z^wI_N. There is an integer a such that det r(g)=det(g)^a and wN=an.
Exact mathematical passage
r:\GL_n(\CC)\longrightarrow\GL_N(\CC)
\operatorname{GL}_nUse in source · 1 passage
Search aliases: GL_n operatorname
Use in source · formal_appendix.tex, line 565
The root constructed over Q_p is integral. Its center is integral, and every correction entry has norm at most p^-m≤1. If d>0 and U is a unit, the root F is also a unit. Indeed, F commutes with U=F^d and hence with U^-1, and F^d-1U^-1 is a two-sided inverse of F. Thus the unit version has an actual output in GL_n( Z_p).
Exact mathematical passage
\operatorname{GL}_n(\mathbb Z_p)
\mathrm{GO}Use in source · 1 passage
Search aliases: GO mathrm
Use in source · body.tex, line 45
The solvable two dimensional sectors treated by Langlands and Tunnell, further examples such as the solvable GO(4) range, and other established cases show that the conjecture is far from an empty expectation [source reference]. A recent preprint of Wang states the generalized octahedral GL_3 case with projective image C_3^2rtimesSL(2,3), together with a primitive solvable six dimensional consequence [source reference]. Its stated results have been checked against the submitted source. Its full p…
Exact mathematical passage
\mathrm{GO}(4)
hUse in source · 61 passages
Search aliases: h
Use in source · body.tex, line 370
Choose g∈ G whose class generates G/H, so g^p∈ H. Stability gives an invertible intertwiner A with Aτ(h)A^-1=τ(ghg^-1) (h∈ H). Iterate this identity p times. The operator τ(g^p)^-1A^p commutes with τ(H), so Schur's lemma makes it a nonzero scalar cI. Also A commutes with τ(g^p), by the same intertwining identity at h=g^p. Choose b∈CC^× with b^p=c^-1 and put R=bA. Then Rτ(h)R^-1=τ(ghg^-1), R^p=τ(g^p). These are exactly the conjugation and closing relations for the coset presentation of G. Assigni…
Exact mathematical passage
A\tau(h)A^{-1}=\tau(ghg^{-1})\qquad(h\in H).
H^1Use in source · 4 passages
Search aliases: H^1
Use in source · formal_appendix.tex, line 47
For a zero-cochain y∈ M, its coboundary at γ is (T-1)y. Taking the quotient by these changes gives the precise presentation H^1(C_e,M)≃ ker N/im(T-1). If the norm operator is identically zero, this reduces to M/im(T-1). That reduction has a hypothesis. Dropping the kernel from the general formula introduces classes that cannot satisfy the cyclic relation.
Exact mathematical passage
H^1(C_e,M)\simeq \ker N/\operatorname{im}(T-1). \label{eq:appendix-cyclic-H1}
H^1_{\mathrm{cont}}Use in source · 1 passage
Search aliases: H^1_cont mathrm
Use in source · formal_appendix.tex, line 218
After taking the local coboundary quotients, the remaining class belongs to coker( H^1_cont(G,I)⟶ ∏_i H^1_cont(G_i,I)). Here the local groups, their maps into G, and the precise continuous coefficient action must be supplied. The recovered source constructs the corresponding quotients directly in its square-zero data.
Exact mathematical passage
\operatorname{coker}\left( H^1_{\mathrm{cont}}(G,I)\longrightarrow \prod_i H^1_{\mathrm{cont}}(G_i,I)\right). \label{eq:appendix-gauge-cokernel}
H^2Use in source · 3 passages
Search aliases: H^2
Use in source · formal_appendix.tex, line 144
hosen lifts to r'_g=(1+b(g))r_g, where b:G→ I and b(1)=0. Expanding without commuting the factors gives r'_gr'_h =(1+b(g)+g· b(h)+c(g,h))r_gh. To compare with r'_gh, multiply by 1-b(gh). The new defect is c'(g,h)=c(g,h)+g· b(h)-b(gh)+b(g). Thus changing the section adds a two-coboundary. The class [c]∈ H^2(G,I) does not depend on the section.
Exact mathematical passage
[c]\in H^2(G,I)
h^2Use in source · 3 passages
Search aliases: h^2
Use in source · formal_appendix.tex, line 306
inear system F(x)=0, a relation among the residual Jacobian rows is only a first-order identity. Consider F(x)=(x,x^2), J_x(h)=(h,2xh). At the origin the derivative has horizontal image. The functional selecting the second coordinate is a residual syzygy, but it does not annihilate F(x) for general x. The exact relation is the moving relation (-x,1)F(x)=0. Furthermore, F(x+h)-F(x)-J_x(h)=(0,h^2). Modulo 8, taking x=0 and h=2 gives remainder (0,4), outside the horizontal target. Keeping only the …
Exact mathematical passage
F(x+h)-F(x)-J_x(h)=(0,h^2).
h^3Use in source · 1 passage
Search aliases: h^3
Use in source · formal_appendix.tex, line 401
For degree three, the linear term is D_A,3(h)=A^2h+AhA+hA^2, and the nonlinear remainder is R_A,3(h)=Ah^2+hAh+h^2A+h^3. The middle word hAh cannot be absorbed into Ah^2 in a matrix ring. The remainder estimate keeps all four words and bounds each by its norm, so it respects the ordered expansion. This also explains why the estimate is not an assertion of commutative binomial calculus in disguise. The scalar bound remains uniform because the norm of a finite ultrametric sum is controlled by its l…
Exact mathematical passage
R_{A,3}(h)=Ah^2+hAh+h^2A+h^3.
h_0Use in source · 1 passage
Search aliases: h_0
Use in source · formal_appendix.tex, line 506
The proof gives a controlled approximation process. Start at h_0=0 and iterate h_j+1=Φ(h_j). Then h_1=QΔ, and normh_j+1-h_j≤ (Kr)^jnormh_1. For t>j, ultrametricity bounds normh_t-h_j by the largest of the intervening increments, which is at most (Kr)^jnormh_1. The limit therefore satisfies the same bound for its distance from h_j. With r=p^-m and K=p^s, every iteration gains at least m-s digits in this estimate. This is a quantitative consequence after the inverse and target test have been certi…
Exact mathematical passage
h_0=0
h_1Use in source · 3 passages
Search aliases: h_1
Use in source · formal_appendix.tex, line 507
The proof gives a controlled approximation process. Start at h_0=0 and iterate h_j+1=Φ(h_j). Then h_1=QΔ, and normh_j+1-h_j≤ (Kr)^jnormh_1. For t>j, ultrametricity bounds normh_t-h_j by the largest of the intervening increments, which is at most (Kr)^jnormh_1. The limit therefore satisfies the same bound for its distance from h_j. With r=p^-m and K=p^s, every iteration gains at least m-s digits in this estimate. This is a quantitative consequence after the inverse and target test have been certi…
Exact mathematical passage
h_1=Q\Delta
H_iUse in source · 2 passages
Search aliases: H_i
Use in source · body.tex, line 1505
[Positive saturation criterion] Let χ be a nontrivial irreducible character of G. Suppose there are a positive integer m, nonnegative integers a_i, subgroups H_i, and characters θ_i∈calE such that mχ = ∑_i a_iInd_H_i^Gθ_i. Then L(s,χ) is entire.
Exact mathematical passage
H_i
h_jUse in source · 4 passages
Search aliases: h_j
Use in source · formal_appendix.tex, line 507
The proof gives a controlled approximation process. Start at h_0=0 and iterate h_j+1=Φ(h_j). Then h_1=QΔ, and normh_j+1-h_j≤ (Kr)^jnormh_1. For t>j, ultrametricity bounds normh_t-h_j by the largest of the intervening increments, which is at most (Kr)^jnormh_1. The limit therefore satisfies the same bound for its distance from h_j. With r=p^-m and K=p^s, every iteration gains at least m-s digits in this estimate. This is a quantitative consequence after the inverse and target test have been certi…
Exact mathematical passage
h_{j+1}=\Phi(h_j)
h_{j+1}Use in source · 2 passages
Search aliases: h_j+1
Use in source · formal_appendix.tex, line 507
The proof gives a controlled approximation process. Start at h_0=0 and iterate h_j+1=Φ(h_j). Then h_1=QΔ, and normh_j+1-h_j≤ (Kr)^jnormh_1. For t>j, ultrametricity bounds normh_t-h_j by the largest of the intervening increments, which is at most (Kr)^jnormh_1. The limit therefore satisfies the same bound for its distance from h_j. With r=p^-m and K=p^s, every iteration gains at least m-s digits in this estimate. This is a quantitative consequence after the inverse and target test have been certi…
Exact mathematical passage
h_{j+1}=\Phi(h_j)
h_tUse in source · 1 passage
Search aliases: h_t
Use in source · formal_appendix.tex, line 511
The proof gives a controlled approximation process. Start at h_0=0 and iterate h_j+1=Φ(h_j). Then h_1=QΔ, and normh_j+1-h_j≤ (Kr)^jnormh_1. For t>j, ultrametricity bounds normh_t-h_j by the largest of the intervening increments, which is at most (Kr)^jnormh_1. The limit therefore satisfies the same bound for its distance from h_j. With r=p^-m and K=p^s, every iteration gains at least m-s digits in this estimate. This is a quantitative consequence after the inverse and target test have been certi…
Exact mathematical passage
\norm{h_t-h_j}
I^2Use in source · 2 passages
Search aliases: I^2
Use in source · formal_appendix.tex, line 96
Let q:R'→ R be a ring map with kernel I, and assume I^2=0. Let ρ:G→ R^× be a group representation. Choose unit-valued lifts r_g∈ (R')^× with q(r_g)=ρ(g) and r_1=1. Such a choice is an input to this calculation. It is not already a multiplicative lift. Define its normalized defect by c(g,h)=(r_gr_h-r_gh)r_gh^-1∈ I. Equivalently, r_gr_h=(1+c(g,h))r_gh. The coefficients are elements of the actual ring kernel, with its additive group, rather than an unrelated vector space of the same dimension.
Exact mathematical passage
I^2=0
I_2Use in source · 3 passages
Search aliases: I_2
Use in source · formal_appendix.tex, line 600
The residual derivative is singular. In characteristic two, D_A,2(I_2)=2A=0, while I_2≠0. Nevertheless J is invertible over Q_2, with determinant 16. Residual singularity and singularity over the fraction field are different tests.
Exact mathematical passage
D_{A,2}(I_2)=2A=0
I_4Use in source · 1 passage
Search aliases: I_4
Use in source · formal_appendix.tex, line 595
0 · 0 · -1 · 2 . Direct multiplication gives BJ=JB=4I_4. Therefore Q=tfrac14 B is the exact inverse and s=2. The entry tfrac14 in this inverse prevents clearing its denominators with only one factor of 2.
Exact mathematical passage
BJ=JB=4I_4
I_NUse in source · 1 passage
Search aliases: I_N
Use in source · body.tex, line 245
[Determinant character of an algebraic representation] Let r:GL_n(CC)⟶GL_N(CC) be an irreducible algebraic representation. Suppose the center acts with weight w, so r(zI_n)=z^wI_N. There is an integer a such that det r(g)=det(g)^a and wN=an.
Exact mathematical passage
r(zI_n)=z^wI_N.
\imUse in source · 1 passage
Search aliases: im
Use in source · body.tex, line 943
[Uniform bound for a residual Galois image] Let overline r:G_F⟶GL_n(overlineFF_ℓ) be continuous and semisimple, with ℓ>n. If its Frobenius traces belong to a set of size at most t on a Chebotarev dense set, then absimoverline r≤ t^n^2.
Exact mathematical passage
\boxed{ \abs{\im\overline r}\leq t^{n^2}. }
\operatorname{im}Use in source · 7 passages
Search aliases: im operatorname
Use in source · formal_appendix.tex, line 47
For a zero-cochain y∈ M, its coboundary at γ is (T-1)y. Taking the quotient by these changes gives the precise presentation H^1(C_e,M)≃ ker N/im(T-1). If the norm operator is identically zero, this reduces to M/im(T-1). That reduction has a hypothesis. Dropping the kernel from the general formula introduces classes that cannot satisfy the cyclic relation.
Exact mathematical passage
H^1(C_e,M)\simeq \ker N/\operatorname{im}(T-1). \label{eq:appendix-cyclic-H1}
\IndUse in source · 5 passages
Search aliases: Ind
Use in source · body.tex, line 317
[The total induced object is torsor blind] Let Htriangleleft G with cyclic quotient character group X, and let V be an extension of an H representation. Then ⊕_χ∈ X V⊗χ ≅ Ind_H^GRes_HV. The left side is unchanged when V is replaced by another member of its extension torsor. Likewise, for cyclic base change, mathopboxplus_χ∈ X π⊗χ ≅ AI_E/F(BC_E/Fπ) inside the established automorphic induction formalism. Therefore comparison of the total induced objects cannot determine the cyclic attachment offse…
Exact mathematical passage
\boxed{ \bigoplus_{\chi\in X} V\otimes\chi \cong \Ind_H^G\Res_HV. }
\Ind_{G_E}^{G_F}Use in source · 1 passage
Search aliases: Ind_G_E^G_F Ind
Use in source · body.tex, line 1568
The automorphic induction of the attached representation over E is already attached to Ind_G_E^G_Fτ ≅ ⊕_j=0^p-1 ρ⊗ϵ^j. It is also the isobaric sum of all descents. By [source reference], this total object is invariant under rotating either torsor. It proves the direct sum correspondence but cannot identify one constituent with the target extension.
Exact mathematical passage
\Ind_{G_E}^{G_F}\tau \cong \bigoplus_{j=0}^{p-1} \rho\otimes\epsilon^j.
\Ind_H^GUse in source · 3 passages
Search aliases: Ind_H^G Ind
Use in source · body.tex, line 317
[The total induced object is torsor blind] Let Htriangleleft G with cyclic quotient character group X, and let V be an extension of an H representation. Then ⊕_χ∈ X V⊗χ ≅ Ind_H^GRes_HV. The left side is unchanged when V is replaced by another member of its extension torsor. Likewise, for cyclic base change, mathopboxplus_χ∈ X π⊗χ ≅ AI_E/F(BC_E/Fπ) inside the established automorphic induction formalism. Therefore comparison of the total induced objects cannot determine the cyclic attachment offse…
Exact mathematical passage
\boxed{ \bigoplus_{\chi\in X} V\otimes\chi \cong \Ind_H^G\Res_HV. }
\Ind_{H_i}^GUse in source · 1 passage
Search aliases: Ind_H_i^G Ind
Use in source · body.tex, line 1506
[Positive saturation criterion] Let χ be a nontrivial irreducible character of G. Suppose there are a positive integer m, nonnegative integers a_i, subgroups H_i, and characters θ_i∈calE such that mχ = ∑_i a_iInd_H_i^Gθ_i. Then L(s,χ) is entire.
Exact mathematical passage
\boxed{ m\chi = \sum_i a_i\Ind_{H_i}^G\theta_i. }
JUse in source · 16 passages
Search aliases: J
Use in source · formal_appendix.tex, line 250
Let J:V→ W be a linear map over a field k. The correction equation Ju+e=0 has a solution precisely when e∈imJ. A useful description of this image uses all the dual relations among the equations. Let S=ker(J^*:W^*→ V^*), C={w∈ W:ϕ(w)=0 for every ϕ∈ S}. Every Ju belongs to C. Conversely, if w does not belong to imJ, its nonzero class in W/imJ is detected by an algebraic linear functional. Pulling that functional back to W gives an element of S that does not annihilate w. Hence imJ=C. This algebrai…
Exact mathematical passage
J:V\to W
J'Use in source · 1 passage
Search aliases: J'
Use in source · formal_appendix.tex, line 286
If a third variable t is added so that the derivative becomes J'(u,v,t)=(u,v,u+v+t), the enlarged map is surjective. It cancels the second error with (u,v,t)=(0,0,-1). But the nonzero value of t records exactly the failed original relation. This enlarged solution cannot be declared a solution of the original equations by forgetting t. One must show that the additional variable is removable, or that the target error already lies in the original compatible plane.
Exact mathematical passage
J'(u,v,t)=(u,v,u+v+t)
J^*Use in source · 1 passage
Search aliases: J^*
Use in source · formal_appendix.tex, line 254
Let J:V→ W be a linear map over a field k. The correction equation Ju+e=0 has a solution precisely when e∈imJ. A useful description of this image uses all the dual relations among the equations. Let S=ker(J^*:W^*→ V^*), C={w∈ W:ϕ(w)=0 for every ϕ∈ S}. Every Ju belongs to C. Conversely, if w does not belong to imJ, its nonzero class in W/imJ is detected by an algebraic linear functional. Pulling that functional back to W gives an element of S that does not annihilate w. Hence imJ=C. This algebrai…
Exact mathematical passage
S=\ker(J^*:W^*\to V^*),\qquad C=\{w\in W:\phi(w)=0\text{ for every }\phi\in S\}.
j_0Use in source · 1 passage
Search aliases: j_0
Use in source · body.tex, line 477
If j_0 is one solution, every other solution differs from it by a rotation stabilizing every spectrum polynomial. The converse is immediate.
Exact mathematical passage
j_0
j_{\det}Use in source · 2 passages
Search aliases: j_det det
Use in source · body.tex, line 669
[Determinant chosen candidate] If pnmid n, determinant and central character select one candidate j_det. The cyclic Strong Artin step is then equivalent to j_det∈calJ_v for every v.
Exact mathematical passage
j_{\det}
J_xUse in source · 2 passages
Search aliases: J_x
Use in source · formal_appendix.tex, line 296
For a nonlinear system F(x)=0, a relation among the residual Jacobian rows is only a first-order identity. Consider F(x)=(x,x^2), J_x(h)=(h,2xh). At the origin the derivative has horizontal image. The functional selecting the second coordinate is a residual syzygy, but it does not annihilate F(x) for general x. The exact relation is the moving relation (-x,1)F(x)=0. Furthermore, F(x+h)-F(x)-J_x(h)=(0,h^2). Modulo 8, taking x=0 and h=2 gives remainder (0,4), outside the horizontal target. Keeping…
Exact mathematical passage
F(x)=(x,x^2),\qquad J_x(h)=(h,2xh).
j_\lambdaUse in source · 1 passage
Search aliases: j_λ lambda
Use in source · body.tex, line 1052
[Only finitely many local checks remain] Under [source reference], all local attachment to ρ is equivalent to checking the label -j_λ at every exceptional finite place excluded from the unramified comparison and at every archimedean place.
Exact mathematical passage
-j_\lambda
j_{\lambda}Use in source · 6 passages
Search aliases: j_λ lambda
Use in source · body.tex, line 1006
labelthm:one_prime_attachment Use the cyclic Artin setup of [source reference]. For this unramified theorem it is enough that the lower cuspidal representation Π is weakly attached to τ, with π a cuspidal cyclic descent. Let λ be a good separating prime. Suppose π has a residual realization overline r_π,λ. Then there is a unique index j_λ∈ZZ/pZZ such that P_π,v(T) = P_ρ⊗ϵ^j_λ,v(T) for every finite place v outside a finite set. Consequently, π⊗ϵ^-j_λ is weakly attached to ρ.
Exact mathematical passage
j_{\lambda}\in\ZZ/p\ZZ
j_\tauUse in source · 1 passage
Search aliases: j_τ tau
Use in source · worked_appendix.tex, line 72
ma, the first block demands shift one and the second demands shift two. The two restrictions to H have the same multiplicities. Each block separately is a quotient twist of the matching source block. The full representation is not one quotient twist, because the required shifts disagree. This is the reason the spectrum criterion quantifies over one common j, rather than an independent j_τ for each type.
Exact mathematical passage
j_\tau
k^2Use in source · 1 passage
Search aliases: k^2
Use in source · formal_appendix.tex, line 277
[A reachable target with a visible failed correction] Let V=k^2, W=k^3, and J(u,v)=(u,v,u+v). Its image is the plane w_3=w_1+w_2. The dual kernel is generated by the functional ϕ(w_1,w_2,w_3)=w_3-w_1-w_2. The error (1,2,3) is compatible in every characteristic and is cancelled by (u,v)=(-1,-2). The error (0,0,1) is incompatible in every field, since its value under ϕ is 1. No choice of the two correction variables can cancel it.
Exact mathematical passage
V=k^2
k^3Use in source · 1 passage
Search aliases: k^3
Use in source · formal_appendix.tex, line 277
[A reachable target with a visible failed correction] Let V=k^2, W=k^3, and J(u,v)=(u,v,u+v). Its image is the plane w_3=w_1+w_2. The dual kernel is generated by the functional ϕ(w_1,w_2,w_3)=w_3-w_1-w_2. The error (1,2,3) is compatible in every characteristic and is cancelled by (u,v)=(-1,-2). The error (0,0,1) is incompatible in every field, since its value under ϕ is 1. No choice of the two correction variables can cancel it.
Exact mathematical passage
W=k^3
k^{d_i}Use in source · 1 passage
Search aliases: k^d_i
Use in source · body.tex, line 909
oof Because products of group matrices are group matrices, A is a finite dimensional subalgebra of M_n(k). An A submodule is the same thing as a Γ submodule. The given module is semisimple and faithful for A. The Jacobson radical annihilates every simple constituent, hence annihilates this faithful module. It is therefore zero. The split semisimple algebra has the form A≅⊕_iM_d_i(k), k^n≅⊕_i(k^d_i)^⊕ m_i, ∑_i d_im_i=n. Faithfulness gives m_i≥1 for every block. The trace pairing B(x,y)=Tr_k^n(xy)…
Exact mathematical passage
A\cong\bigoplus_iM_{d_i}(k),\qquad k^n\cong\bigoplus_i(k^{d_i})^{\oplus m_i}, \qquad\sum_i d_im_i=n.
k^nUse in source · 3 passages
Search aliases: k^n
Use in source · body.tex, line 895
[Finite trace image bound] Let k be algebraically closed of characteristic zero or characteristic ℓ>n. Let Γ⊆GL_n(k) be a subgroup, and assume its defining module k^n is semisimple. Suppose its trace set is finite, with t=#{Tr(γ):γ∈Γ}. Then Γ is finite and absΓ≤ t^dim_k A≤ t^n^2, A=span_kΓ. In particular this bounds a finite semisimple residual matrix group in characteristic greater than its rank.
Exact mathematical passage
k^n
\kerUse in source · 5 passages
Search aliases: ker
Use in source · formal_appendix.tex, line 43
If x=a(γ), repeated application of the cocycle equation gives a(γ^r)=∑_j=0^r-1T^j x (1≤ r≤ e). The group relation γ^e=1 therefore forces Nx=0. Conversely, if Nx=0, this formula defines a cocycle. Whenever exponents wrap around e, the discrepancy is a translate of Nx, which is zero. Thus evaluating at the chosen generator identifies cocycles with ker N.
Exact mathematical passage
\ker N
L_3Use in source · 1 passage
Search aliases: L_3
Use in source · bibliography.tex, line 87
bibitemWang2026 J. Wang, Strong Artin conjecture for generalized octahedral representations in GL_3, https://arxiv.org/abs/2609.38231arXiv:2609.38231v1, 28 September 2026. Theorem 1.1 and Corollary 5.4 state the special three and six dimensional cases mentioned here. Statements checked. The full proof has not been independently audited for this reader edition.
Exact mathematical passage
GL_3
L_nUse in source · 2 passages
Search aliases: L_n
Use in source · bibliography.tex, line 12
bibitemKimYamauchi2016 H. H. Kim and T. Yamauchi, A uniform structure on subgroups of GL_n( F_q) and its application to a conditional construction of Artin representations of GL_n, Journal of the Ramanujan Mathematical Society 32(1) (2017), 75--99. Checked preprint https://arxiv.org/abs/1403.5535arXiv:1403.5535v3, 7 April 2016. Theorem 1.1 is conditional on its five inputs. The inherited citation key records the preprint year.
Exact mathematical passage
GL_n(\mathbb F_q)
\operatorname{lcm}Use in source · 2 passages
Search aliases: lcm operatorname
Use in source · body.tex, line 974
Let t≥1 be the number of trace values in the source alphabet and put B=t^n^2. First fix a finite normal extension L/QQ containing K and the roots of unity of orders dividing D_B=lcm(1,2,…,B). Every polynomial in calQ_n,B, the bounded cyclotomic alphabet, has coefficients in L. Form the union of all Galois conjugates in L of calP∪calQ_n,B. This is a fixed finite set of integral polynomials. For each distinct pair choose one nonzero coefficient difference. Exclude the rational primes below the pri…
Exact mathematical passage
D_B=\operatorname{lcm}(1,2,\ldots,B).
\LongleftrightarrowUse in source · 2 passages
Search aliases: Longleftrightarrow
Use in source · formal_appendix.tex, line 547
Take r=p^-m with m>s. Then Kr=p^s-m<1, and normQ(U-A^d)≤ p^-m · Longleftrightarrow normB(U-A^d)≤ p^-(m+s)notag · Longleftrightarrow Bvec(U-A^d)=0pmod p^m+s. The last congruence is entrywise. It depends only on the reductions of the target, center, and integral operator modulo p^m+s. The theorem therefore decides the existence and uniqueness of a root in A+p^mMat_n( Z_p) by one finite target test, once the exact inverse certificate is supplied.
Exact mathematical passage
\norm{Q(U-A^d)}\le p^{-m} &\Longleftrightarrow \norm{B(U-A^d)}\le p^{-(m+s)}\notag\\ &\Longleftrightarrow B\operatorname{vec}(U-A^d)=0\pmod {p^{m+s}}. \label{eq:appendix-finite-criterion}
\lvertUse in source · 1 passage
Search aliases: lvert
Use in source · body.tex, line 1254
from a simple unitary Kottwitz datum and a cuspidal representation whose infinity component is an X holomorphic limit of discrete series [source reference]. At a coefficient prime satisfying the good reduction and reflex field conditions, it supplies a semisimple l adic representation over the CM field. Its local compatibility is stated on the density one split prime set with the twist lvert·rvert^(1-n)/2 in the source normalization. This is a concrete realization range. It is not a theorem for …
Exact mathematical passage
\lvert\cdot\rvert^{(1-n)/2}
m_aUse in source · 2 passages
Search aliases: m_a
Use in source · body.tex, line 489
Fix a primitive pth root ζ. For P(z)=∑_a∈ZZ/pZZm_az^a, define widehat P(r) = ∑_a∈ZZ/pZZm_aζ^ar, r∈ZZ/pZZ.
Exact mathematical passage
P(z)=\sum_{a\in\ZZ/p\ZZ}m_az^a,
M_{d_i}Use in source · 1 passage
Search aliases: M_d_i
Use in source · body.tex, line 909
oof Because products of group matrices are group matrices, A is a finite dimensional subalgebra of M_n(k). An A submodule is the same thing as a Γ submodule. The given module is semisimple and faithful for A. The Jacobson radical annihilates every simple constituent, hence annihilates this faithful module. It is therefore zero. The split semisimple algebra has the form A≅⊕_iM_d_i(k), k^n≅⊕_i(k^d_i)^⊕ m_i, ∑_i d_im_i=n. Faithfulness gives m_i≥1 for every block. The trace pairing B(x,y)=Tr_k^n(xy)…
Exact mathematical passage
A\cong\bigoplus_iM_{d_i}(k),\qquad k^n\cong\bigoplus_i(k^{d_i})^{\oplus m_i}, \qquad\sum_i d_im_i=n.
m_iUse in source · 6 passages
Search aliases: m_i
Use in source · body.tex, line 909
oof Because products of group matrices are group matrices, A is a finite dimensional subalgebra of M_n(k). An A submodule is the same thing as a Γ submodule. The given module is semisimple and faithful for A. The Jacobson radical annihilates every simple constituent, hence annihilates this faithful module. It is therefore zero. The split semisimple algebra has the form A≅⊕_iM_d_i(k), k^n≅⊕_i(k^d_i)^⊕ m_i, ∑_i d_im_i=n. Faithfulness gives m_i≥1 for every block. The trace pairing B(x,y)=Tr_k^n(xy)…
Exact mathematical passage
A\cong\bigoplus_iM_{d_i}(k),\qquad k^n\cong\bigoplus_i(k^{d_i})^{\oplus m_i}, \qquad\sum_i d_im_i=n.
M_nUse in source · 2 passages
Search aliases: M_n
Use in source · body.tex, line 869
Absolute irreducibility and Burnside's theorem give group elements g_1,…,g_n^2 whose matrices form a K basis of M_n(K). Vectorize these matrices in the fixed basis. Their determinant is nonzero. Exclude places at which that determinant is zero or has a denominator, together with denominators of the finitely many group matrices. At a remaining place the coordinate lattice is stable, and the reduced matrices span the full matrix algebra over the residue field. They continue to span it after extend…
Exact mathematical passage
M_n(K)
\operatorname{Mat}Use in source · 4 passages
Search aliases: Mat operatorname
Use in source · formal_appendix.tex, line 67
[The same generator action gives different cyclic cohomology] Over F_3, let M=Mat_2( F_3) and let T be conjugation by F=diag(1,-1). The diagonal entries of a matrix form the +1 eigenspace of T. The off-diagonal entries form its -1 eigenspace. Both spaces have dimension two.
Exact mathematical passage
M=\operatorname{Mat}_2(\mathbb F_3)
\operatorname{Mat}_2Use in source · 2 passages
Search aliases: Mat_2 operatorname
Use in source · formal_appendix.tex, line 67
[The same generator action gives different cyclic cohomology] Over F_3, let M=Mat_2( F_3) and let T be conjugation by F=diag(1,-1). The diagonal entries of a matrix form the +1 eigenspace of T. The off-diagonal entries form its -1 eigenspace. Both spaces have dimension two.
Exact mathematical passage
M=\operatorname{Mat}_2(\mathbb F_3)
\operatorname{Mat}_nUse in source · 2 passages
Search aliases: Mat_n operatorname
Use in source · formal_appendix.tex, line 446
For E=Mat_n( Q_p) use the entrywise supremum norm and normalize abs p_p=p^-1. Matrix multiplication satisfies abs(XY)_ij_p =|∑_tX_itY_tj|_p ≤max_tabsX_it_pabsY_tj_p ≤norm Xnorm Y. Taking the maximum over the finite set of entries proves submultiplicativity. The same ultrametric calculation removes a matrix dimension factor. Completeness is inherited from the finite product of copies of Q_p. These are the hypotheses required by the actual matrix application, rather than properties borrowed from a…
Exact mathematical passage
E=\operatorname{Mat}_n(\mathbb Q_p)
\mathopUse in source · 1 passage
Search aliases: mathop
Use in source · body.tex, line 326
ced object is torsor blind] Let Htriangleleft G with cyclic quotient character group X, and let V be an extension of an H representation. Then ⊕_χ∈ X V⊗χ ≅ Ind_H^GRes_HV. The left side is unchanged when V is replaced by another member of its extension torsor. Likewise, for cyclic base change, mathopboxplus_χ∈ X π⊗χ ≅ AI_E/F(BC_E/Fπ) inside the established automorphic induction formalism. Therefore comparison of the total induced objects cannot determine the cyclic attachment offset.
Exact mathematical passage
\boxed{ \mathop{\boxplus}_{\chi\in X} \pi\otimes\chi \cong \AI_{E/F}(\BC_{E/F}\pi) }
\maxUse in source · 4 passages
Search aliases: max
Use in source · body.tex, line 977
polynomial in calQ_n,B, the bounded cyclotomic alphabet, has coefficients in L. Form the union of all Galois conjugates in L of calP∪calQ_n,B. This is a fixed finite set of integral polynomials. For each distinct pair choose one nonzero coefficient difference. Exclude the rational primes below the prime divisors of all these differences, and exclude the rational primes ℓ≤max(n,B). This defines a finite exceptional set independent of any residual representation or choice of lift. Outside it, redu…
Exact mathematical passage
\ell\le\max(n,B)
\max_tUse in source · 1 passage
Search aliases: max_t max
Use in source · formal_appendix.tex, line 448
For E=Mat_n( Q_p) use the entrywise supremum norm and normalize abs p_p=p^-1. Matrix multiplication satisfies abs(XY)_ij_p =|∑_tX_itY_tj|_p ≤max_tabsX_it_pabsY_tj_p ≤norm Xnorm Y. Taking the maximum over the finite set of entries proves submultiplicativity. The same ultrametric calculation removes a matrix dimension factor. Completeness is inherited from the finite product of copies of Q_p. These are the hypotheses required by the actual matrix application, rather than properties borrowed from a…
Exact mathematical passage
\abs{(XY)_{ij}}_p =\left|\sum_tX_{it}Y_{tj}\right|_p \le\max_t\abs{X_{it}}_p\abs{Y_{tj}}_p \le\norm X\norm Y.
\midUse in source · 11 passages
Search aliases: mid
Use in source · body.tex, line 176
By [source reference], the stabilizer of the characteristic polynomial is {ζ∈μ_p:ζ^k=1 for every k∈ S(A)}. Since p is prime, a nontrivial pth root satisfies ζ^k=1 exactly when pmid k.
Exact mathematical passage
p\mid k
\mathbb NUse in source · 1 passage
Search aliases: N mathbb
Use in source · formal_appendix.tex, line 527
For integral matrices suppose the inverse has the certified form Q=p^-sB, s∈ N, where B is an integral linear operator on the matrix entries. In a fixed ordering of the n^2 entries it is an n^2 by n^2 matrix over Z_p. This is an operator on the matrix space, not ordinary left multiplication by an n by n matrix. Integral coefficients imply normBz≤norm z, so K=p^s is a valid bound.
Exact mathematical passage
Q=p^{-s}B,\qquad s\in\mathbb N,
n^2Use in source · 10 passages
Search aliases: n^2
Use in source · body.tex, line 869
Absolute irreducibility and Burnside's theorem give group elements g_1,…,g_n^2 whose matrices form a K basis of M_n(K). Vectorize these matrices in the fixed basis. Their determinant is nonzero. Exclude places at which that determinant is zero or has a denominator, together with denominators of the finitely many group matrices. At a remaining place the coordinate lattice is stable, and the reduced matrices span the full matrix algebra over the residue field. They continue to span it after extend…
Exact mathematical passage
g_1,\ldots,g_{n^2}
N_vUse in source · 2 passages
Search aliases: N_v
Use in source · body.tex, line 700
Fix v and wmid v, and write the local Weil Deligne parameter as (ϕ_v,N_v). Base change restricts ϕ_v to W_E_w and keeps the monodromy operator. Since the parameter of Π_w has zero monodromy, N_v=0.
Exact mathematical passage
(\phi_v,N_v)
\nmidUse in source · 11 passages
Search aliases: nmid
Use in source · body.tex, line 180
[Trace and determinant selectors] A nonzero trace separates all cyclic twists. The determinant separates all cyclic twists when pnmid n.
Exact mathematical passage
p\nmid n
\normUse in source · 25 passages
Search aliases: norm
Use in source · formal_appendix.tex, line 370
Let E be a complete normed ring with norm1=1, a submultiplicative norm, and the ultrametric inequality normx+y≤max(normx,normy). Assume norm A≤1. For a nonnegative integer d, define D_A,d(h) · =∑_i=0^d-1A^ihA^d-1-i,
Exact mathematical passage
\norm{1}=1
OUse in source · 1 passage
Search aliases: O
Use in source · bibliography.tex, line 15
bibitemRamakrishnan2002 D. Ramakrishnan, Modularity of solvable Artin representations of GO(4) type, International Mathematics Research Notices 2002(1), 1--54. DOI: https://doi.org/10.1155/S107379280200001610.1155/S1073792802000016.
Exact mathematical passage
GO(4)
\OOUse in source · 5 passages
Search aliases: OO
Use in source · body.tex, line 825
Let K be a number field containing the normalized unramified Hecke coefficients of a finite Satake descent, the character values of the target Artin representation, and the values of the quotient character. Enlarge it to a splitting field in which the target and its restriction have matrix realizations. Let OO_K be its ring of integers.
Exact mathematical passage
\OO_K
\OO_KUse in source · 4 passages
Search aliases: OO_K OO
Use in source · body.tex, line 825
Let K be a number field containing the normalized unramified Hecke coefficients of a finite Satake descent, the character values of the target Artin representation, and the values of the quotient character. Enlarge it to a splitting field in which the target and its restriction have matrix realizations. Let OO_K be its ring of integers.
Exact mathematical passage
\OO_K
\overline BUse in source · 2 passages
Search aliases: overline B overline
Use in source · formal_appendix.tex, line 315
The graded correction argument retains an exact moving relation B_xF(x)=0 and a residual exactness statement imoverline J=keroverline B. If F(x)=π^r e, the moving identity gives π^r B_xe=0. A faithful leading layer allows this equality to imply overline B overline e=0. Equation [source reference] then supplies a residual correction overline u with overline J overline u=-overline e. Lift overline u to u and put x'=x+π^r u. The linear part of F(x') is divisible by π^r+1. For polynomial systems the…
Exact mathematical passage
\operatorname{im}\overline J=\ker\overline B. \label{eq:appendix-residual-exactness}
\overline eUse in source · 2 passages
Search aliases: overline e overline
Use in source · formal_appendix.tex, line 321
The graded correction argument retains an exact moving relation B_xF(x)=0 and a residual exactness statement imoverline J=keroverline B. If F(x)=π^r e, the moving identity gives π^r B_xe=0. A faithful leading layer allows this equality to imply overline B overline e=0. Equation [source reference] then supplies a residual correction overline u with overline J overline u=-overline e. Lift overline u to u and put x'=x+π^r u. The linear part of F(x') is divisible by π^r+1. For polynomial systems the…
Exact mathematical passage
\overline B\,\overline e=0
\overline FUse in source · 1 passage
Search aliases: overline F overline
Use in source · body.tex, line 9
Let F be a number field, let G_F=Gal(overline F/F), and let ρ:G_F⟶GL_n(CC) be a continuous representation with finite image. Continuity is taken for the usual complex topology. Such a representation factors through the Galois group of a finite extension. Throughout the conjecture discussion, ρ is irreducible. The trivial one dimensional representation is excluded when discussing entireness of an Artin L function, since its L function is the Dedekind zeta function.
Exact mathematical passage
G_F=\Gal(\overline F/F)
\overline JUse in source · 2 passages
Search aliases: overline J overline
Use in source · formal_appendix.tex, line 315
The graded correction argument retains an exact moving relation B_xF(x)=0 and a residual exactness statement imoverline J=keroverline B. If F(x)=π^r e, the moving identity gives π^r B_xe=0. A faithful leading layer allows this equality to imply overline B overline e=0. Equation [source reference] then supplies a residual correction overline u with overline J overline u=-overline e. Lift overline u to u and put x'=x+π^r u. The linear part of F(x') is divisible by π^r+1. For polynomial systems the…
Exact mathematical passage
\operatorname{im}\overline J=\ker\overline B. \label{eq:appendix-residual-exactness}
\overline PUse in source · 2 passages
Search aliases: overline P overline
Use in source · body.tex, line 1124
[Frobenius transform after reduction] Let P have integral coefficients at a place of residue characteristic p. Then overlineP^[p](T) = Frob(overline P(T)), where Frobenius raises every coefficient to its pth power. In particular, the map overline P⟼overlineP^[p] is injective over the residue field.
Exact mathematical passage
\boxed{ \overline{P^{[p]}}(T) = \operatorname{Frob}\bigl(\overline P(T)\bigr), }
\overline r_\lambdaUse in source · 3 passages
Search aliases: overline r_λ overline lambda
Use in source · body.tex, line 966
There is a finite set Σ(calP) of rational primes, fixed from the alphabet before any realization is chosen, with the following property. Let λ be a coefficient place whose residue characteristic ℓ is outside Σ(calP). Suppose a continuous semisimple representation overline r_λ:G_F⟶GL_n(overlineFF_ℓ) is unramified outside S and the places above ℓ, and has characteristic polynomial overlineP_v at every remaining unramified place. Then there is a finite image semisimple complex representation r with…
Exact mathematical passage
\overline r_\lambda:G_F\longrightarrow\GL_n(\overline\FF_\ell)
\overline uUse in source · 3 passages
Search aliases: overline u overline
Use in source · formal_appendix.tex, line 323
on B_xF(x)=0 and a residual exactness statement imoverline J=keroverline B. If F(x)=π^r e, the moving identity gives π^r B_xe=0. A faithful leading layer allows this equality to imply overline B overline e=0. Equation [source reference] then supplies a residual correction overline u with overline J overline u=-overline e. Lift overline u to u and put x'=x+π^r u. The linear part of F(x') is divisible by π^r+1. For polynomial systems the remainder contains at least two copies of π^r, so it is divi…
Exact mathematical passage
\overline u
\overline VUse in source · 1 passage
Search aliases: overline V overline
Use in source · body.tex, line 1081
llapse of cyclic extension labels] Let G be finite, let Htriangleleft G with G/H≅ C_p, and let V,W be semisimple representations over a number field K with isomorphic restrictions to H. Choose stable lattices at a coefficient place above p. Over an algebraic closure of the residue field their semisimplified reductions satisfy overline V^ss≅overline W^ss. The same statement applies to finite image representations of an arbitrary group after passing to a common finite quotient that retains the cyc…
Exact mathematical passage
\overline V^{\mathrm{ss}}\cong\overline W^{\mathrm{ss}}.
\overline V^{\mathrm{ss}}Use in source · 1 passage
Search aliases: overline V^ss overline mathrm
Use in source · body.tex, line 1081
llapse of cyclic extension labels] Let G be finite, let Htriangleleft G with G/H≅ C_p, and let V,W be semisimple representations over a number field K with isomorphic restrictions to H. Choose stable lattices at a coefficient place above p. Over an algebraic closure of the residue field their semisimplified reductions satisfy overline V^ss≅overline W^ss. The same statement applies to finite image representations of an arbitrary group after passing to a common finite quotient that retains the cyc…
Exact mathematical passage
\overline V^{\mathrm{ss}}\cong\overline W^{\mathrm{ss}}.
\overline WUse in source · 1 passage
Search aliases: overline W overline
Use in source · body.tex, line 1081
llapse of cyclic extension labels] Let G be finite, let Htriangleleft G with G/H≅ C_p, and let V,W be semisimple representations over a number field K with isomorphic restrictions to H. Choose stable lattices at a coefficient place above p. Over an algebraic closure of the residue field their semisimplified reductions satisfy overline V^ss≅overline W^ss. The same statement applies to finite image representations of an arbitrary group after passing to a common finite quotient that retains the cyc…
Exact mathematical passage
\overline V^{\mathrm{ss}}\cong\overline W^{\mathrm{ss}}.
\overline W^{\mathrm{ss}}Use in source · 1 passage
Search aliases: overline W^ss overline mathrm
Use in source · body.tex, line 1081
llapse of cyclic extension labels] Let G be finite, let Htriangleleft G with G/H≅ C_p, and let V,W be semisimple representations over a number field K with isomorphic restrictions to H. Choose stable lattices at a coefficient place above p. Over an algebraic closure of the residue field their semisimplified reductions satisfy overline V^ss≅overline W^ss. The same statement applies to finite image representations of an arbitrary group after passing to a common finite quotient that retains the cyc…
Exact mathematical passage
\overline V^{\mathrm{ss}}\cong\overline W^{\mathrm{ss}}.
\overline\FFUse in source · 2 passages
Search aliases: overlineFF overline FF
Use in source · body.tex, line 939
[Uniform bound for a residual Galois image] Let overline r:G_F⟶GL_n(overlineFF_ℓ) be continuous and semisimple, with ℓ>n. If its Frobenius traces belong to a set of size at most t on a Chebotarev dense set, then absimoverline r≤ t^n^2.
Exact mathematical passage
\overline r:G_F\longrightarrow\GL_n(\overline\FF_{\ell})
\overline\FF_\ellUse in source · 1 passage
Search aliases: overlineFF_ℓ overline FF ell
Use in source · body.tex, line 966
There is a finite set Σ(calP) of rational primes, fixed from the alphabet before any realization is chosen, with the following property. Let λ be a coefficient place whose residue characteristic ℓ is outside Σ(calP). Suppose a continuous semisimple representation overline r_λ:G_F⟶GL_n(overlineFF_ℓ) is unramified outside S and the places above ℓ, and has characteristic polynomial overlineP_v at every remaining unramified place. Then there is a finite image semisimple complex representation r with…
Exact mathematical passage
\overline r_\lambda:G_F\longrightarrow\GL_n(\overline\FF_\ell)
\overline\FF_{\ell}Use in source · 1 passage
Search aliases: overlineFF_ℓ overline FF ell
Use in source · body.tex, line 939
[Uniform bound for a residual Galois image] Let overline r:G_F⟶GL_n(overlineFF_ℓ) be continuous and semisimple, with ℓ>n. If its Frobenius traces belong to a set of size at most t on a Chebotarev dense set, then absimoverline r≤ t^n^2.
Exact mathematical passage
\overline r:G_F\longrightarrow\GL_n(\overline\FF_{\ell})
\overline{P^{[p]}}Use in source · 3 passages
Search aliases: overlineP^[p] overline
Use in source · body.tex, line 1124
[Frobenius transform after reduction] Let P have integral coefficients at a place of residue characteristic p. Then overlineP^[p](T) = Frob(overline P(T)), where Frobenius raises every coefficient to its pth power. In particular, the map overline P⟼overlineP^[p] is injective over the residue field.
Exact mathematical passage
\boxed{ \overline{P^{[p]}}(T) = \operatorname{Frob}\bigl(\overline P(T)\bigr), }
\overline{P_v}Use in source · 1 passage
Search aliases: overlineP_v overline
Use in source · body.tex, line 969
from the alphabet before any realization is chosen, with the following property. Let λ be a coefficient place whose residue characteristic ℓ is outside Σ(calP). Suppose a continuous semisimple representation overline r_λ:G_F⟶GL_n(overlineFF_ℓ) is unramified outside S and the places above ℓ, and has characteristic polynomial overlineP_v at every remaining unramified place. Then there is a finite image semisimple complex representation r with polynomial P_v at those places. If overline r_λ is abso…
Exact mathematical passage
\overline{P_v}
\overline{P_{\pi,v}}Use in source · 1 passage
Search aliases: overlineP_π,v overline pi
Use in source · body.tex, line 1258
gincorollary[A normalized torsion realization closes the weak step] Use the cyclic Artin setup and a good separating prime λ. Suppose the reduced eigensystem of a descent occurs in the cohomology range of [source reference], with all source hypotheses met, and an explicit normalization comparison identifies the source Frobenius polynomials with overlineP_π,v. Then the cyclic step is weakly attached after one quotient twist.
Exact mathematical passage
\overline{P_{\pi,v}}
\overline\QQUse in source · 1 passage
Search aliases: overlineQQ overline QQ
Use in source · body.tex, line 1230
[One semisimple l adic realization is enough] Let π be a finite Satake cuspidal representation with algebraic unramified Hecke polynomials. Fix an embedding of their number field into overlineQQ_ℓ. Suppose there is a continuous semisimple representation r:G_F→GL_n(L), for a finite extension L/QQ_ℓ, unramified outside a finite set, whose Frobenius polynomials agree with the embedded Hecke polynomials at almost all places. Then r has finite image and determines a finite image complex representatio…
Exact mathematical passage
\overline\QQ_\ell
\overline\QQ_\ellUse in source · 1 passage
Search aliases: overlineQQ_ℓ overline QQ ell
Use in source · body.tex, line 1230
[One semisimple l adic realization is enough] Let π be a finite Satake cuspidal representation with algebraic unramified Hecke polynomials. Fix an embedding of their number field into overlineQQ_ℓ. Suppose there is a continuous semisimple representation r:G_F→GL_n(L), for a finite extension L/QQ_ℓ, unramified outside a finite set, whose Frobenius polynomials agree with the embedded Hecke polynomials at almost all places. Then r has finite image and determines a finite image complex representatio…
Exact mathematical passage
\overline\QQ_\ell
\overline\alphaUse in source · 1 passage
Search aliases: overlineα overline alpha
Use in source · body.tex, line 1140
Over an algebraic closure of the residue field, overlineP^[p](T) = ∏_i(T-overlineα_i^p). The elementary symmetric functions of the pth powers are the pth powers of the elementary symmetric functions in characteristic p. Finite residue fields are perfect, so Frobenius is injective.
Exact mathematical passage
\overline{P^{[p]}}(T) = \prod_i(T-\overline\alpha_i^p).
\overline\alpha_i^pUse in source · 1 passage
Search aliases: overlineα_i^p overline alpha
Use in source · body.tex, line 1140
Over an algebraic closure of the residue field, overlineP^[p](T) = ∏_i(T-overlineα_i^p). The elementary symmetric functions of the pth powers are the pth powers of the elementary symmetric functions in characteristic p. Finite residue fields are perfect, so Frobenius is injective.
Exact mathematical passage
\overline{P^{[p]}}(T) = \prod_i(T-\overline\alpha_i^p).
\overline\epsilonUse in source · 1 passage
Search aliases: overlineϵ overline epsilon
Use in source · body.tex, line 1036
The representations overline r_π,λ and overlineρ are therefore two extensions of the same absolutely irreducible representation of G_E. Their ratio is a character of G_F/G_E. Since the residue characteristic is different from p, the reduced quotient character has order p, and there is a unique index j_λ with overline r_π,λ ≅ overlineρ⊗overlineϵ^j_λ. At every good unramified place, P_π,v ≡ P_ρ⊗ϵ^j_λ,v pmodλ. Both polynomials belong to the finite alphabet calA. The prime is separating, so the cong…
Exact mathematical passage
\overline r_{\pi,\lambda} \cong \overline\rho\otimes\overline\epsilon^{j_{\lambda}}.
\overline\epsilon^{j_{\lambda}}Use in source · 1 passage
Search aliases: overlineϵ^j_λ overline epsilon lambda
Use in source · body.tex, line 1036
The representations overline r_π,λ and overlineρ are therefore two extensions of the same absolutely irreducible representation of G_E. Their ratio is a character of G_F/G_E. Since the residue characteristic is different from p, the reduced quotient character has order p, and there is a unique index j_λ with overline r_π,λ ≅ overlineρ⊗overlineϵ^j_λ. At every good unramified place, P_π,v ≡ P_ρ⊗ϵ^j_λ,v pmodλ. Both polynomials belong to the finite alphabet calA. The prime is separating, so the cong…
Exact mathematical passage
\overline r_{\pi,\lambda} \cong \overline\rho\otimes\overline\epsilon^{j_{\lambda}}.
\overline\rhoUse in source · 3 passages
Search aliases: overlineρ overline rho
Use in source · body.tex, line 1035
The representations overline r_π,λ and overlineρ are therefore two extensions of the same absolutely irreducible representation of G_E. Their ratio is a character of G_F/G_E. Since the residue characteristic is different from p, the reduced quotient character has order p, and there is a unique index j_λ with overline r_π,λ ≅ overlineρ⊗overlineϵ^j_λ. At every good unramified place, P_π,v ≡ P_ρ⊗ϵ^j_λ,v pmodλ. Both polynomials belong to the finite alphabet calA. The prime is separating, so the cong…
Exact mathematical passage
\overline\rho
\overline{\rho\otimes\epsilon^j}Use in source · 1 passage
Search aliases: overlineρ⊗ϵ^j overline rho otimes epsilon
Use in source · body.tex, line 1175
[All cyclic twists collapse residually] For every j∈ZZ/pZZ, overlineρ⊗ϵ^j ≅ overlineρ at the semisimple character level modulo λ_p. Likewise, all automorphic descents have the same mod λ_p unramified Hecke eigensystem.
Exact mathematical passage
\overline{\rho\otimes\epsilon^j} \cong \overline\rho
\overline\tauUse in source · 3 passages
Search aliases: overlineτ overline tau
Use in source · body.tex, line 1025
Restrict the residual realization to G_E. At every unramified place w of E above a good place v of F, cyclic base change identifies the characteristic polynomial of the restriction with the reduction of the Hecke polynomial of Π_w. Since Π is attached to τ, this is the characteristic polynomial of overlineτ(Frob_w). Chebotarev density and Brauer Nesbitt give ( overline r_π,λ|_G_E )^ss ≅ overlineτ. The right side is absolutely irreducible and has the full dimension n. Hence the restriction itself…
Exact mathematical passage
\overline\tau(\Frob_w)
p^{-(m+s)}Use in source · 1 passage
Search aliases: p^-(m+s)
Use in source · formal_appendix.tex, line 547
Take r=p^-m with m>s. Then Kr=p^s-m<1, and normQ(U-A^d)≤ p^-m · Longleftrightarrow normB(U-A^d)≤ p^-(m+s)notag · Longleftrightarrow Bvec(U-A^d)=0pmod p^m+s. The last congruence is entrywise. It depends only on the reductions of the target, center, and integral operator modulo p^m+s. The theorem therefore decides the existence and uniqueness of a root in A+p^mMat_n( Z_p) by one finite target test, once the exact inverse certificate is supplied.
Exact mathematical passage
\norm{Q(U-A^d)}\le p^{-m} &\Longleftrightarrow \norm{B(U-A^d)}\le p^{-(m+s)}\notag\\ &\Longleftrightarrow B\operatorname{vec}(U-A^d)=0\pmod {p^{m+s}}. \label{eq:appendix-finite-criterion}
p^{-1}Use in source · 1 passage
Search aliases: p^-1
Use in source · formal_appendix.tex, line 447
For E=Mat_n( Q_p) use the entrywise supremum norm and normalize abs p_p=p^-1. Matrix multiplication satisfies abs(XY)_ij_p =|∑_tX_itY_tj|_p ≤max_tabsX_it_pabsY_tj_p ≤norm Xnorm Y. Taking the maximum over the finite set of entries proves submultiplicativity. The same ultrametric calculation removes a matrix dimension factor. Completeness is inherited from the finite product of copies of Q_p. These are the hypotheses required by the actual matrix application, rather than properties borrowed from a…
Exact mathematical passage
\abs p_p=p^{-1}
p^{-m}Use in source · 4 passages
Search aliases: p^-m
Use in source · formal_appendix.tex, line 514
The proof gives a controlled approximation process. Start at h_0=0 and iterate h_j+1=Φ(h_j). Then h_1=QΔ, and normh_j+1-h_j≤ (Kr)^jnormh_1. For t>j, ultrametricity bounds normh_t-h_j by the largest of the intervening increments, which is at most (Kr)^jnormh_1. The limit therefore satisfies the same bound for its distance from h_j. With r=p^-m and K=p^s, every iteration gains at least m-s digits in this estimate. This is a quantitative consequence after the inverse and target test have been certi…
Exact mathematical passage
r=p^{-m}
p^{-s}Use in source · 1 passage
Search aliases: p^-s
Use in source · formal_appendix.tex, line 527
For integral matrices suppose the inverse has the certified form Q=p^-sB, s∈ N, where B is an integral linear operator on the matrix entries. In a fixed ordering of the n^2 entries it is an n^2 by n^2 matrix over Z_p. This is an operator on the matrix space, not ordinary left multiplication by an n by n matrix. Integral coefficients imply normBz≤norm z, so K=p^s is a valid bound.
Exact mathematical passage
Q=p^{-s}B,\qquad s\in\mathbb N,
p^mUse in source · 1 passage
Search aliases: p^m
Use in source · formal_appendix.tex, line 557
· Longleftrightarrow Bvec(U-A^d)=0pmod p^m+s. The last congruence is entrywise. It depends only on the reductions of the target, center, and integral operator modulo p^m+s. The theorem therefore decides the existence and uniqueness of a root in A+p^mMat_n( Z_p) by one finite target test, once the exact inverse certificate is supplied.
Exact mathematical passage
A+p^m\operatorname{Mat}_n(\mathbb Z_p)
p^{m+s}Use in source · 2 passages
Search aliases: p^m+s
Use in source · formal_appendix.tex, line 547
Take r=p^-m with m>s. Then Kr=p^s-m<1, and normQ(U-A^d)≤ p^-m · Longleftrightarrow normB(U-A^d)≤ p^-(m+s)notag · Longleftrightarrow Bvec(U-A^d)=0pmod p^m+s. The last congruence is entrywise. It depends only on the reductions of the target, center, and integral operator modulo p^m+s. The theorem therefore decides the existence and uniqueness of a root in A+p^mMat_n( Z_p) by one finite target test, once the exact inverse certificate is supplied.
Exact mathematical passage
\norm{Q(U-A^d)}\le p^{-m} &\Longleftrightarrow \norm{B(U-A^d)}\le p^{-(m+s)}\notag\\ &\Longleftrightarrow B\operatorname{vec}(U-A^d)=0\pmod {p^{m+s}}. \label{eq:appendix-finite-criterion}
p^rUse in source · 1 passage
Search aliases: p^r
Use in source · formal_appendix.tex, line 673
This matters for simultaneous matrix prescriptions. At level r, a node can contain a shared matrix F_r together with one local conjugator z_i,r for each member of a finite family. The equations are z_i,rF_r^d_iz_i,r^-1=U_ipmod p^r, with the prescribed prefix of F_r and prescribed residual identity for each conjugator. These are a single joint predicate on a finite tuple. The arithmetic equations and prefixes are preserved by reduction. Compactness therefore produces compatible towers for the sha…
Exact mathematical passage
z_{i,r}F_r^{d_i}z_{i,r}^{-1}=U_i\pmod {p^r},
p^sUse in source · 2 passages
Search aliases: p^s
Use in source · formal_appendix.tex, line 514
he proof gives a controlled approximation process. Start at h_0=0 and iterate h_j+1=Φ(h_j). Then h_1=QΔ, and normh_j+1-h_j≤ (Kr)^jnormh_1. For t>j, ultrametricity bounds normh_t-h_j by the largest of the intervening increments, which is at most (Kr)^jnormh_1. The limit therefore satisfies the same bound for its distance from h_j. With r=p^-m and K=p^s, every iteration gains at least m-s digits in this estimate. This is a quantitative consequence after the inverse and target test have been certif…
Exact mathematical passage
K=p^s
p^{s-m}Use in source · 1 passage
Search aliases: p^s-m
Use in source · formal_appendix.tex, line 546
Take r=p^-m with m>s. Then Kr=p^s-m<1, and normQ(U-A^d)≤ p^-m · Longleftrightarrow normB(U-A^d)≤ p^-(m+s)notag
Exact mathematical passage
Kr=p^{s-m}<1
p_pUse in source · 1 passage
Search aliases: p_p
Use in source · formal_appendix.tex, line 447
For E=Mat_n( Q_p) use the entrywise supremum norm and normalize abs p_p=p^-1. Matrix multiplication satisfies abs(XY)_ij_p =|∑_tX_itY_tj|_p ≤max_tabsX_it_pabsY_tj_p ≤norm Xnorm Y. Taking the maximum over the finite set of entries proves submultiplicativity. The same ultrametric calculation removes a matrix dimension factor. Completeness is inherited from the finite product of copies of Q_p. These are the hypotheses required by the actual matrix application, rather than properties borrowed from a…
Exact mathematical passage
\abs p_p=p^{-1}
P_rUse in source · 2 passages
Search aliases: P_r
Use in source · formal_appendix.tex, line 653
Let X_r be finite sets with genuine reduction maps π_r,t:X_t→ X_r for r≤ t. Assume identity and composition laws for these maps. Let P_r⊆ X_r be solution sets, and assume reduction preserves solutions. Then (∀ r, P_r≠∅) Longleftrightarrow (∃ (x_r)_r, x_r∈ P_r and π_r,t(x_t)=x_r for r≤ t). The nontrivial implication is finite branch compactness. For each depth, a solution at that depth gives a finite path of solutions at all earlier depths. There are arbitrarily long paths. The resulting tree has…
Exact mathematical passage
P_r\subseteq X_r
P_vUse in source · 4 passages
Search aliases: P_v
Use in source · body.tex, line 963
[One large residual realization gives an Artin representation] Let K be a number field, let S be a finite set of places of F, and let calP⊆OO_K[T] be a nonempty finite set of monic degree n polynomials. Assume all their roots are roots of unity. For v∉ S, choose P_v∈calP.
Exact mathematical passage
P_v\in\calP
P_{\pi,v}^{[p]}Use in source · 1 passage
Search aliases: P_π,v^[p] pi
Use in source · body.tex, line 1165
Assume that v is unramified and nonsplit. Since the extension has prime degree, its residue degree is p. Unramified base change raises the Satake conjugacy class of π_v to its pth power. Restriction of ρ_v to W_E_w does the same to its Frobenius class. Since Π_w is attached to ρ|_W_E_w, P_π,v^[p](T) = P_ρ,v^[p](T). Reduce modulo λ_p and use [source reference]. Injectivity of residue Frobenius gives the desired congruence. The finitely many ramified and excluded places do not affect residual atta…
Exact mathematical passage
P_{\pi,v}^{[p]}(T) = P_{\rho,v}^{[p]}(T).
P_{\rho,v}^{[p]}Use in source · 1 passage
Search aliases: P_ρ,v^[p] rho
Use in source · body.tex, line 1165
Assume that v is unramified and nonsplit. Since the extension has prime degree, its residue degree is p. Unramified base change raises the Satake conjugacy class of π_v to its pth power. Restriction of ρ_v to W_E_w does the same to its Frobenius class. Since Π_w is attached to ρ|_W_E_w, P_π,v^[p](T) = P_ρ,v^[p](T). Reduce modulo λ_p and use [source reference]. Injectivity of residue Frobenius gives the desired congruence. The finitely many ramified and excluded places do not affect residual atta…
Exact mathematical passage
P_{\pi,v}^{[p]}(T) = P_{\rho,v}^{[p]}(T).
P_{\rho\otimes\epsilon^{j_{\lambda}},v}Use in source · 2 passages
Search aliases: P_ρ⊗ϵ^j_λ,v rho otimes epsilon lambda
Use in source · body.tex, line 1010
cyclic Artin setup of [source reference]. For this unramified theorem it is enough that the lower cuspidal representation Π is weakly attached to τ, with π a cuspidal cyclic descent. Let λ be a good separating prime. Suppose π has a residual realization overline r_π,λ. Then there is a unique index j_λ∈ZZ/pZZ such that P_π,v(T) = P_ρ⊗ϵ^j_λ,v(T) for every finite place v outside a finite set. Consequently, π⊗ϵ^-j_λ is weakly attached to ρ.
Exact mathematical passage
\boxed{ P_{\pi,v}(T) = P_{\rho\otimes\epsilon^{j_{\lambda}},v}(T) }
P_{\tau,V_0}Use in source · 1 passage
Search aliases: P_τ,V_0 tau
Use in source · body.tex, line 585
[A stable cyclic type] Let G=C_3× C_2, H=C_3. Fix a nontrivial character τ of H. It is stable under G. After choosing the extension with trivial C_2 coordinate, the two one dimensional extensions have spectra P_τ,V_0(z)=1, P_τ,V_1(z)=z. The alignment set is the singleton containing the unique quotient twist carrying one extension to the other.
Exact mathematical passage
P_{\tau,V_0}(z)=1, \qquad P_{\tau,V_1}(z)=z.
P_{\tau,V_1}Use in source · 1 passage
Search aliases: P_τ,V_1 tau
Use in source · body.tex, line 585
[A stable cyclic type] Let G=C_3× C_2, H=C_3. Fix a nontrivial character τ of H. It is stable under G. After choosing the extension with trivial C_2 coordinate, the two one dimensional extensions have spectra P_τ,V_0(z)=1, P_τ,V_1(z)=z. The alignment set is the singleton containing the unique quotient twist carrying one extension to the other.
Exact mathematical passage
P_{\tau,V_0}(z)=1, \qquad P_{\tau,V_1}(z)=z.
P_{\tau,V\otimes\xi^j}Use in source · 1 passage
Search aliases: P_τ,V⊗ξ^j tau otimes xi
Use in source · body.tex, line 408
[Effect of quotient twisting] For every j∈ZZ/pZZ, P_τ,V⊗ξ^j(z) = z^jP_τ,V(z). Moving orbit blocks are unchanged up to isomorphism by quotient twisting.
Exact mathematical passage
\boxed{ P_{\tau,V\otimes\xi^j}(z) = z^jP_{\tau,V}(z). }
\pmodUse in source · 6 passages
Search aliases: pmod
Use in source · body.tex, line 1042
two extensions of the same absolutely irreducible representation of G_E. Their ratio is a character of G_F/G_E. Since the residue characteristic is different from p, the reduced quotient character has order p, and there is a unique index j_λ with overline r_π,λ ≅ overlineρ⊗overlineϵ^j_λ. At every good unramified place, P_π,v ≡ P_ρ⊗ϵ^j_λ,v pmodλ. Both polynomials belong to the finite alphabet calA. The prime is separating, so the congruence is equality in characteristic zero. Uniqueness follows f…
Exact mathematical passage
P_{\pi,v} \equiv P_{\rho\otimes\epsilon^{j_{\lambda}},v} \pmod\lambda.
QUse in source · 17 passages
Search aliases: Q
Use in source · body.tex, line 861
For every ordered pair of distinct polynomials P,Q∈calA, choose one nonzero coefficient difference. Only finitely many finite places divide one of these finitely many nonzero algebraic integers. Outside their union, no two polynomials have the same reduction.
Exact mathematical passage
P,Q\in\calA
\mathbb QUse in source · 7 passages
Search aliases: Q mathbb
Use in source · formal_appendix.tex, line 446
For E=Mat_n( Q_p) use the entrywise supremum norm and normalize abs p_p=p^-1. Matrix multiplication satisfies abs(XY)_ij_p =|∑_tX_itY_tj|_p ≤max_tabsX_it_pabsY_tj_p ≤norm Xnorm Y. Taking the maximum over the finite set of entries proves submultiplicativity. The same ultrametric calculation removes a matrix dimension factor. Completeness is inherited from the finite product of copies of Q_p. These are the hypotheses required by the actual matrix application, rather than properties borrowed from a…
Exact mathematical passage
E=\operatorname{Mat}_n(\mathbb Q_p)
qUse in source · 4 passages
Search aliases: q
Use in source · bibliography.tex, line 12
bibitemKimYamauchi2016 H. H. Kim and T. Yamauchi, A uniform structure on subgroups of GL_n( F_q) and its application to a conditional construction of Artin representations of GL_n, Journal of the Ramanujan Mathematical Society 32(1) (2017), 75--99. Checked preprint https://arxiv.org/abs/1403.5535arXiv:1403.5535v3, 7 April 2016. Theorem 1.1 is conditional on its five inputs. The inherited citation key records the preprint year.
Exact mathematical passage
GL_n(\mathbb F_q)
\mathbb Q_2Use in source · 1 passage
Search aliases: Q_2 mathbb
Use in source · formal_appendix.tex, line 601
The residual derivative is singular. In characteristic two, D_A,2(I_2)=2A=0, while I_2≠0. Nevertheless J is invertible over Q_2, with determinant 16. Residual singularity and singularity over the fraction field are different tests.
Exact mathematical passage
\mathbb Q_2
\mathbb Q_pUse in source · 6 passages
Search aliases: Q_p mathbb
Use in source · formal_appendix.tex, line 446
For E=Mat_n( Q_p) use the entrywise supremum norm and normalize abs p_p=p^-1. Matrix multiplication satisfies abs(XY)_ij_p =|∑_tX_itY_tj|_p ≤max_tabsX_it_pabsY_tj_p ≤norm Xnorm Y. Taking the maximum over the finite set of entries proves submultiplicativity. The same ultrametric calculation removes a matrix dimension factor. Completeness is inherited from the finite product of copies of Q_p. These are the hypotheses required by the actual matrix application, rather than properties borrowed from a…
Exact mathematical passage
E=\operatorname{Mat}_n(\mathbb Q_p)
q_v^{i(n+1)/2}Use in source · 1 passage
Search aliases: q_v^i(n+1)/2
Use in source · body.tex, line 1250
t[A torsion cohomology realization range] Scholze's Corollary 5.4.3 supplies a unique continuous semisimple residual representation for a Hecke eigensystem actually occurring in the stated cohomology of a locally symmetric space, under the field, level, coefficient, and theorem dependencies of that source [source reference]. The specified polynomial has coefficients q_v^i(n+1)/2T_i,v, rather than an unspecified normalization of the eigenvalues. A fully stated unconditional subrange uses a CM fie…
Exact mathematical passage
q_v^{i(n+1)/2}T_{i,v}
R'Use in source · 2 passages
Search aliases: R'
Use in source · formal_appendix.tex, line 96
Let q:R'→ R be a ring map with kernel I, and assume I^2=0. Let ρ:G→ R^× be a group representation. Choose unit-valued lifts r_g∈ (R')^× with q(r_g)=ρ(g) and r_1=1. Such a choice is an input to this calculation. It is not already a multiplicative lift. Define its normalized defect by c(g,h)=(r_gr_h-r_gh)r_gh^-1∈ I. Equivalently, r_gr_h=(1+c(g,h))r_gh. The coefficients are elements of the actual ring kernel, with its additive group, rather than an unrelated vector space of the same dimension.
Exact mathematical passage
q:R'\to R
r'_gUse in source · 2 passages
Search aliases: r'_g
Use in source · formal_appendix.tex, line 133
Change the chosen lifts to r'_g=(1+b(g))r_g, where b:G→ I and b(1)=0. Expanding without commuting the factors gives r'_gr'_h =(1+b(g)+g· b(h)+c(g,h))r_gh. To compare with r'_gh, multiply by 1-b(gh). The new defect is c'(g,h)=c(g,h)+g· b(h)-b(gh)+b(g). Thus changing the section adds a two-coboundary. The class [c]∈ H^2(G,I) does not depend on the section.
Exact mathematical passage
r'_g=(1+b(g))r_g
r'_{gh}Use in source · 1 passage
Search aliases: r'_gh
Use in source · formal_appendix.tex, line 139
Change the chosen lifts to r'_g=(1+b(g))r_g, where b:G→ I and b(1)=0. Expanding without commuting the factors gives r'_gr'_h =(1+b(g)+g· b(h)+c(g,h))r_gh. To compare with r'_gh, multiply by 1-b(gh). The new defect is c'(g,h)=c(g,h)+g· b(h)-b(gh)+b(g). Thus changing the section adds a two-coboundary. The class [c]∈ H^2(G,I) does not depend on the section.
Exact mathematical passage
r'_{gh}
r'_hUse in source · 1 passage
Search aliases: r'_h
Use in source · formal_appendix.tex, line 135
Change the chosen lifts to r'_g=(1+b(g))r_g, where b:G→ I and b(1)=0. Expanding without commuting the factors gives r'_gr'_h =(1+b(g)+g· b(h)+c(g,h))r_gh. To compare with r'_gh, multiply by 1-b(gh). The new defect is c'(g,h)=c(g,h)+g· b(h)-b(gh)+b(g). Thus changing the section adds a two-coboundary. The class [c]∈ H^2(G,I) does not depend on the section.
Exact mathematical passage
r'_gr'_h =(1+b(g)+g\cdot b(h)+c(g,h))r_{gh}.
R^{-1}Use in source · 1 passage
Search aliases: R^-1
Use in source · body.tex, line 374
ves an invertible intertwiner A with Aτ(h)A^-1=τ(ghg^-1) (h∈ H). Iterate this identity p times. The operator τ(g^p)^-1A^p commutes with τ(H), so Schur's lemma makes it a nonzero scalar cI. Also A commutes with τ(g^p), by the same intertwining identity at h=g^p. Choose b∈CC^× with b^p=c^-1 and put R=bA. Then Rτ(h)R^-1=τ(ghg^-1), R^p=τ(g^p). These are exactly the conjugation and closing relations for the coset presentation of G. Assigning g↦ R and h↦τ(h) therefore defines a representation of G ext…
Exact mathematical passage
R\tau(h)R^{-1}=\tau(ghg^{-1}),\qquad R^p=\tau(g^p).
r^{-1}Use in source · 2 passages
Search aliases: r^-1
Use in source · body.tex, line 1375
The Lie group GL_n(CC) has a neighborhood U of the identity that contains no nontrivial subgroup. Continuity gives a neighborhood r^-1(U) of the identity in G. A profinite group has a basis of open normal subgroups, so one such subgroup N lies inside r^-1(U). Its image is a subgroup contained in U, hence is trivial. Therefore r factors through the finite quotient G/N. This proof uses a group neighborhood theorem. It does not use the false general topological assertion that an arbitrary continuou…
Exact mathematical passage
r^{-1}(U)
R^pUse in source · 1 passage
Search aliases: R^p
Use in source · body.tex, line 374
ves an invertible intertwiner A with Aτ(h)A^-1=τ(ghg^-1) (h∈ H). Iterate this identity p times. The operator τ(g^p)^-1A^p commutes with τ(H), so Schur's lemma makes it a nonzero scalar cI. Also A commutes with τ(g^p), by the same intertwining identity at h=g^p. Choose b∈CC^× with b^p=c^-1 and put R=bA. Then Rτ(h)R^-1=τ(ghg^-1), R^p=τ(g^p). These are exactly the conjugation and closing relations for the coset presentation of G. Assigning g↦ R and h↦τ(h) therefore defines a representation of G ext…
Exact mathematical passage
R\tau(h)R^{-1}=\tau(ghg^{-1}),\qquad R^p=\tau(g^p).
R^\timesUse in source · 1 passage
Search aliases: R^× times
Use in source · formal_appendix.tex, line 97
Let q:R'→ R be a ring map with kernel I, and assume I^2=0. Let ρ:G→ R^× be a group representation. Choose unit-valued lifts r_g∈ (R')^× with q(r_g)=ρ(g) and r_1=1. Such a choice is an input to this calculation. It is not already a multiplicative lift. Define its normalized defect by c(g,h)=(r_gr_h-r_gh)r_gh^-1∈ I. Equivalently, r_gr_h=(1+c(g,h))r_gh. The coefficients are elements of the actual ring kernel, with its additive group, rather than an unrelated vector space of the same dimension.
Exact mathematical passage
\rho:G\to R^\times
r_1Use in source · 3 passages
Search aliases: r_1
Use in source · formal_appendix.tex, line 98
Let q:R'→ R be a ring map with kernel I, and assume I^2=0. Let ρ:G→ R^× be a group representation. Choose unit-valued lifts r_g∈ (R')^× with q(r_g)=ρ(g) and r_1=1. Such a choice is an input to this calculation. It is not already a multiplicative lift. Define its normalized defect by c(g,h)=(r_gr_h-r_gh)r_gh^-1∈ I. Equivalently, r_gr_h=(1+c(g,h))r_gh. The coefficients are elements of the actual ring kernel, with its additive group, rather than an unrelated vector space of the same dimension.
Exact mathematical passage
r_1=1
R_{A,3}Use in source · 1 passage
Search aliases: R_A,3
Use in source · formal_appendix.tex, line 401
For degree three, the linear term is D_A,3(h)=A^2h+AhA+hA^2, and the nonlinear remainder is R_A,3(h)=Ah^2+hAh+h^2A+h^3. The middle word hAh cannot be absorbed into Ah^2 in a matrix ring. The remainder estimate keeps all four words and bounds each by its norm, so it respects the ordered expansion. This also explains why the estimate is not an assertion of commutative binomial calculus in disguise. The scalar bound remains uniform because the norm of a finite ultrametric sum is controlled by its l…
Exact mathematical passage
R_{A,3}(h)=Ah^2+hAh+h^2A+h^3.
R_{A,d}Use in source · 11 passages
Search aliases: R_A,d
Use in source · formal_appendix.tex, line 376
Let E be a complete normed ring with norm1=1, a submultiplicative norm, and the ultrametric inequality normx+y≤max(normx,normy). Assume norm A≤1. For a nonnegative integer d, define D_A,d(h) · =∑_i=0^d-1A^ihA^d-1-i, R_A,d(h) · =(A+h)^d-A^d-D_A,d(h). For d=0 the sum is empty and both maps are zero. The order of the factors in [source reference] matters. Replacing this operator by dA^d-1h requires A and h to commute. The replacement is invalid for general matrices.
Exact mathematical passage
D_{A,d}(h)&=\sum_{i=0}^{d-1}A^ihA^{d-1-i}, \label{eq:appendix-ordered-derivative}\\ R_{A,d}(h)&=(A+h)^d-A^d-D_{A,d}(h). \label{eq:appendix-ordered-remainder}
R_{A,d+1}Use in source · 1 passage
Search aliases: R_A,d+1
Use in source · formal_appendix.tex, line 388
The recursive identities are D_A,d+1(h) · =D_A,d(h)A+A^dh, R_A,d+1(h) · =R_A,d(h)(A+h)+D_A,d(h)h. They follow by multiplying (A+h)^d on the right by A+h and collecting the terms with zero, one, and at least two occurrences of h. This derivation does not rearrange any noncommuting factors.
Exact mathematical passage
D_{A,d+1}(h)&=D_{A,d}(h)A+A^dh,\\ R_{A,d+1}(h)&=R_{A,d}(h)(A+h)+D_{A,d}(h)h.
R_dUse in source · 3 passages
Search aliases: R_d
Use in source · formal_appendix.tex, line 429
Induct on d for the second estimate. The case d=0 is immediate. Write D_d=D_A,d and R_d=R_A,d. Subtracting the remainder recursions gives R_d+1(h)-R_d+1(k) = · (R_d(h)-R_d(k))(A+h)
Exact mathematical passage
R_d=R_{A,d}
R_{d+1}Use in source · 1 passage
Search aliases: R_d+1
Use in source · formal_appendix.tex, line 430
Induct on d for the second estimate. The case d=0 is immediate. Write D_d=D_A,d and R_d=R_A,d. Subtracting the remainder recursions gives R_d+1(h)-R_d+1(k) = · (R_d(h)-R_d(k))(A+h) · +R_d(k)(h-k) · +(D_d(h)-D_d(k))h+D_d(k)(h-k). The first term has norm at most rnormh-k by induction and normA+h≤1. Taking h=k and the other argument zero in the inductive estimate gives normR_d(k)≤ rnorm k≤ r, so the second term has the same bound. Additivity gives normD_d(h)-D_d(k)≤normh-k, which bounds the third t…
Exact mathematical passage
R_{d+1}(h)-R_{d+1}(k) ={}&(R_d(h)-R_d(k))(A+h)\\ &+R_d(k)(h-k)\\ &+(D_d(h)-D_d(k))h+D_d(k)(h-k).
r_gUse in source · 12 passages
Search aliases: r_g
Use in source · formal_appendix.tex, line 98
Let q:R'→ R be a ring map with kernel I, and assume I^2=0. Let ρ:G→ R^× be a group representation. Choose unit-valued lifts r_g∈ (R')^× with q(r_g)=ρ(g) and r_1=1. Such a choice is an input to this calculation. It is not already a multiplicative lift. Define its normalized defect by c(g,h)=(r_gr_h-r_gh)r_gh^-1∈ I. Equivalently, r_gr_h=(1+c(g,h))r_gh. The coefficients are elements of the actual ring kernel, with its additive group, rather than an unrelated vector space of the same dimension.
Exact mathematical passage
r_g\in (R')^\times
r_g^{-1}Use in source · 1 passage
Search aliases: r_g^-1
Use in source · formal_appendix.tex, line 109
The action of G on I is g· x=r_gxr_g^-1. It is independent of the chosen unit lift. Indeed, any other unit lift has the form (1+b)r_g with b∈ I. Since (1+b)^-1=1-b and products of two elements of I vanish, conjugating x∈ I by 1+b leaves x unchanged. The same observation shows that the defect in the multiplication of the r_g does not prevent these conjugations from forming a genuine group action on I.
Exact mathematical passage
g\cdot x=r_gxr_g^{-1}
r_{gh}Use in source · 3 passages
Search aliases: r_gh
Use in source · formal_appendix.tex, line 101
Let q:R'→ R be a ring map with kernel I, and assume I^2=0. Let ρ:G→ R^× be a group representation. Choose unit-valued lifts r_g∈ (R')^× with q(r_g)=ρ(g) and r_1=1. Such a choice is an input to this calculation. It is not already a multiplicative lift. Define its normalized defect by c(g,h)=(r_gr_h-r_gh)r_gh^-1∈ I. Equivalently, r_gr_h=(1+c(g,h))r_gh. The coefficients are elements of the actual ring kernel, with its additive group, rather than an unrelated vector space of the same dimension.
Exact mathematical passage
c(g,h)=(r_gr_h-r_{gh})r_{gh}^{-1}\in I. \label{eq:appendix-normalized-defect}
r_{gh}^{-1}Use in source · 1 passage
Search aliases: r_gh^-1
Use in source · formal_appendix.tex, line 101
Let q:R'→ R be a ring map with kernel I, and assume I^2=0. Let ρ:G→ R^× be a group representation. Choose unit-valued lifts r_g∈ (R')^× with q(r_g)=ρ(g) and r_1=1. Such a choice is an input to this calculation. It is not already a multiplicative lift. Define its normalized defect by c(g,h)=(r_gr_h-r_gh)r_gh^-1∈ I. Equivalently, r_gr_h=(1+c(g,h))r_gh. The coefficients are elements of the actual ring kernel, with its additive group, rather than an unrelated vector space of the same dimension.
Exact mathematical passage
c(g,h)=(r_gr_h-r_{gh})r_{gh}^{-1}\in I. \label{eq:appendix-normalized-defect}
r_{gh\ell}Use in source · 3 passages
Search aliases: r_ghℓ ell
Use in source · formal_appendix.tex, line 117
Compute (r_gr_h)r_ℓ and r_g(r_hr_ℓ). After expressing both in terms of r_ghℓ, the first is (1+c(g,h)+c(gh,ℓ))r_ghℓ, while the second is (1+g· c(h,ℓ)+c(g,hℓ))r_ghℓ. All omitted cross products belong to I^2. Associativity therefore yields g· c(h,ℓ)-c(gh,ℓ)+c(g,hℓ)-c(g,h)=0. This is the usual inhomogeneous two-cocycle identity. Normalization at the identity follows from r_1=1.
Exact mathematical passage
r_{gh\ell}
r_hUse in source · 4 passages
Search aliases: r_h
Use in source · formal_appendix.tex, line 101
Let q:R'→ R be a ring map with kernel I, and assume I^2=0. Let ρ:G→ R^× be a group representation. Choose unit-valued lifts r_g∈ (R')^× with q(r_g)=ρ(g) and r_1=1. Such a choice is an input to this calculation. It is not already a multiplicative lift. Define its normalized defect by c(g,h)=(r_gr_h-r_gh)r_gh^-1∈ I. Equivalently, r_gr_h=(1+c(g,h))r_gh. The coefficients are elements of the actual ring kernel, with its additive group, rather than an unrelated vector space of the same dimension.
Exact mathematical passage
c(g,h)=(r_gr_h-r_{gh})r_{gh}^{-1}\in I. \label{eq:appendix-normalized-defect}
r_\gammaUse in source · 1 passage
Search aliases: r_γ gamma
Use in source · formal_appendix.tex, line 184
The defect can also be seen without an existence argument. Choose the section r_1=1, r_γ=4, and r_γ^2=16 modulo 27. These reduce to the required units modulo 9. The products of consecutive powers have no defect until the exponent passes through three. At that point 4^3=10=1+9 modulo 27. Consequently the normalized defect is 9 when the exponents wrap around and zero otherwise. Under the identification I≃ F_3, this is the ordinary carry cocycle.
Exact mathematical passage
r_\gamma=4
r_{\gamma^2}Use in source · 1 passage
Search aliases: r_γ^2 gamma
Use in source · formal_appendix.tex, line 184
The defect can also be seen without an existence argument. Choose the section r_1=1, r_γ=4, and r_γ^2=16 modulo 27. These reduce to the required units modulo 9. The products of consecutive powers have no defect until the exponent passes through three. At that point 4^3=10=1+9 modulo 27. Consequently the normalized defect is 9 when the exponents wrap around and zero otherwise. Under the identification I≃ F_3, this is the ordinary carry cocycle.
Exact mathematical passage
r_{\gamma^2}=16
r_\ellUse in source · 2 passages
Search aliases: r_ℓ ell
Use in source · formal_appendix.tex, line 116
Compute (r_gr_h)r_ℓ and r_g(r_hr_ℓ). After expressing both in terms of r_ghℓ, the first is (1+c(g,h)+c(gh,ℓ))r_ghℓ, while the second is (1+g· c(h,ℓ)+c(g,hℓ))r_ghℓ. All omitted cross products belong to I^2. Associativity therefore yields g· c(h,ℓ)-c(gh,ℓ)+c(g,hℓ)-c(g,h)=0. This is the usual inhomogeneous two-cocycle identity. Normalization at the identity follows from r_1=1.
Exact mathematical passage
(r_gr_h)r_\ell
\ResUse in source · 7 passages
Search aliases: Res
Use in source · body.tex, line 317
[The total induced object is torsor blind] Let Htriangleleft G with cyclic quotient character group X, and let V be an extension of an H representation. Then ⊕_χ∈ X V⊗χ ≅ Ind_H^GRes_HV. The left side is unchanged when V is replaced by another member of its extension torsor. Likewise, for cyclic base change, mathopboxplus_χ∈ X π⊗χ ≅ AI_E/F(BC_E/Fπ) inside the established automorphic induction formalism. Therefore comparison of the total induced objects cannot determine the cyclic attachment offse…
Exact mathematical passage
\boxed{ \bigoplus_{\chi\in X} V\otimes\chi \cong \Ind_H^G\Res_HV. }
\operatorname{res}Use in source · 1 passage
Search aliases: res operatorname
Use in source · formal_appendix.tex, line 230
To see the criterion, let a_i be the local mismatch after b_0. A match up to the allowed local conjugations exists exactly when a_i=res_i(z)+δ x_i for a global continuous one-cocycle z and kernel elements x_i. This is exactly the assertion that the image of (a_i)_i in [source reference] is zero. If b_1 is another absolute primitive, then b_1-b_0 is a global cocycle. Its restriction disappears in the cokernel, so the remaining class does not depend on which primitive was used.
Exact mathematical passage
a_i=\operatorname{res}_i(z)+\delta x_i
\Res_HUse in source · 6 passages
Search aliases: Res_H Res
Use in source · body.tex, line 317
[The total induced object is torsor blind] Let Htriangleleft G with cyclic quotient character group X, and let V be an extension of an H representation. Then ⊕_χ∈ X V⊗χ ≅ Ind_H^GRes_HV. The left side is unchanged when V is replaced by another member of its extension torsor. Likewise, for cyclic base change, mathopboxplus_χ∈ X π⊗χ ≅ AI_E/F(BC_E/Fπ) inside the established automorphic induction formalism. Therefore comparison of the total induced objects cannot determine the cyclic attachment offse…
Exact mathematical passage
\boxed{ \bigoplus_{\chi\in X} V\otimes\chi \cong \Ind_H^G\Res_HV. }
\operatorname{res}_iUse in source · 1 passage
Search aliases: res_i operatorname
Use in source · formal_appendix.tex, line 230
To see the criterion, let a_i be the local mismatch after b_0. A match up to the allowed local conjugations exists exactly when a_i=res_i(z)+δ x_i for a global continuous one-cocycle z and kernel elements x_i. This is exactly the assertion that the image of (a_i)_i in [source reference] is zero. If b_1 is another absolute primitive, then b_1-b_0 is a global cocycle. Its restriction disappears in the cokernel, so the remaining class does not depend on which primitive was used.
Exact mathematical passage
a_i=\operatorname{res}_i(z)+\delta x_i
\Res_{W_{E_w}}Use in source · 1 passage
Search aliases: Res_W_E_w Res
Use in source · body.tex, line 623
For each place v of F, put R_v=ρ|_W_F_v, A_v=rec_F_v(π_0,v). Cyclic base change compatibility gives Res_W_E_wR_v ≅ Res_W_E_wA_v.
Exact mathematical passage
\Res_{W_{E_w}}R_v \cong \Res_{W_{E_w}}A_v.
\rtimesUse in source · 1 passage
Search aliases: rtimes
Use in source · body.tex, line 45
The solvable two dimensional sectors treated by Langlands and Tunnell, further examples such as the solvable GO(4) range, and other established cases show that the conjecture is far from an empty expectation [source reference]. A recent preprint of Wang states the generalized octahedral GL_3 case with projective image C_3^2rtimesSL(2,3), together with a primitive solvable six dimensional consequence [source reference]. Its stated results have been checked against the submitted source. Its full p…
Exact mathematical passage
C_3^2\rtimes\mathrm{SL}(2,3)
\rvertUse in source · 1 passage
Search aliases: rvert
Use in source · body.tex, line 1254
from a simple unitary Kottwitz datum and a cuspidal representation whose infinity component is an X holomorphic limit of discrete series [source reference]. At a coefficient prime satisfying the good reduction and reflex field conditions, it supplies a semisimple l adic representation over the CM field. Its local compatibility is stated on the density one split prime set with the twist lvert·rvert^(1-n)/2 in the source normalization. This is a concrete realization range. It is not a theorem for …
Exact mathematical passage
\lvert\cdot\rvert^{(1-n)/2}
\rvert^{(1-n)/2}Use in source · 1 passage
Search aliases: rvert^(1-n)/2 rvert
Use in source · body.tex, line 1254
from a simple unitary Kottwitz datum and a cuspidal representation whose infinity component is an X holomorphic limit of discrete series [source reference]. At a coefficient prime satisfying the good reduction and reflex field conditions, it supplies a semisimple l adic representation over the CM field. Its local compatibility is stated on the density one split prime set with the twist lvert·rvert^(1-n)/2 in the source normalization. This is a concrete realization range. It is not a theorem for …
Exact mathematical passage
\lvert\cdot\rvert^{(1-n)/2}
S_3Use in source · 2 passages
Search aliases: S_3
Use in source · body.tex, line 598
[A moving type and self twist] Let G=S_3, let H=A_3, and let V be the standard two dimensional representation. Its restriction is Res_HV=τ⊕τ^-1 for the two nontrivial characters of C_3. These characters form one moving orbit under G/H≅ C_2. Hence there is no stable extension spectrum, and V⊗sgn≅ V.
Exact mathematical passage
G=S_3
\operatorname{sgn}Use in source · 1 passage
Search aliases: sgn operatorname
Use in source · body.tex, line 603
[A moving type and self twist] Let G=S_3, let H=A_3, and let V be the standard two dimensional representation. Its restriction is Res_HV=τ⊕τ^-1 for the two nontrivial characters of C_3. These characters form one moving orbit under G/H≅ C_2. Hence there is no stable extension spectrum, and V⊗sgn≅ V.
Exact mathematical passage
V\otimes\operatorname{sgn}\cong V.
\mathrm{SL}Use in source · 1 passage
Search aliases: SL mathrm
Use in source · body.tex, line 45
The solvable two dimensional sectors treated by Langlands and Tunnell, further examples such as the solvable GO(4) range, and other established cases show that the conjecture is far from an empty expectation [source reference]. A recent preprint of Wang states the generalized octahedral GL_3 case with projective image C_3^2rtimesSL(2,3), together with a primitive solvable six dimensional consequence [source reference]. Its stated results have been checked against the submitted source. Its full p…
Exact mathematical passage
C_3^2\rtimes\mathrm{SL}(2,3)
\operatorname{span}Use in source · 1 passage
Search aliases: span operatorname
Use in source · body.tex, line 900
[Finite trace image bound] Let k be algebraically closed of characteristic zero or characteristic ℓ>n. Let Γ⊆GL_n(k) be a subgroup, and assume its defining module k^n is semisimple. Suppose its trace set is finite, with t=#{Tr(γ):γ∈Γ}. Then Γ is finite and absΓ≤ t^dim_k A≤ t^n^2, A=span_kΓ. In particular this bounds a finite semisimple residual matrix group in characteristic greater than its rank.
Exact mathematical passage
\abs\Gamma\le t^{\dim_k A}\le t^{n^2}, \qquad A=\operatorname{span}_k\Gamma.
\operatorname{span}_kUse in source · 1 passage
Search aliases: span_k operatorname
Use in source · body.tex, line 900
[Finite trace image bound] Let k be algebraically closed of characteristic zero or characteristic ℓ>n. Let Γ⊆GL_n(k) be a subgroup, and assume its defining module k^n is semisimple. Suppose its trace set is finite, with t=#{Tr(γ):γ∈Γ}. Then Γ is finite and absΓ≤ t^dim_k A≤ t^n^2, A=span_kΓ. In particular this bounds a finite semisimple residual matrix group in characteristic greater than its rank.
Exact mathematical passage
\abs\Gamma\le t^{\dim_k A}\le t^{n^2}, \qquad A=\operatorname{span}_k\Gamma.
\mathrm{ss}Use in source · 2 passages
Search aliases: ss mathrm
Use in source · body.tex, line 1026
e residual realization to G_E. At every unramified place w of E above a good place v of F, cyclic base change identifies the characteristic polynomial of the restriction with the reduction of the Hecke polynomial of Π_w. Since Π is attached to τ, this is the characteristic polynomial of overlineτ(Frob_w). Chebotarev density and Brauer Nesbitt give ( overline r_π,λ|_G_E )^ss ≅ overlineτ. The right side is absolutely irreducible and has the full dimension n. Hence the restriction itself has one co…
Exact mathematical passage
\left( \overline r_{\pi,\lambda}|_{G_E} \right)^{\mathrm{ss}} \cong \overline\tau.
tUse in source · 27 passages
Search aliases: t
Use in source · body.tex, line 896
[Finite trace image bound] Let k be algebraically closed of characteristic zero or characteristic ℓ>n. Let Γ⊆GL_n(k) be a subgroup, and assume its defining module k^n is semisimple. Suppose its trace set is finite, with t=#{Tr(γ):γ∈Γ}. Then Γ is finite and absΓ≤ t^dim_k A≤ t^n^2, A=span_kΓ. In particular this bounds a finite semisimple residual matrix group in characteristic greater than its rank.
Exact mathematical passage
t=\#\{\Tr(\gamma):\gamma\in\Gamma\}.
T^2Use in source · 2 passages
Search aliases: T^2
Use in source · worked_appendix.tex, line 34
Let ω be a primitive cube root of unity and put A=diag(1,1,ω). Expanding directly gives det(TI-A) =T^3-(2+ω)T^2+(1+2ω)T-ω. The three elementary symmetric coefficients are e_1=2+ω, e_2=1+2ω, e_3=ω. For scalar twist ω^jA, they become ω^j(2+ω), ω^2j(1+2ω), ω^3jω=ω. The determinant is constant for j=0,1,2, but the first coefficient distinguishes the three labels because 2+ω≠0. Thus rank divisibility causes blindness of the determinant, rather than blindness of every coefficient.
Exact mathematical passage
\det(TI-A) =T^3-(2+\omega)T^2+(1+2\omega)T-\omega.
T^3Use in source · 2 passages
Search aliases: T^3
Use in source · worked_appendix.tex, line 34
Let ω be a primitive cube root of unity and put A=diag(1,1,ω). Expanding directly gives det(TI-A) =T^3-(2+ω)T^2+(1+2ω)T-ω. The three elementary symmetric coefficients are e_1=2+ω, e_2=1+2ω, e_3=ω. For scalar twist ω^jA, they become ω^j(2+ω), ω^2j(1+2ω), ω^3jω=ω. The determinant is constant for j=0,1,2, but the first coefficient distinguishes the three labels because 2+ω≠0. Thus rank divisibility causes blindness of the determinant, rather than blindness of every coefficient.
Exact mathematical passage
\det(TI-A) =T^3-(2+\omega)T^2+(1+2\omega)T-\omega.
t^{\dim_k A}Use in source · 1 passage
Search aliases: t^dim_k A dim
Use in source · body.tex, line 900
[Finite trace image bound] Let k be algebraically closed of characteristic zero or characteristic ℓ>n. Let Γ⊆GL_n(k) be a subgroup, and assume its defining module k^n is semisimple. Suppose its trace set is finite, with t=#{Tr(γ):γ∈Γ}. Then Γ is finite and absΓ≤ t^dim_k A≤ t^n^2, A=span_kΓ. In particular this bounds a finite semisimple residual matrix group in characteristic greater than its rank.
Exact mathematical passage
\abs\Gamma\le t^{\dim_k A}\le t^{n^2}, \qquad A=\operatorname{span}_k\Gamma.
T^jUse in source · 2 passages
Search aliases: T^j
Use in source · formal_appendix.tex, line 37
A one-cocycle is a function a:C_e→ M satisfying a(gh)=a(g)+g· a(h). Putting g=h=1 gives a(1)=0. If x=a(γ), repeated application of the cocycle equation gives a(γ^r)=∑_j=0^r-1T^j x (1≤ r≤ e). The group relation γ^e=1 therefore forces Nx=0. Conversely, if Nx=0, this formula defines a cocycle. Whenever exponents wrap around e, the discrepancy is a translate of Nx, which is zero. Thus evaluating at the chosen generator identifies cocycles with ker N.
Exact mathematical passage
a(\gamma^r)=\sum_{j=0}^{r-1}T^j x\qquad(1\le r\le e).
T^{n-k}Use in source · 1 passage
Search aliases: T^n-k
Use in source · body.tex, line 141
Let ξ be a generator of X, and identify its values with pth roots of unity. For a matrix A∈GL_n(CC), write det(TI-A) = ∑_k=0^n(-1)^k e_k(A)T^n-k.
Exact mathematical passage
\det(TI-A) = \sum_{k=0}^{n}(-1)^k e_k(A)T^{n-k}.
t^{n^2}Use in source · 5 passages
Search aliases: t^n^2
Use in source · body.tex, line 900
[Finite trace image bound] Let k be algebraically closed of characteristic zero or characteristic ℓ>n. Let Γ⊆GL_n(k) be a subgroup, and assume its defining module k^n is semisimple. Suppose its trace set is finite, with t=#{Tr(γ):γ∈Γ}. Then Γ is finite and absΓ≤ t^dim_k A≤ t^n^2, A=span_kΓ. In particular this bounds a finite semisimple residual matrix group in characteristic greater than its rank.
Exact mathematical passage
\abs\Gamma\le t^{\dim_k A}\le t^{n^2}, \qquad A=\operatorname{span}_k\Gamma.
t^rUse in source · 1 passage
Search aliases: t^r
Use in source · body.tex, line 925
t to Φ(γ)=(Tr(γγ_1),…, Tr(γγ_r)). All coordinates lie in the trace set, because γγ_i∈Γ. If two tuples agree, then γ-γ' pairs to zero with every basis element, hence every element of A. Nondegeneracy gives γ=γ'. Therefore Φ injects Γ into a set of size t^r. This proves finiteness as well as the bound. Prior finiteness of Γ was not needed in this argument.
Exact mathematical passage
t^r
T_{i,v}Use in source · 1 passage
Search aliases: T_i,v
Use in source · body.tex, line 1250
t[A torsion cohomology realization range] Scholze's Corollary 5.4.3 supplies a unique continuous semisimple residual representation for a Hecke eigensystem actually occurring in the stated cohomology of a locally symmetric space, under the field, level, coefficient, and theorem dependencies of that source [source reference]. The specified polynomial has coefficients q_v^i(n+1)/2T_i,v, rather than an unspecified normalization of the eigenvalues. A fully stated unconditional subrange uses a CM fie…
Exact mathematical passage
q_v^{i(n+1)/2}T_{i,v}
\TrUse in source · 13 passages
Search aliases: Tr
Use in source · body.tex, line 503
[Fourier trace interpretation] Let g be a lift of a generator of G/H. For a stable type τ, widehat P_τ,V(r) = Tr(g^rmid M_τ(V)). If V is twisted by ξ^j, then widehat P_τ,V⊗ξ^j(r) = ζ^jrwidehat P_τ,V(r).
Exact mathematical passage
\boxed{ \widehat P_{\tau,V}(r) = \Tr\left(g^r\mid M_\tau(V)\right). }
\Tr_{k^n}Use in source · 1 passage
Search aliases: Tr_k^n Tr
Use in source · body.tex, line 915
le. The given module is semisimple and faithful for A. The Jacobson radical annihilates every simple constituent, hence annihilates this faithful module. It is therefore zero. The split semisimple algebra has the form A≅⊕_iM_d_i(k), k^n≅⊕_i(k^d_i)^⊕ m_i, ∑_i d_im_i=n. Faithfulness gives m_i≥1 for every block. The trace pairing B(x,y)=Tr_k^n(xy) restricts on the ith block to m_iTr(XY). Matrix trace pairing is nondegenerate, as is seen by pairing matrix units E_ab with E_ba. Each m_i is nonzero in…
Exact mathematical passage
B(x,y)=\Tr_{k^n}(xy)
\triangleleftUse in source · 6 passages
Search aliases: triangleleft
Use in source · body.tex, line 49
The finite representation arguments have a parallel shape. If Htriangleleft G and G/H≅ C_p, an irreducible representation either restricts to a full orbit of p subgroup types or restricts irreducibly to one invariant type. In the first case induction reconstructs the representation. In the second case an extension must be selected. The second case is where two torsors meet. Each torsor exists, and each has p elements. There is no preferred origin in the statement that they are torsors under the …
Exact mathematical passage
H\triangleleft G
UUse in source · 22 passages
Search aliases: U
Use in source · body.tex, line 1375
The Lie group GL_n(CC) has a neighborhood U of the identity that contains no nontrivial subgroup. Continuity gives a neighborhood r^-1(U) of the identity in G. A profinite group has a basis of open normal subgroups, so one such subgroup N lies inside r^-1(U). Its image is a subgroup contained in U, hence is trivial. Therefore r factors through the finite quotient G/N. This proof uses a group neighborhood theorem. It does not use the false general topological assertion that an arbitrary continuou…
Exact mathematical passage
U
uUse in source · 8 passages
Search aliases: u
Use in source · formal_appendix.tex, line 251
Let J:V→ W be a linear map over a field k. The correction equation Ju+e=0 has a solution precisely when e∈imJ. A useful description of this image uses all the dual relations among the equations. Let S=ker(J^*:W^*→ V^*), C={w∈ W:ϕ(w)=0 for every ϕ∈ S}. Every Ju belongs to C. Conversely, if w does not belong to imJ, its nonzero class in W/imJ is detected by an algebraic linear functional. Pulling that functional back to W gives an element of S that does not annihilate w. Hence imJ=C. This algebrai…
Exact mathematical passage
Ju+e=0
U^{-1}Use in source · 2 passages
Search aliases: U^-1
Use in source · formal_appendix.tex, line 563
The root constructed over Q_p is integral. Its center is integral, and every correction entry has norm at most p^-m≤1. If d>0 and U is a unit, the root F is also a unit. Indeed, F commutes with U=F^d and hence with U^-1, and F^d-1U^-1 is a two-sided inverse of F. Thus the unit version has an actual output in GL_n( Z_p).
Exact mathematical passage
U^{-1}
U_{\mathrm{bad}}Use in source · 2 passages
Search aliases: U_bad mathrm
Use in source · formal_appendix.tex, line 625
For a negative control, take U_bad=1 · 10 0 · 1. The difference from A^2 is (0,8,0,0) in entry coordinates, and B sends it to (0,16,0,0). This is nonzero modulo 32, so no square root of U_bad lies in that ball. This is a failure for the specified prefix. It is not a claim that the target has no square root anywhere in a larger coefficient field.
Exact mathematical passage
U_{\mathrm{bad}}=\begin{pmatrix}1&10\\0&1\end{pmatrix}.
U_iUse in source · 1 passage
Search aliases: U_i
Use in source · formal_appendix.tex, line 673
This matters for simultaneous matrix prescriptions. At level r, a node can contain a shared matrix F_r together with one local conjugator z_i,r for each member of a finite family. The equations are z_i,rF_r^d_iz_i,r^-1=U_ipmod p^r, with the prescribed prefix of F_r and prescribed residual identity for each conjugator. These are a single joint predicate on a finite tuple. The arithmetic equations and prefixes are preserved by reduction. Compactness therefore produces compatible towers for the sha…
Exact mathematical passage
z_{i,r}F_r^{d_i}z_{i,r}^{-1}=U_i\pmod {p^r},
V'Use in source · 6 passages
Search aliases: V'
Use in source · body.tex, line 289
[Stable equality of epsilon factors] Let V,V' be finite dimensional complex Weil representations with finite inertia image, of equal dimension and determinant. For every sufficiently ramified auxiliary character η, ϵ(V⊗η,Ψ)=ϵ(V'⊗η,Ψ). Here Ψ is a fixed nontrivial additive character and the same normalization is used on both sides. Deligne and Henniart's stability theorem applies to the virtual representation B=V-V', which has dimension zero and determinant one. Their Theorem 4.6 gives the ratio …
Exact mathematical passage
V,V'
V^*Use in source · 1 passage
Search aliases: V^*
Use in source · formal_appendix.tex, line 254
Let J:V→ W be a linear map over a field k. The correction equation Ju+e=0 has a solution precisely when e∈imJ. A useful description of this image uses all the dual relations among the equations. Let S=ker(J^*:W^*→ V^*), C={w∈ W:ϕ(w)=0 for every ϕ∈ S}. Every Ju belongs to C. Conversely, if w does not belong to imJ, its nonzero class in W/imJ is detected by an algebraic linear functional. Pulling that functional back to W gives an element of S that does not annihilate w. Hence imJ=C. This algebrai…
Exact mathematical passage
S=\ker(J^*:W^*\to V^*),\qquad C=\{w\in W:\phi(w)=0\text{ for every }\phi\in S\}.
V_0Use in source · 1 passage
Search aliases: V_0
Use in source · body.tex, line 585
[A stable cyclic type] Let G=C_3× C_2, H=C_3. Fix a nontrivial character τ of H. It is stable under G. After choosing the extension with trivial C_2 coordinate, the two one dimensional extensions have spectra P_τ,V_0(z)=1, P_τ,V_1(z)=z. The alignment set is the singleton containing the unique quotient twist carrying one extension to the other.
Exact mathematical passage
P_{\tau,V_0}(z)=1, \qquad P_{\tau,V_1}(z)=z.
V_1Use in source · 1 passage
Search aliases: V_1
Use in source · body.tex, line 585
[A stable cyclic type] Let G=C_3× C_2, H=C_3. Fix a nontrivial character τ of H. It is stable under G. After choosing the extension with trivial C_2 coordinate, the two one dimensional extensions have spectra P_τ,V_0(z)=1, P_τ,V_1(z)=z. The alignment set is the singleton containing the unique quotient twist carrying one extension to the other.
Exact mathematical passage
P_{\tau,V_0}(z)=1, \qquad P_{\tau,V_1}(z)=z.
v_pUse in source · 1 passage
Search aliases: v_p
Use in source · formal_appendix.tex, line 568
The value of s measures denominators of the whole inverse derivative. It is not automatically v_p(d). The proof makes no statement about shallow balls with m≤ s, where its contraction estimate is insufficient. It also does not provide automatic certificate discovery for arbitrary p-adic input. Exact rational examples can supply certificates by ordinary linear algebra, but a general certified implementation is an additional task.
Exact mathematical passage
v_p(d)
\operatorname{vec}Use in source · 2 passages
Search aliases: vec operatorname
Use in source · formal_appendix.tex, line 547
Take r=p^-m with m>s. Then Kr=p^s-m<1, and normQ(U-A^d)≤ p^-m · Longleftrightarrow normB(U-A^d)≤ p^-(m+s)notag · Longleftrightarrow Bvec(U-A^d)=0pmod p^m+s. The last congruence is entrywise. It depends only on the reductions of the target, center, and integral operator modulo p^m+s. The theorem therefore decides the existence and uniqueness of a root in A+p^mMat_n( Z_p) by one finite target test, once the exact inverse certificate is supplied.
Exact mathematical passage
\norm{Q(U-A^d)}\le p^{-m} &\Longleftrightarrow \norm{B(U-A^d)}\le p^{-(m+s)}\notag\\ &\Longleftrightarrow B\operatorname{vec}(U-A^d)=0\pmod {p^{m+s}}. \label{eq:appendix-finite-criterion}
wUse in source · 31 passages
Search aliases: w
Use in source · body.tex, line 244
[Determinant character of an algebraic representation] Let r:GL_n(CC)⟶GL_N(CC) be an irreducible algebraic representation. Suppose the center acts with weight w, so r(zI_n)=z^wI_N. There is an integer a such that det r(g)=det(g)^a and wN=an.
Exact mathematical passage
w
W^*Use in source · 1 passage
Search aliases: W^*
Use in source · formal_appendix.tex, line 254
Let J:V→ W be a linear map over a field k. The correction equation Ju+e=0 has a solution precisely when e∈imJ. A useful description of this image uses all the dual relations among the equations. Let S=ker(J^*:W^*→ V^*), C={w∈ W:ϕ(w)=0 for every ϕ∈ S}. Every Ju belongs to C. Conversely, if w does not belong to imJ, its nonzero class in W/imJ is detected by an algebraic linear functional. Pulling that functional back to W gives an element of S that does not annihilate w. Hence imJ=C. This algebrai…
Exact mathematical passage
S=\ker(J^*:W^*\to V^*),\qquad C=\{w\in W:\phi(w)=0\text{ for every }\phi\in S\}.
w_1Use in source · 2 passages
Search aliases: w_1
Use in source · formal_appendix.tex, line 278
[A reachable target with a visible failed correction] Let V=k^2, W=k^3, and J(u,v)=(u,v,u+v). Its image is the plane w_3=w_1+w_2. The dual kernel is generated by the functional ϕ(w_1,w_2,w_3)=w_3-w_1-w_2. The error (1,2,3) is compatible in every characteristic and is cancelled by (u,v)=(-1,-2). The error (0,0,1) is incompatible in every field, since its value under ϕ is 1. No choice of the two correction variables can cancel it.
Exact mathematical passage
w_3=w_1+w_2
w_2Use in source · 2 passages
Search aliases: w_2
Use in source · formal_appendix.tex, line 278
[A reachable target with a visible failed correction] Let V=k^2, W=k^3, and J(u,v)=(u,v,u+v). Its image is the plane w_3=w_1+w_2. The dual kernel is generated by the functional ϕ(w_1,w_2,w_3)=w_3-w_1-w_2. The error (1,2,3) is compatible in every characteristic and is cancelled by (u,v)=(-1,-2). The error (0,0,1) is incompatible in every field, since its value under ϕ is 1. No choice of the two correction variables can cancel it.
Exact mathematical passage
w_3=w_1+w_2
w_3Use in source · 2 passages
Search aliases: w_3
Use in source · formal_appendix.tex, line 278
[A reachable target with a visible failed correction] Let V=k^2, W=k^3, and J(u,v)=(u,v,u+v). Its image is the plane w_3=w_1+w_2. The dual kernel is generated by the functional ϕ(w_1,w_2,w_3)=w_3-w_1-w_2. The error (1,2,3) is compatible in every characteristic and is cancelled by (u,v)=(-1,-2). The error (0,0,1) is incompatible in every field, since its value under ϕ is 1. No choice of the two correction variables can cancel it.
Exact mathematical passage
w_3=w_1+w_2
W_{E_w}Use in source · 8 passages
Search aliases: W_E_w
Use in source · body.tex, line 623
For each place v of F, put R_v=ρ|_W_F_v, A_v=rec_F_v(π_0,v). Cyclic base change compatibility gives Res_W_E_wR_v ≅ Res_W_E_wA_v.
Exact mathematical passage
\Res_{W_{E_w}}R_v \cong \Res_{W_{E_w}}A_v.
\widehat P_{\tau,V\otimes\xi^j}Use in source · 1 passage
Search aliases: widehat P_τ,V⊗ξ^j widehat tau otimes xi
Use in source · body.tex, line 511
[Fourier trace interpretation] Let g be a lift of a generator of G/H. For a stable type τ, widehat P_τ,V(r) = Tr(g^rmid M_τ(V)). If V is twisted by ξ^j, then widehat P_τ,V⊗ξ^j(r) = ζ^jrwidehat P_τ,V(r).
Exact mathematical passage
\boxed{ \widehat P_{\tau,V\otimes\xi^j}(r) = \zeta^{jr}\widehat P_{\tau,V}(r). }
x'Use in source · 2 passages
Search aliases: x'
Use in source · formal_appendix.tex, line 324
J=keroverline B. If F(x)=π^r e, the moving identity gives π^r B_xe=0. A faithful leading layer allows this equality to imply overline B overline e=0. Equation [source reference] then supplies a residual correction overline u with overline J overline u=-overline e. Lift overline u to u and put x'=x+π^r u. The linear part of F(x') is divisible by π^r+1. For polynomial systems the remainder contains at least two copies of π^r, so it is divisible by π^2r and hence by π^r+1 when r≥1. This produces th…
Exact mathematical passage
x'=x+\pi^r u
x^2Use in source · 1 passage
Search aliases: x^2
Use in source · formal_appendix.tex, line 296
For a nonlinear system F(x)=0, a relation among the residual Jacobian rows is only a first-order identity. Consider F(x)=(x,x^2), J_x(h)=(h,2xh). At the origin the derivative has horizontal image. The functional selecting the second coordinate is a residual syzygy, but it does not annihilate F(x) for general x. The exact relation is the moving relation (-x,1)F(x)=0. Furthermore, F(x+h)-F(x)-J_x(h)=(0,h^2). Modulo 8, taking x=0 and h=2 gives remainder (0,4), outside the horizontal target. Keeping…
Exact mathematical passage
F(x)=(x,x^2),\qquad J_x(h)=(h,2xh).
x_iUse in source · 5 passages
Search aliases: x_i
Use in source · formal_appendix.tex, line 213
Suppose an absolute two-cocycle has been cancelled by a continuous one-cochain b_0. Prescribed local lifts need not agree with the resulting global lift. Their remaining differences are local one-cocycles. Changing b_0 to another absolute primitive changes these differences by the restriction of a global one-cocycle. Conjugating the local lift at index i by a unit 1+x_i, with x_i∈ I, changes the local cocycle by a local coboundary. The conjugators x_i are independently chosen at each local index…
Exact mathematical passage
1+x_i
X_{it}Use in source · 1 passage
Search aliases: X_it
Use in source · formal_appendix.tex, line 448
For E=Mat_n( Q_p) use the entrywise supremum norm and normalize abs p_p=p^-1. Matrix multiplication satisfies abs(XY)_ij_p =|∑_tX_itY_tj|_p ≤max_tabsX_it_pabsY_tj_p ≤norm Xnorm Y. Taking the maximum over the finite set of entries proves submultiplicativity. The same ultrametric calculation removes a matrix dimension factor. Completeness is inherited from the finite product of copies of Q_p. These are the hypotheses required by the actual matrix application, rather than properties borrowed from a…
Exact mathematical passage
\abs{(XY)_{ij}}_p =\left|\sum_tX_{it}Y_{tj}\right|_p \le\max_t\abs{X_{it}}_p\abs{Y_{tj}}_p \le\norm X\norm Y.
X_rUse in source · 3 passages
Search aliases: X_r
Use in source · formal_appendix.tex, line 651
Let X_r be finite sets with genuine reduction maps π_r,t:X_t→ X_r for r≤ t. Assume identity and composition laws for these maps. Let P_r⊆ X_r be solution sets, and assume reduction preserves solutions. Then (∀ r, P_r≠∅) Longleftrightarrow (∃ (x_r)_r, x_r∈ P_r and π_r,t(x_t)=x_r for r≤ t). The nontrivial implication is finite branch compactness. For each depth, a solution at that depth gives a finite path of solutions at all earlier depths. There are arbitrarily long paths. The resulting tree has…
Exact mathematical passage
X_r
x_rUse in source · 1 passage
Search aliases: x_r
Use in source · formal_appendix.tex, line 655
Let X_r be finite sets with genuine reduction maps π_r,t:X_t→ X_r for r≤ t. Assume identity and composition laws for these maps. Let P_r⊆ X_r be solution sets, and assume reduction preserves solutions. Then (∀ r, P_r≠∅) Longleftrightarrow (∃ (x_r)_r, x_r∈ P_r and π_r,t(x_t)=x_r for r≤ t). The nontrivial implication is finite branch compactness. For each depth, a solution at that depth gives a finite path of solutions at all earlier depths. There are arbitrarily long paths. The resulting tree has…
Exact mathematical passage
\left(\forall r,\ P_r\ne\varnothing\right) \Longleftrightarrow \left(\exists (x_r)_r,\ x_r\in P_r\text{ and } \pi_{r,t}(x_t)=x_r\text{ for }r\le t\right). \label{eq:appendix-branch-criterion}
X_tUse in source · 1 passage
Search aliases: X_t
Use in source · formal_appendix.tex, line 652
Let X_r be finite sets with genuine reduction maps π_r,t:X_t→ X_r for r≤ t. Assume identity and composition laws for these maps. Let P_r⊆ X_r be solution sets, and assume reduction preserves solutions. Then (∀ r, P_r≠∅) Longleftrightarrow (∃ (x_r)_r, x_r∈ P_r and π_r,t(x_t)=x_r for r≤ t). The nontrivial implication is finite branch compactness. For each depth, a solution at that depth gives a finite path of solutions at all earlier depths. There are arbitrarily long paths. The resulting tree has…
Exact mathematical passage
\pi_{r,t}:X_t\to X_r
x_tUse in source · 1 passage
Search aliases: x_t
Use in source · formal_appendix.tex, line 655
Let X_r be finite sets with genuine reduction maps π_r,t:X_t→ X_r for r≤ t. Assume identity and composition laws for these maps. Let P_r⊆ X_r be solution sets, and assume reduction preserves solutions. Then (∀ r, P_r≠∅) Longleftrightarrow (∃ (x_r)_r, x_r∈ P_r and π_r,t(x_t)=x_r for r≤ t). The nontrivial implication is finite branch compactness. For each depth, a solution at that depth gives a finite path of solutions at all earlier depths. There are arbitrarily long paths. The resulting tree has…
Exact mathematical passage
\left(\forall r,\ P_r\ne\varnothing\right) \Longleftrightarrow \left(\exists (x_r)_r,\ x_r\in P_r\text{ and } \pi_{r,t}(x_t)=x_r\text{ for }r\le t\right). \label{eq:appendix-branch-criterion}
YUse in source · 2 passages
Search aliases: Y
Use in source · body.tex, line 918
Jacobson radical annihilates every simple constituent, hence annihilates this faithful module. It is therefore zero. The split semisimple algebra has the form A≅⊕_iM_d_i(k), k^n≅⊕_i(k^d_i)^⊕ m_i, ∑_i d_im_i=n. Faithfulness gives m_i≥1 for every block. The trace pairing B(x,y)=Tr_k^n(xy) restricts on the ith block to m_iTr(XY). Matrix trace pairing is nondegenerate, as is seen by pairing matrix units E_ab with E_ba. Each m_i is nonzero in k: this is automatic in characteristic zero, and follows f…
Exact mathematical passage
m_i\Tr(XY)
yUse in source · 7 passages
Search aliases: y
Use in source · body.tex, line 915
le. The given module is semisimple and faithful for A. The Jacobson radical annihilates every simple constituent, hence annihilates this faithful module. It is therefore zero. The split semisimple algebra has the form A≅⊕_iM_d_i(k), k^n≅⊕_i(k^d_i)^⊕ m_i, ∑_i d_im_i=n. Faithfulness gives m_i≥1 for every block. The trace pairing B(x,y)=Tr_k^n(xy) restricts on the ith block to m_iTr(XY). Matrix trace pairing is nondegenerate, as is seen by pairing matrix units E_ab with E_ba. Each m_i is nonzero in…
Exact mathematical passage
B(x,y)=\Tr_{k^n}(xy)
Y_{tj}Use in source · 1 passage
Search aliases: Y_tj
Use in source · formal_appendix.tex, line 448
For E=Mat_n( Q_p) use the entrywise supremum norm and normalize abs p_p=p^-1. Matrix multiplication satisfies abs(XY)_ij_p =|∑_tX_itY_tj|_p ≤max_tabsX_it_pabsY_tj_p ≤norm Xnorm Y. Taking the maximum over the finite set of entries proves submultiplicativity. The same ultrametric calculation removes a matrix dimension factor. Completeness is inherited from the finite product of copies of Q_p. These are the hypotheses required by the actual matrix application, rather than properties borrowed from a…
Exact mathematical passage
\abs{(XY)_{ij}}_p =\left|\sum_tX_{it}Y_{tj}\right|_p \le\max_t\abs{X_{it}}_p\abs{Y_{tj}}_p \le\norm X\norm Y.
\mathbb ZUse in source · 10 passages
Search aliases: Z mathbb
Use in source · formal_appendix.tex, line 167
[An obstruction that really does not vanish] Reduce Z/27 Z to Z/9 Z. Its kernel is 9 Z/27 Z, whose square is zero. Map a generator of C_3 to the unit 4 modulo 9. This defines a representation because 4^3=1 modulo 9.
Exact mathematical passage
\mathbb Z/27\mathbb Z
z^2Use in source · 4 passages
Search aliases: z^2
Use in source · worked_appendix.tex, line 67
Take G=C_2× C_3, let H=C_2, and let ξ be the quotient character taking a generator of C_3 to ω. Let τ be the nontrivial character of H, extended with trivial quotient action. Consider a semisimple representation with spectrum P_τ,V(z)=2+z. Its quotient multiplicity vector is (2,1,0). Twisting by ξ gives zP_τ,V(z)=2z+z^2, whose vector is (0,2,1). Twisting by ξ^2 gives (1,0,2). These are ordered vectors. Knowing that each has total multiplicity three does not retain the extension label.
Exact mathematical passage
zP_{\tau,V}(z)=2z+z^2,
z^{an}Use in source · 1 passage
Search aliases: z^an
Use in source · body.tex, line 263
Every algebraic character of GL_n is an integral power of the determinant. Thus det∘ r=det^a for one integer a. Evaluate both sides at zI_n. The left side is z^wN, while the right side is z^an.
Exact mathematical passage
z^{an}
z^wUse in source · 1 passage
Search aliases: z^w
Use in source · body.tex, line 245
[Determinant character of an algebraic representation] Let r:GL_n(CC)⟶GL_N(CC) be an irreducible algebraic representation. Suppose the center acts with weight w, so r(zI_n)=z^wI_N. There is an integer a such that det r(g)=det(g)^a and wN=an.
Exact mathematical passage
r(zI_n)=z^wI_N.
z^{wN}Use in source · 1 passage
Search aliases: z^wN
Use in source · body.tex, line 263
Every algebraic character of GL_n is an integral power of the determinant. Thus det∘ r=det^a for one integer a. Evaluate both sides at zI_n. The left side is z^wN, while the right side is z^an.
Exact mathematical passage
z^{wN}
\mathbb Z_2Use in source · 2 passages
Search aliases: Z_2 mathbb
Use in source · formal_appendix.tex, line 577
Work over Z_2 and take A=1 · 1
Exact mathematical passage
\mathbb Z_2
z_{i,r}Use in source · 2 passages
Search aliases: z_i,r
Use in source · formal_appendix.tex, line 672
This matters for simultaneous matrix prescriptions. At level r, a node can contain a shared matrix F_r together with one local conjugator z_i,r for each member of a finite family. The equations are z_i,rF_r^d_iz_i,r^-1=U_ipmod p^r, with the prescribed prefix of F_r and prescribed residual identity for each conjugator. These are a single joint predicate on a finite tuple. The arithmetic equations and prefixes are preserved by reduction. Compactness therefore produces compatible towers for the sha…
Exact mathematical passage
z_{i,r}
z_{i,r}^{-1}Use in source · 1 passage
Search aliases: z_i,r^-1
Use in source · formal_appendix.tex, line 673
This matters for simultaneous matrix prescriptions. At level r, a node can contain a shared matrix F_r together with one local conjugator z_i,r for each member of a finite family. The equations are z_i,rF_r^d_iz_i,r^-1=U_ipmod p^r, with the prescribed prefix of F_r and prescribed residual identity for each conjugator. These are a single joint predicate on a finite tuple. The arithmetic equations and prefixes are preserved by reduction. Compactness therefore produces compatible towers for the sha…
Exact mathematical passage
z_{i,r}F_r^{d_i}z_{i,r}^{-1}=U_i\pmod {p^r},
\mathbb Z_pUse in source · 4 passages
Search aliases: Z_p mathbb
Use in source · formal_appendix.tex, line 532
For integral matrices suppose the inverse has the certified form Q=p^-sB, s∈ N, where B is an integral linear operator on the matrix entries. In a fixed ordering of the n^2 entries it is an n^2 by n^2 matrix over Z_p. This is an operator on the matrix space, not ordinary left multiplication by an n by n matrix. Integral coefficients imply normBz≤norm z, so K=p^s is a valid bound.
Exact mathematical passage
\mathbb Z_p
|_HUse in source · 5 passages
Search aliases: |_H
Use in source · worked_appendix.tex, line 10
[The two prime cyclic cases] Let G be finite, let Htriangleleft G with G/H≅ C_p, and let V be irreducible over CC. Exactly one of the following occurs: item V|_H is irreducible and invariant under G, and V is one of its p quotient character extensions. item V|_H is a sum of p pairwise nonisomorphic conjugate irreducible representations, and V is induced from any one of them.
Exact mathematical passage
V|_H
|_pUse in source · 1 passage
Search aliases: |_p
Use in source · formal_appendix.tex, line 448
For E=Mat_n( Q_p) use the entrywise supremum norm and normalize abs p_p=p^-1. Matrix multiplication satisfies abs(XY)_ij_p =|∑_tX_itY_tj|_p ≤max_tabsX_it_pabsY_tj_p ≤norm Xnorm Y. Taking the maximum over the finite set of entries proves submultiplicativity. The same ultrametric calculation removes a matrix dimension factor. Completeness is inherited from the finite product of copies of Q_p. These are the hypotheses required by the actual matrix application, rather than properties borrowed from a…
Exact mathematical passage
\abs{(XY)_{ij}}_p =\left|\sum_tX_{it}Y_{tj}\right|_p \le\max_t\abs{X_{it}}_p\abs{Y_{tj}}_p \le\norm X\norm Y.
|_{W_{E_w}}Use in source · 1 passage
Search aliases: |_W_E_w
Use in source · body.tex, line 1164
Assume that v is unramified and nonsplit. Since the extension has prime degree, its residue degree is p. Unramified base change raises the Satake conjugacy class of π_v to its pth power. Restriction of ρ_v to W_E_w does the same to its Frobenius class. Since Π_w is attached to ρ|_W_E_w, P_π,v^[p](T) = P_ρ,v^[p](T). Reduce modulo λ_p and use [source reference]. Injectivity of residue Frobenius gives the desired congruence. The finitely many ramified and excluded places do not affect residual atta…
Exact mathematical passage
\rho|_{W_{E_w}}
\cdotUse in source · 7 passages
Search aliases: · cdot
Use in source · body.tex, line 1254
from a simple unitary Kottwitz datum and a cuspidal representation whose infinity component is an X holomorphic limit of discrete series [source reference]. At a coefficient prime satisfying the good reduction and reflex field conditions, it supplies a semisimple l adic representation over the CM field. Its local compatibility is stated on the density one split prime set with the twist lvert·rvert^(1-n)/2 in the source normalization. This is a concrete realization range. It is not a theorem for …
Exact mathematical passage
\lvert\cdot\rvert^{(1-n)/2}
\alpha_1Use in source · 1 passage
Search aliases: α_1 alpha
Use in source · body.tex, line 735
Choose wmid v and let f=f(w/v). If the normalized Satake roots of π_v are α_1,…,α_n, unramified base change replaces the Satake conjugacy class by its fth power. Hence the roots of Π_w are α_i^f. Since Π_w is attached to a representation of exponent dividing M, (α_i^f)^M=1. Thus α_i∈μ_fM. Since fmid d, this group is contained in μ_dM.
Exact mathematical passage
\alpha_1,\ldots,\alpha_n
\alpha_i^fUse in source · 2 passages
Search aliases: α_i^f alpha
Use in source · body.tex, line 735
Choose wmid v and let f=f(w/v). If the normalized Satake roots of π_v are α_1,…,α_n, unramified base change replaces the Satake conjugacy class by its fth power. Hence the roots of Π_w are α_i^f. Since Π_w is attached to a representation of exponent dividing M, (α_i^f)^M=1. Thus α_i∈μ_fM. Since fmid d, this group is contained in μ_dM.
Exact mathematical passage
\alpha_i^f
\alpha_nUse in source · 1 passage
Search aliases: α_n alpha
Use in source · body.tex, line 735
Choose wmid v and let f=f(w/v). If the normalized Satake roots of π_v are α_1,…,α_n, unramified base change replaces the Satake conjugacy class by its fth power. Hence the roots of Π_w are α_i^f. Since Π_w is attached to a representation of exponent dividing M, (α_i^f)^M=1. Thus α_i∈μ_fM. Since fmid d, this group is contained in μ_dM.
Exact mathematical passage
\alpha_1,\ldots,\alpha_n
\GammaUse in source · 20 passages
Search aliases: Γ Gamma
Use in source · body.tex, line 895
[Finite trace image bound] Let k be algebraically closed of characteristic zero or characteristic ℓ>n. Let Γ⊆GL_n(k) be a subgroup, and assume its defining module k^n is semisimple. Suppose its trace set is finite, with t=#{Tr(γ):γ∈Γ}. Then Γ is finite and absΓ≤ t^dim_k A≤ t^n^2, A=span_kΓ. In particular this bounds a finite semisimple residual matrix group in characteristic greater than its rank.
Exact mathematical passage
\Gamma\subseteq\GL_n(k)
\gamma'Use in source · 2 passages
Search aliases: γ' gamma
Use in source · body.tex, line 925
Choose a basis γ_1,…,γ_r of A from the spanning set Γ, where r=dim_k A≤ n^2. Send an image element to Φ(γ)=(Tr(γγ_1),…, Tr(γγ_r)). All coordinates lie in the trace set, because γγ_i∈Γ. If two tuples agree, then γ-γ' pairs to zero with every basis element, hence every element of A. Nondegeneracy gives γ=γ'. Therefore Φ injects Γ into a set of size t^r. This proves finiteness as well as the bound. Prior finiteness of Γ was not needed in this argument.
Exact mathematical passage
\gamma-\gamma'
\gamma^2Use in source · 1 passage
Search aliases: γ^2 gamma
Use in source · formal_appendix.tex, line 184
The defect can also be seen without an existence argument. Choose the section r_1=1, r_γ=4, and r_γ^2=16 modulo 27. These reduce to the required units modulo 9. The products of consecutive powers have no defect until the exponent passes through three. At that point 4^3=10=1+9 modulo 27. Consequently the normalized defect is 9 when the exponents wrap around and zero otherwise. Under the identification I≃ F_3, this is the ordinary carry cocycle.
Exact mathematical passage
r_{\gamma^2}=16
\gamma^eUse in source · 1 passage
Search aliases: γ^e gamma
Use in source · formal_appendix.tex, line 40
A one-cocycle is a function a:C_e→ M satisfying a(gh)=a(g)+g· a(h). Putting g=h=1 gives a(1)=0. If x=a(γ), repeated application of the cocycle equation gives a(γ^r)=∑_j=0^r-1T^j x (1≤ r≤ e). The group relation γ^e=1 therefore forces Nx=0. Conversely, if Nx=0, this formula defines a cocycle. Whenever exponents wrap around e, the discrepancy is a translate of Nx, which is zero. Thus evaluating at the chosen generator identifies cocycles with ker N.
Exact mathematical passage
\gamma^e=1
\gamma^iUse in source · 1 passage
Search aliases: γ^i gamma
Use in source · formal_appendix.tex, line 192
For a proposed correction b, sum the equations c(γ^i,γ)=b(γ)-b(γ^i+1)+b(γ^i) for i=0,1,2. The right side telescopes to 3b(γ)=0, using b(1)=0 and the trivial action. The left side is 9≠0 in I. There is therefore no primitive. This explicit calculation recovers the nonzero obstruction class directly from its normalized representative.
Exact mathematical passage
c(\gamma^i,\gamma)=b(\gamma)-b(\gamma^{i+1})+b(\gamma^i)
\gamma^{i+1}Use in source · 1 passage
Search aliases: γ^i+1 gamma
Use in source · formal_appendix.tex, line 192
For a proposed correction b, sum the equations c(γ^i,γ)=b(γ)-b(γ^i+1)+b(γ^i) for i=0,1,2. The right side telescopes to 3b(γ)=0, using b(1)=0 and the trivial action. The left side is 9≠0 in I. There is therefore no primitive. This explicit calculation recovers the nonzero obstruction class directly from its normalized representative.
Exact mathematical passage
c(\gamma^i,\gamma)=b(\gamma)-b(\gamma^{i+1})+b(\gamma^i)
\gamma^rUse in source · 1 passage
Search aliases: γ^r gamma
Use in source · formal_appendix.tex, line 37
A one-cocycle is a function a:C_e→ M satisfying a(gh)=a(g)+g· a(h). Putting g=h=1 gives a(1)=0. If x=a(γ), repeated application of the cocycle equation gives a(γ^r)=∑_j=0^r-1T^j x (1≤ r≤ e). The group relation γ^e=1 therefore forces Nx=0. Conversely, if Nx=0, this formula defines a cocycle. Whenever exponents wrap around e, the discrepancy is a translate of Nx, which is zero. Thus evaluating at the chosen generator identifies cocycles with ker N.
Exact mathematical passage
a(\gamma^r)=\sum_{j=0}^{r-1}T^j x\qquad(1\le r\le e).
\gamma_0Use in source · 1 passage
Search aliases: γ_0 gamma
Use in source · worked_appendix.tex, line 92
Let Γ={diag(ω^a,ω^-a):a=0,1,2}. The trace alphabet is {2,-1}. The image algebra has dimension two, with basis the identity and γ_1=diag(ω,ω^2). The trace coordinate map from [source reference] gives Φ(γ_0)=(2,-1), Φ(γ_1)=(-1,-1), Φ(γ_2)=(-1,2). The tuples are distinct. The sharper algebra bound is 3≤2^2=4, while the rank only bound is 3≤2^2^2=16. The latter is deliberately rough. Its role is a uniform bound independent of the coefficient prime, not an efficient classification of the group.
Exact mathematical passage
\Phi(\gamma_0)=(2,-1),\qquad \Phi(\gamma_1)=(-1,-1),\qquad \Phi(\gamma_2)=(-1,2).
\gamma_1Use in source · 4 passages
Search aliases: γ_1 gamma
Use in source · body.tex, line 920
Choose a basis γ_1,…,γ_r of A from the spanning set Γ, where r=dim_k A≤ n^2. Send an image element to Φ(γ)=(Tr(γγ_1),…, Tr(γγ_r)). All coordinates lie in the trace set, because γγ_i∈Γ. If two tuples agree, then γ-γ' pairs to zero with every basis element, hence every element of A. Nondegeneracy gives γ=γ'. Therefore Φ injects Γ into a set of size t^r. This proves finiteness as well as the bound. Prior finiteness of Γ was not needed in this argument.
Exact mathematical passage
\gamma_1,\ldots,\gamma_r
\gamma_2Use in source · 1 passage
Search aliases: γ_2 gamma
Use in source · worked_appendix.tex, line 92
Let Γ={diag(ω^a,ω^-a):a=0,1,2}. The trace alphabet is {2,-1}. The image algebra has dimension two, with basis the identity and γ_1=diag(ω,ω^2). The trace coordinate map from [source reference] gives Φ(γ_0)=(2,-1), Φ(γ_1)=(-1,-1), Φ(γ_2)=(-1,2). The tuples are distinct. The sharper algebra bound is 3≤2^2=4, while the rank only bound is 3≤2^2^2=16. The latter is deliberately rough. Its role is a uniform bound independent of the coefficient prime, not an efficient classification of the group.
Exact mathematical passage
\Phi(\gamma_0)=(2,-1),\qquad \Phi(\gamma_1)=(-1,-1),\qquad \Phi(\gamma_2)=(-1,2).
\gamma_iUse in source · 1 passage
Search aliases: γ_i gamma
Use in source · body.tex, line 925
Choose a basis γ_1,…,γ_r of A from the spanning set Γ, where r=dim_k A≤ n^2. Send an image element to Φ(γ)=(Tr(γγ_1),…, Tr(γγ_r)). All coordinates lie in the trace set, because γγ_i∈Γ. If two tuples agree, then γ-γ' pairs to zero with every basis element, hence every element of A. Nondegeneracy gives γ=γ'. Therefore Φ injects Γ into a set of size t^r. This proves finiteness as well as the bound. Prior finiteness of Γ was not needed in this argument.
Exact mathematical passage
\gamma\gamma_i\in\Gamma
\gamma_rUse in source · 2 passages
Search aliases: γ_r gamma
Use in source · body.tex, line 920
Choose a basis γ_1,…,γ_r of A from the spanning set Γ, where r=dim_k A≤ n^2. Send an image element to Φ(γ)=(Tr(γγ_1),…, Tr(γγ_r)). All coordinates lie in the trace set, because γγ_i∈Γ. If two tuples agree, then γ-γ' pairs to zero with every basis element, hence every element of A. Nondegeneracy gives γ=γ'. Therefore Φ injects Γ into a set of size t^r. This proves finiteness as well as the bound. Prior finiteness of Γ was not needed in this argument.
Exact mathematical passage
\gamma_1,\ldots,\gamma_r
\DeltaUse in source · 9 passages
Search aliases: Δ Delta
Use in source · formal_appendix.tex, line 465
Assume that D=D_A,d has an additive two-sided inverse Q, and that normQz≤ Knorm z, with K≥0. Choose 0≤ r≤1 with Kr<1. For a target U, set Δ=U-A^d. The proposed correction map is Φ(h)=Q(Δ-R_A,d(h)). Both inverse identities will be used. DQ=1 turns a fixed point into a root. QD=1 proves the necessary condition from an existing root.
Exact mathematical passage
\Delta=U-A^d
\deltaUse in source · 5 passages
Search aliases: δ delta
Use in source · body.tex, line 217
The base change of ω_π_0 is the central character of Π. Since Π is attached to τ=ρ|_G_E, that character equals detτ. The base change of detρ is also detτ. Hence δ=(detρ)ω_π_0^-1 is a character killed by base change to E, so δ∈ X. We need χ^n=δ. Since X has order p and pnmid n, the nth power map on X is bijective. Thus there is one and only one such χ.
Exact mathematical passage
\delta=(\det\rho)\omega_{\pi_0}^{-1}
\epsilon^{-j_{\lambda}}Use in source · 1 passage
Search aliases: ϵ^-j_λ epsilon lambda
Use in source · body.tex, line 1018
is weakly attached to τ, with π a cuspidal cyclic descent. Let λ be a good separating prime. Suppose π has a residual realization overline r_π,λ. Then there is a unique index j_λ∈ZZ/pZZ such that P_π,v(T) = P_ρ⊗ϵ^j_λ,v(T) for every finite place v outside a finite set. Consequently, π⊗ϵ^-j_λ is weakly attached to ρ.
Exact mathematical passage
\pi\otimes\epsilon^{-j_{\lambda}}
\epsilon^{j_{\lambda}}Use in source · 2 passages
Search aliases: ϵ^j_λ epsilon lambda
Use in source · body.tex, line 1010
cyclic Artin setup of [source reference]. For this unramified theorem it is enough that the lower cuspidal representation Π is weakly attached to τ, with π a cuspidal cyclic descent. Let λ be a good separating prime. Suppose π has a residual realization overline r_π,λ. Then there is a unique index j_λ∈ZZ/pZZ such that P_π,v(T) = P_ρ⊗ϵ^j_λ,v(T) for every finite place v outside a finite set. Consequently, π⊗ϵ^-j_λ is weakly attached to ρ.
Exact mathematical passage
\boxed{ P_{\pi,v}(T) = P_{\rho\otimes\epsilon^{j_{\lambda}},v}(T) }
\epsilon_vUse in source · 3 passages
Search aliases: ϵ_v epsilon
Use in source · body.tex, line 1471
cient separator] Let E/F be cyclic of prime degree p. Let π be cuspidal and assume its base change to E is cuspidal. Let ϵ be the quotient character. Then there is an unramified finite place v and an index k with pnmid k such that the degree k coefficient of the normalized Hecke polynomial of π_v is nonzero and ϵ_v(Frob_v)≠1.
Exact mathematical passage
\epsilon_v(\Frob_v)\neq1
\zeta^aUse in source · 1 passage
Search aliases: ζ^a zeta
Use in source · body.tex, line 521
The quotient generator acts on the ξ^a summand by ζ^a. Taking the trace of its rth power gives the Fourier sum. The twist multiplies the action of g^r by ζ^jr.
Exact mathematical passage
\zeta^a
\zeta^{ar}Use in source · 1 passage
Search aliases: ζ^ar zeta
Use in source · body.tex, line 493
Fix a primitive pth root ζ. For P(z)=∑_a∈ZZ/pZZm_az^a, define widehat P(r) = ∑_a∈ZZ/pZZm_aζ^ar, r∈ZZ/pZZ.
Exact mathematical passage
\widehat P(r) = \sum_{a\in\ZZ/p\ZZ}m_a\zeta^{ar}, \qquad r\in\ZZ/p\ZZ.
\zeta^kUse in source · 4 passages
Search aliases: ζ^k zeta
Use in source · body.tex, line 149
[Coefficient rotation] For ζ∈CC^×, e_k(ζ A)=ζ^k e_k(A).
Exact mathematical passage
\boxed{ e_k(\zeta A)=\zeta^k e_k(A). }
\etaUse in source · 5 passages
Search aliases: η eta
Use in source · body.tex, line 289
[Stable equality of epsilon factors] Let V,V' be finite dimensional complex Weil representations with finite inertia image, of equal dimension and determinant. For every sufficiently ramified auxiliary character η, ϵ(V⊗η,Ψ)=ϵ(V'⊗η,Ψ). Here Ψ is a fixed nontrivial additive character and the same normalization is used on both sides. Deligne and Henniart's stability theorem applies to the virtual representation B=V-V', which has dimension zero and determinant one. Their Theorem 4.6 gives the ratio …
Exact mathematical passage
\eta
\theta_iUse in source · 3 passages
Search aliases: θ_i theta
Use in source · body.tex, line 1505
[Positive saturation criterion] Let χ be a nontrivial irreducible character of G. Suppose there are a positive integer m, nonnegative integers a_i, subgroups H_i, and characters θ_i∈calE such that mχ = ∑_i a_iInd_H_i^Gθ_i. Then L(s,χ) is entire.
Exact mathematical passage
\theta_i\in\calE
\lambda_pUse in source · 9 passages
Search aliases: λ_p lambda
Use in source · body.tex, line 1077
Let K be a number field containing the values of the order p quotient characters, let λ_p be a place above p, and let OO be the corresponding valuation ring.
Exact mathematical passage
\lambda_p
\muUse in source · 16 passages
Search aliases: μ mu
Use in source · body.tex, line 168
[One polynomial separator] Let ζ range over μ_p. The twists ζ A have pairwise distinct characteristic polynomials exactly when S(A) contains a degree not divisible by p.
Exact mathematical passage
\mu_p
\mu_{dM}Use in source · 9 passages
Search aliases: μ_dM mu
Use in source · body.tex, line 727
[Finite Satake roots] At every finite place v where E/F, π, and Π are unramified and the stated Artin polynomial comparison holds above v, every normalized Satake root of π_v belongs to μ_dM.
Exact mathematical passage
\boxed{ \mu_{dM}. }
\mu_{fM}Use in source · 1 passage
Search aliases: μ_fM mu
Use in source · body.tex, line 739
Choose wmid v and let f=f(w/v). If the normalized Satake roots of π_v are α_1,…,α_n, unramified base change replaces the Satake conjugacy class by its fth power. Hence the roots of Π_w are α_i^f. Since Π_w is attached to a representation of exponent dividing M, (α_i^f)^M=1. Thus α_i∈μ_fM. Since fmid d, this group is contained in μ_dM.
Exact mathematical passage
\alpha_i\in\mu_{fM}
\mu_{\operatorname{lcm}(1,\ldots,B)}Use in source · 1 passage
Search aliases: μ_lcm(1,…,B) mu operatorname ldots
Use in source · worked_appendix.tex, line 123
The large residual theorem uses two alphabets. One is the supplied automorphic alphabet. The other contains every degree n polynomial whose roots have orders at most B=t^n^2. A field containing μ_lcm(1,…,B) contains all their coefficients. Both the field and the exceptional coefficient primes are fixed before choosing a residual realization. This order matters. A proof that chose a separation field from the coefficients of an as yet unknown lift and then called the excluded primes a predetermine…
Exact mathematical passage
\mu_{\operatorname{lcm}(1,\ldots,B)}
\mu_NUse in source · 3 passages
Search aliases: μ_N mu
Use in source · body.tex, line 803
A polynomial is determined by the multiset of its n Satake roots. Those roots lie in the N element set μ_N. The number of multisets of size n selected from N elements is the displayed binomial coefficient.
Exact mathematical passage
\mu_N
\mu_pUse in source · 2 passages
Search aliases: μ_p mu
Use in source · body.tex, line 168
[One polynomial separator] Let ζ range over μ_p. The twists ζ A have pairwise distinct characteristic polynomials exactly when S(A) contains a degree not divisible by p.
Exact mathematical passage
\mu_p
\xi^2Use in source · 1 passage
Search aliases: ξ^2 xi
Use in source · worked_appendix.tex, line 70
d let ξ be the quotient character taking a generator of C_3 to ω. Let τ be the nontrivial character of H, extended with trivial quotient action. Consider a semisimple representation with spectrum P_τ,V(z)=2+z. Its quotient multiplicity vector is (2,1,0). Twisting by ξ gives zP_τ,V(z)=2z+z^2, whose vector is (0,2,1). Twisting by ξ^2 gives (1,0,2). These are ordered vectors. Knowing that each has total multiplicity three does not retain the extension label.
Exact mathematical passage
\xi^2
\xi^jUse in source · 8 passages
Search aliases: ξ^j xi
Use in source · body.tex, line 408
[Effect of quotient twisting] For every j∈ZZ/pZZ, P_τ,V⊗ξ^j(z) = z^jP_τ,V(z). Moving orbit blocks are unchanged up to isomorphism by quotient twisting.
Exact mathematical passage
\boxed{ P_{\tau,V\otimes\xi^j}(z) = z^jP_{\tau,V}(z). }
\pi^{2r}Use in source · 1 passage
Search aliases: π^2r pi
Use in source · formal_appendix.tex, line 326
equality to imply overline B overline e=0. Equation [source reference] then supplies a residual correction overline u with overline J overline u=-overline e. Lift overline u to u and put x'=x+π^r u. The linear part of F(x') is divisible by π^r+1. For polynomial systems the remainder contains at least two copies of π^r, so it is divisible by π^2r and hence by π^r+1 when r≥1. This produces the next layer.
Exact mathematical passage
\pi^{2r}
\pi^NUse in source · 1 passage
Search aliases: π^N pi
Use in source · formal_appendix.tex, line 330
Over a suitable truncated principal coefficient ring, iteration terminates at π^N=0 and produces a root of the original system. The recovered v0.25 modules derive the necessary faithful-layer property under their stated principal residue and nilpotence hypotheses. They still require the moving identity and exactness. Equality of selected dimensions alone does not prove either one. A correct dimension count can help prove exactness after an inclusion is established, but cannot replace that inclus…
Exact mathematical passage
\pi^N=0
\pi^rUse in source · 9 passages
Search aliases: π^r pi
Use in source · formal_appendix.tex, line 319
The graded correction argument retains an exact moving relation B_xF(x)=0 and a residual exactness statement imoverline J=keroverline B. If F(x)=π^r e, the moving identity gives π^r B_xe=0. A faithful leading layer allows this equality to imply overline B overline e=0. Equation [source reference] then supplies a residual correction overline u with overline J overline u=-overline e. Lift overline u to u and put x'=x+π^r u. The linear part of F(x') is divisible by π^r+1. For polynomial systems the…
Exact mathematical passage
F(x)=\pi^r e
\pi^{r+1}Use in source · 2 passages
Search aliases: π^r+1 pi
Use in source · formal_appendix.tex, line 325
If F(x)=π^r e, the moving identity gives π^r B_xe=0. A faithful leading layer allows this equality to imply overline B overline e=0. Equation [source reference] then supplies a residual correction overline u with overline J overline u=-overline e. Lift overline u to u and put x'=x+π^r u. The linear part of F(x') is divisible by π^r+1. For polynomial systems the remainder contains at least two copies of π^r, so it is divisible by π^2r and hence by π^r+1 when r≥1. This produces the next layer.
Exact mathematical passage
\pi^{r+1}
\pi_0Use in source · 6 passages
Search aliases: π_0 pi
Use in source · body.tex, line 204
[Unique determinant matched descent] Use the cyclic Artin setup of [source reference], and assume pnmid n. Let π_0 be any cuspidal descent of Π. Then there is a unique quotient character χ∈ X such that ω_π_0⊗χ = detρ under global class field theory.
Exact mathematical passage
\pi_0
\pi_{0,v}Use in source · 2 passages
Search aliases: π_0,v pi
Use in source · body.tex, line 617
For each place v of F, put R_v=ρ|_W_F_v, A_v=rec_F_v(π_0,v). Cyclic base change compatibility gives Res_W_E_wR_v ≅ Res_W_E_wA_v.
Exact mathematical passage
R_v=\rho|_{W_{F_v}}, \qquad A_v=\rec_{F_v}(\pi_{0,v}).
\pi_{r,t}Use in source · 2 passages
Search aliases: π_r,t pi
Use in source · formal_appendix.tex, line 652
Let X_r be finite sets with genuine reduction maps π_r,t:X_t→ X_r for r≤ t. Assume identity and composition laws for these maps. Let P_r⊆ X_r be solution sets, and assume reduction preserves solutions. Then (∀ r, P_r≠∅) Longleftrightarrow (∃ (x_r)_r, x_r∈ P_r and π_r,t(x_t)=x_r for r≤ t). The nontrivial implication is finite branch compactness. For each depth, a solution at that depth gives a finite path of solutions at all earlier depths. There are arbitrarily long paths. The resulting tree has…
Exact mathematical passage
\pi_{r,t}:X_t\to X_r
\Pi_wUse in source · 5 passages
Search aliases: Π_w Pi
Use in source · body.tex, line 700
Fix v and wmid v, and write the local Weil Deligne parameter as (ϕ_v,N_v). Base change restricts ϕ_v to W_E_w and keeps the monodromy operator. Since the parameter of Π_w has zero monodromy, N_v=0.
Exact mathematical passage
\Pi_w
\rho'Use in source · 2 passages
Search aliases: ρ' rho
Use in source · body.tex, line 90
Fix one extension ρ. Every χ∈ X is trivial on H, so ρ⊗χ restricts to τ. Conversely, let ρ' be another extension. Since both restrictions are irreducible and isomorphic, Schur's lemma shows that after one choice of an H intertwiner, the ratio between the two actions is scalar on each coset. Those scalars define a character of G/H, so ρ'≅ρ⊗χ.
Exact mathematical passage
\rho'
\rho_vUse in source · 2 passages
Search aliases: ρ_v rho
Use in source · body.tex, line 115
Local class field theory identifies a quotient character χ∈ X with the corresponding Hecke character. Local Langlands is compatible with character twisting, [source reference]. Hence rec(π_v⊗χ_v) ≅ rec(π_v)⊗χ_v ≅ ρ_v⊗χ_v. The result holds at every place.
Exact mathematical passage
\rec(\pi_v\otimes\chi_v) \cong \rec(\pi_v)\otimes\chi_v \cong \rho_v\otimes\chi_v.
\SigmaUse in source · 2 passages
Search aliases: Σ Sigma
Use in source · body.tex, line 965
There is a finite set Σ(calP) of rational primes, fixed from the alphabet before any realization is chosen, with the following property. Let λ be a coefficient place whose residue characteristic ℓ is outside Σ(calP). Suppose a continuous semisimple representation overline r_λ:G_F⟶GL_n(overlineFF_ℓ) is unramified outside S and the places above ℓ, and has characteristic polynomial overlineP_v at every remaining unramified place. Then there is a finite image semisimple complex representation r with…
Exact mathematical passage
\Sigma(\calP)
\sigmaUse in source · 2 passages
Search aliases: σ sigma
Use in source · worked_appendix.tex, line 72
Suppose a second stable H type σ occurs with spectrum one. If the target has spectra 2z+z^2 for τ and z^2 for σ, the first block demands shift one and the second demands shift two. The two restrictions to H have the same multiplicities. Each block separately is a quotient twist of the matching source block. The full representation is not one quotient twist, because the required shifts disagree. This is the reason the spectrum criterion quantifies over one common j, rather than an independent j_τ…
Exact mathematical passage
\sigma
\tau^{-1}Use in source · 1 passage
Search aliases: τ^-1 tau
Use in source · body.tex, line 599
[A moving type and self twist] Let G=S_3, let H=A_3, and let V be the standard two dimensional representation. Its restriction is Res_HV=τ⊕τ^-1 for the two nontrivial characters of C_3. These characters form one moving orbit under G/H≅ C_2. Hence there is no stable extension spectrum, and V⊗sgn≅ V.
Exact mathematical passage
\Res_HV=\tau\oplus\tau^{-1}
\PhiUse in source · 7 passages
Search aliases: Φ Phi
Use in source · body.tex, line 921
Choose a basis γ_1,…,γ_r of A from the spanning set Γ, where r=dim_k A≤ n^2. Send an image element to Φ(γ)=(Tr(γγ_1),…, Tr(γγ_r)). All coordinates lie in the trace set, because γγ_i∈Γ. If two tuples agree, then γ-γ' pairs to zero with every basis element, hence every element of A. Nondegeneracy gives γ=γ'. Therefore Φ injects Γ into a set of size t^r. This proves finiteness as well as the bound. Prior finiteness of Γ was not needed in this argument.
Exact mathematical passage
\Phi(\gamma)=\bigl(\Tr(\gamma\gamma_1),\ldots, \Tr(\gamma\gamma_r)\bigr).
\phiUse in source · 8 passages
Search aliases: ϕ phi
Use in source · body.tex, line 700
Fix v and wmid v, and write the local Weil Deligne parameter as (ϕ_v,N_v). Base change restricts ϕ_v to W_E_w and keeps the monodromy operator. Since the parameter of Π_w has zero monodromy, N_v=0.
Exact mathematical passage
(\phi_v,N_v)
\phi_vUse in source · 5 passages
Search aliases: ϕ_v phi
Use in source · body.tex, line 700
Fix v and wmid v, and write the local Weil Deligne parameter as (ϕ_v,N_v). Base change restricts ϕ_v to W_E_w and keeps the monodromy operator. Since the parameter of Π_w has zero monodromy, N_v=0.
Exact mathematical passage
(\phi_v,N_v)
\chiUse in source · 33 passages
Search aliases: χ chi
Use in source · body.tex, line 90
Fix one extension ρ. Every χ∈ X is trivial on H, so ρ⊗χ restricts to τ. Conversely, let ρ' be another extension. Since both restrictions are irreducible and isomorphic, Schur's lemma shows that after one choice of an H intertwiner, the ratio between the two actions is scalar on each coset. Those scalars define a character of G/H, so ρ'≅ρ⊗χ.
Exact mathematical passage
\chi\in X
\chi^{\dim V}Use in source · 1 passage
Search aliases: χ^dim V chi dim
Use in source · body.tex, line 308
The determinant changes by χ^dim V=1. The two representations have the same dimension and determinant. Apply [source reference].
Exact mathematical passage
\chi^{\dim V}=1
\chi^nUse in source · 4 passages
Search aliases: χ^n chi
Use in source · body.tex, line 193
For a quotient character χ, det(ρ⊗χ)=det(ρ)χ^n. The central character satisfies the identical formula ω_π⊗χ=ω_πχ^n. Since X has prime order p and pnmid n, the map χ↦χ^n is an automorphism of X.
Exact mathematical passage
\det(\rho\otimes\chi)=\det(\rho)\chi^n.
\chi^wUse in source · 1 passage
Search aliases: χ^w chi
Use in source · body.tex, line 277
The scalar twist acts on r(ρ) by χ^w. Hence its determinant changes by χ^wN. By [source reference], wN=an, which is divisible by p. Thus χ^wN=1. Apply the argument to each irreducible summand.
Exact mathematical passage
\chi^w
\chi^{wN}Use in source · 2 passages
Search aliases: χ^wN chi
Use in source · body.tex, line 277
The scalar twist acts on r(ρ) by χ^w. Hence its determinant changes by χ^wN. By [source reference], wN=an, which is divisible by p. Thus χ^wN=1. Apply the argument to each irreducible summand.
Exact mathematical passage
\chi^{wN}
\chi_0Use in source · 1 passage
Search aliases: χ_0 chi
Use in source · body.tex, line 346
The restriction of every quotient character to H is trivial. The regular representation of G/H decomposes as the direct sum of all characters in X. Hence Ind_H^GRes_HV ≅ V⊗CC[G/H] ≅ ⊕_χ∈ XV⊗χ. Replacing V by V⊗χ_0 only permutes the summands. The automorphic identity is the cyclic automorphic induction and base change analogue. It has the same permutation invariance.
Exact mathematical passage
V\otimes\chi_0
\chi_vUse in source · 1 passage
Search aliases: χ_v chi
Use in source · body.tex, line 115
Local class field theory identifies a quotient character χ∈ X with the corresponding Hecke character. Local Langlands is compatible with character twisting, [source reference]. Hence rec(π_v⊗χ_v) ≅ rec(π_v)⊗χ_v ≅ ρ_v⊗χ_v. The result holds at every place.
Exact mathematical passage
\rec(\pi_v\otimes\chi_v) \cong \rec(\pi_v)\otimes\chi_v \cong \rho_v\otimes\chi_v.
\PsiUse in source · 3 passages
Search aliases: Ψ Psi
Use in source · body.tex, line 290
[Stable equality of epsilon factors] Let V,V' be finite dimensional complex Weil representations with finite inertia image, of equal dimension and determinant. For every sufficiently ramified auxiliary character η, ϵ(V⊗η,Ψ)=ϵ(V'⊗η,Ψ). Here Ψ is a fixed nontrivial additive character and the same normalization is used on both sides. Deligne and Henniart's stability theorem applies to the virtual representation B=V-V', which has dimension zero and determinant one. Their Theorem 4.6 gives the ratio …
Exact mathematical passage
\epsilon(V\otimes\eta,\Psi)=\epsilon(V'\otimes\eta,\Psi).
\omegaUse in source · 23 passages
Search aliases: ω omega
Use in source · body.tex, line 197
For a quotient character χ, det(ρ⊗χ)=det(ρ)χ^n. The central character satisfies the identical formula ω_π⊗χ=ω_πχ^n. Since X has prime order p and pnmid n, the map χ↦χ^n is an automorphism of X.
Exact mathematical passage
\omega_{\pi\otimes\chi}=\omega_\pi\chi^n.
\omega^{-a}Use in source · 1 passage
Search aliases: ω^-a omega
Use in source · worked_appendix.tex, line 88
Let Γ={diag(ω^a,ω^-a):a=0,1,2}. The trace alphabet is {2,-1}. The image algebra has dimension two, with basis the identity and γ_1=diag(ω,ω^2). The trace coordinate map from [source reference] gives Φ(γ_0)=(2,-1), Φ(γ_1)=(-1,-1), Φ(γ_2)=(-1,2). The tuples are distinct. The sharper algebra bound is 3≤2^2=4, while the rank only bound is 3≤2^2^2=16. The latter is deliberately rough. Its role is a uniform bound independent of the coefficient prime, not an efficient classification of the group.
Exact mathematical passage
\Gamma=\{\operatorname{diag}(\omega^a,\omega^{-a}):a=0,1,2\}.
\omega^2Use in source · 4 passages
Search aliases: ω^2 omega
Use in source · worked_appendix.tex, line 51
Now compare the balanced matrix A_bal=diag(1,ω,ω^2). Its polynomial is T^3-1. Multiplying all eigenvalues by ω merely permutes their multiset. Every lower coefficient vanishes, and the determinant is fixed. Its full characteristic polynomial is therefore blind to the label. This is an actual local rotational stabilizer. A claim that one characteristic polynomial always detects a quotient twist would fail on this example.
Exact mathematical passage
A_{\mathrm{bal}}=\operatorname{diag}(1,\omega,\omega^2).
\omega^{2j}Use in source · 2 passages
Search aliases: ω^2j omega
Use in source · worked_appendix.tex, line 43
Let ω be a primitive cube root of unity and put A=diag(1,1,ω). Expanding directly gives det(TI-A) =T^3-(2+ω)T^2+(1+2ω)T-ω. The three elementary symmetric coefficients are e_1=2+ω, e_2=1+2ω, e_3=ω. For scalar twist ω^jA, they become ω^j(2+ω), ω^2j(1+2ω), ω^3jω=ω. The determinant is constant for j=0,1,2, but the first coefficient distinguishes the three labels because 2+ω≠0. Thus rank divisibility causes blindness of the determinant, rather than blindness of every coefficient.
Exact mathematical passage
\omega^j(2+\omega),\qquad \omega^{2j}(1+2\omega),\qquad \omega^{3j}\omega=\omega.
\omega^{3j}Use in source · 1 passage
Search aliases: ω^3j omega
Use in source · worked_appendix.tex, line 43
Let ω be a primitive cube root of unity and put A=diag(1,1,ω). Expanding directly gives det(TI-A) =T^3-(2+ω)T^2+(1+2ω)T-ω. The three elementary symmetric coefficients are e_1=2+ω, e_2=1+2ω, e_3=ω. For scalar twist ω^jA, they become ω^j(2+ω), ω^2j(1+2ω), ω^3jω=ω. The determinant is constant for j=0,1,2, but the first coefficient distinguishes the three labels because 2+ω≠0. Thus rank divisibility causes blindness of the determinant, rather than blindness of every coefficient.
Exact mathematical passage
\omega^j(2+\omega),\qquad \omega^{2j}(1+2\omega),\qquad \omega^{3j}\omega=\omega.
\omega^4Use in source · 1 passage
Search aliases: ω^4 omega
Use in source · worked_appendix.tex, line 75
For a balanced multiplicity spectrum 1+z+z^2, both nonzero Fourier modes vanish: 1+ω+ω^2=0, 1+ω^2+ω^4=0. The spectrum 3 has the same total multiplicity, but both nonzero modes equal three. Comparing only ratios for which the source mode is nonzero would omit both decisive comparisons. The zero support clause in [source reference] is therefore necessary. A zero source mode requires a zero target mode. It is not an instruction to skip that mode.
Exact mathematical passage
1+\omega+\omega^2=0,\qquad 1+\omega^2+\omega^4=0.
\omega^aUse in source · 1 passage
Search aliases: ω^a omega
Use in source · worked_appendix.tex, line 88
Let Γ={diag(ω^a,ω^-a):a=0,1,2}. The trace alphabet is {2,-1}. The image algebra has dimension two, with basis the identity and γ_1=diag(ω,ω^2). The trace coordinate map from [source reference] gives Φ(γ_0)=(2,-1), Φ(γ_1)=(-1,-1), Φ(γ_2)=(-1,2). The tuples are distinct. The sharper algebra bound is 3≤2^2=4, while the rank only bound is 3≤2^2^2=16. The latter is deliberately rough. Its role is a uniform bound independent of the coefficient prime, not an efficient classification of the group.
Exact mathematical passage
\Gamma=\{\operatorname{diag}(\omega^a,\omega^{-a}):a=0,1,2\}.
\omega^bUse in source · 1 passage
Search aliases: ω^b omega
Use in source · worked_appendix.tex, line 56
At quotient prime three and rank two, the determinant changes by ω^2j. If a required determinant ratio is ω^b, the unique candidate satisfies 2j=b modulo three, so j=2b. This solves a finite selection problem. It still does not show that every other coefficient of the candidate matches the target.
Exact mathematical passage
\omega^b
\omega^jUse in source · 2 passages
Search aliases: ω^j omega
Use in source · worked_appendix.tex, line 42
Let ω be a primitive cube root of unity and put A=diag(1,1,ω). Expanding directly gives det(TI-A) =T^3-(2+ω)T^2+(1+2ω)T-ω. The three elementary symmetric coefficients are e_1=2+ω, e_2=1+2ω, e_3=ω. For scalar twist ω^jA, they become ω^j(2+ω), ω^2j(1+2ω), ω^3jω=ω. The determinant is constant for j=0,1,2, but the first coefficient distinguishes the three labels because 2+ω≠0. Thus rank divisibility causes blindness of the determinant, rather than blindness of every coefficient.
Exact mathematical passage
\omega^jA
\omega_\piUse in source · 2 passages
Search aliases: ω_π omega pi
Use in source · body.tex, line 197
For a quotient character χ, det(ρ⊗χ)=det(ρ)χ^n. The central character satisfies the identical formula ω_π⊗χ=ω_πχ^n. Since X has prime order p and pnmid n, the map χ↦χ^n is an automorphism of X.
Exact mathematical passage
\omega_{\pi\otimes\chi}=\omega_\pi\chi^n.
\omega_\pi^{dM}Use in source · 1 passage
Search aliases: ω_π^dM omega pi
Use in source · body.tex, line 775
Outside the fixed exceptional set, every unramified determinant value lies in μ_dM. The Hecke character ω_π^dM is therefore trivial at every unramified place outside a finite set. The one dimensional strong multiplicity one statement, or equivalently uniqueness of a Hecke character from its almost everywhere unramified values, makes it the trivial global character. Thus ω_π has finite order. This argument does not require a separate local finiteness assertion at an exceptional place.
Exact mathematical passage
\omega_\pi^{dM}
\omega_{\pi_0}Use in source · 1 passage
Search aliases: ω_π_0 omega pi
Use in source · body.tex, line 216
The base change of ω_π_0 is the central character of Π. Since Π is attached to τ=ρ|_G_E, that character equals detτ. The base change of detρ is also detτ. Hence δ=(detρ)ω_π_0^-1 is a character killed by base change to E, so δ∈ X. We need χ^n=δ. Since X has order p and pnmid n, the nth power map on X is bijective. Thus there is one and only one such χ.
Exact mathematical passage
\omega_{\pi_0}
\omega_{\pi_0}^{-1}Use in source · 1 passage
Search aliases: ω_π_0^-1 omega pi
Use in source · body.tex, line 217
The base change of ω_π_0 is the central character of Π. Since Π is attached to τ=ρ|_G_E, that character equals detτ. The base change of detρ is also detτ. Hence δ=(detρ)ω_π_0^-1 is a character killed by base change to E, so δ∈ X. We need χ^n=δ. Since X has order p and pnmid n, the nth power map on X is bijective. Thus there is one and only one such χ.
Exact mathematical passage
\delta=(\det\rho)\omega_{\pi_0}^{-1}
\omega_{\pi_0\otimes\chi}Use in source · 1 passage
Search aliases: ω_π_0⊗χ omega pi otimes chi
Use in source · body.tex, line 205
[Unique determinant matched descent] Use the cyclic Artin setup of [source reference], and assume pnmid n. Let π_0 be any cuspidal descent of Π. Then there is a unique quotient character χ∈ X such that ω_π_0⊗χ = detρ under global class field theory.
Exact mathematical passage
\boxed{ \omega_{\pi_0\otimes\chi} = \det\rho }
\omega_{\pi\otimes\chi}Use in source · 1 passage
Search aliases: ω_π⊗χ omega pi otimes chi
Use in source · body.tex, line 197
For a quotient character χ, det(ρ⊗χ)=det(ρ)χ^n. The central character satisfies the identical formula ω_π⊗χ=ω_πχ^n. Since X has prime order p and pnmid n, the map χ↦χ^n is an automorphism of X.
Exact mathematical passage
\omega_{\pi\otimes\chi}=\omega_\pi\chi^n.
\toUse in source · 11 passages
Search aliases: → to
Use in source · body.tex, line 981
Use the cross characteristic lifting statement in [source reference], in a splitting modular system compatible with the chosen coefficient place. It gives an ordinary characteristic zero representation of Γ whose reduction is the prescribed residual representation. Compose it with G_F→Γ. Each lifted eigenvalue has order dividing the order of an element of Γ, hence at most B. Every lifted Frobenius polynomial consequently belongs to calQ_n,B.
Exact mathematical passage
G_F\to\Gamma
\mapstoUse in source · 3 passages
Search aliases: ↦ mapsto
Use in source · body.tex, line 200
For a quotient character χ, det(ρ⊗χ)=det(ρ)χ^n. The central character satisfies the identical formula ω_π⊗χ=ω_πχ^n. Since X has prime order p and pnmid n, the map χ↦χ^n is an automorphism of X.
Exact mathematical passage
\chi\mapsto\chi^n
\forallUse in source · 1 passage
Search aliases: ∀ forall
Use in source · formal_appendix.tex, line 655
Let X_r be finite sets with genuine reduction maps π_r,t:X_t→ X_r for r≤ t. Assume identity and composition laws for these maps. Let P_r⊆ X_r be solution sets, and assume reduction preserves solutions. Then (∀ r, P_r≠∅) Longleftrightarrow (∃ (x_r)_r, x_r∈ P_r and π_r,t(x_t)=x_r for r≤ t). The nontrivial implication is finite branch compactness. For each depth, a solution at that depth gives a finite path of solutions at all earlier depths. There are arbitrarily long paths. The resulting tree has…
Exact mathematical passage
\left(\forall r,\ P_r\ne\varnothing\right) \Longleftrightarrow \left(\exists (x_r)_r,\ x_r\in P_r\text{ and } \pi_{r,t}(x_t)=x_r\text{ for }r\le t\right). \label{eq:appendix-branch-criterion}
\existsUse in source · 1 passage
Search aliases: ∃ exists
Use in source · formal_appendix.tex, line 655
Let X_r be finite sets with genuine reduction maps π_r,t:X_t→ X_r for r≤ t. Assume identity and composition laws for these maps. Let P_r⊆ X_r be solution sets, and assume reduction preserves solutions. Then (∀ r, P_r≠∅) Longleftrightarrow (∃ (x_r)_r, x_r∈ P_r and π_r,t(x_t)=x_r for r≤ t). The nontrivial implication is finite branch compactness. For each depth, a solution at that depth gives a finite path of solutions at all earlier depths. There are arbitrarily long paths. The resulting tree has…
Exact mathematical passage
\left(\forall r,\ P_r\ne\varnothing\right) \Longleftrightarrow \left(\exists (x_r)_r,\ x_r\in P_r\text{ and } \pi_{r,t}(x_t)=x_r\text{ for }r\le t\right). \label{eq:appendix-branch-criterion}
\varnothingUse in source · 1 passage
Search aliases: ∅ varnothing
Use in source · formal_appendix.tex, line 655
Let X_r be finite sets with genuine reduction maps π_r,t:X_t→ X_r for r≤ t. Assume identity and composition laws for these maps. Let P_r⊆ X_r be solution sets, and assume reduction preserves solutions. Then (∀ r, P_r≠∅) Longleftrightarrow (∃ (x_r)_r, x_r∈ P_r and π_r,t(x_t)=x_r for r≤ t). The nontrivial implication is finite branch compactness. For each depth, a solution at that depth gives a finite path of solutions at all earlier depths. There are arbitrarily long paths. The resulting tree has…
Exact mathematical passage
\left(\forall r,\ P_r\ne\varnothing\right) \Longleftrightarrow \left(\exists (x_r)_r,\ x_r\in P_r\text{ and } \pi_{r,t}(x_t)=x_r\text{ for }r\le t\right). \label{eq:appendix-branch-criterion}
\notinUse in source · 3 passages
Search aliases: ∉ notin
Use in source · body.tex, line 963
[One large residual realization gives an Artin representation] Let K be a number field, let S be a finite set of places of F, and let calP⊆OO_K[T] be a nonempty finite set of monic degree n polynomials. Assume all their roots are roots of unity. For v∉ S, choose P_v∈calP.
Exact mathematical passage
v\notin S
\prod_iUse in source · 3 passages
Search aliases: ∏_i prod
Use in source · body.tex, line 1140
Over an algebraic closure of the residue field, overlineP^[p](T) = ∏_i(T-overlineα_i^p). The elementary symmetric functions of the pth powers are the pth powers of the elementary symmetric functions in characteristic p. Finite residue fields are perfect, so Frobenius is injective.
Exact mathematical passage
\overline{P^{[p]}}(T) = \prod_i(T-\overline\alpha_i^p).
\sum_iUse in source · 2 passages
Search aliases: ∑_i sum
Use in source · body.tex, line 909
oof Because products of group matrices are group matrices, A is a finite dimensional subalgebra of M_n(k). An A submodule is the same thing as a Γ submodule. The given module is semisimple and faithful for A. The Jacobson radical annihilates every simple constituent, hence annihilates this faithful module. It is therefore zero. The split semisimple algebra has the form A≅⊕_iM_d_i(k), k^n≅⊕_i(k^d_i)^⊕ m_i, ∑_i d_im_i=n. Faithfulness gives m_i≥1 for every block. The trace pairing B(x,y)=Tr_k^n(xy)…
Exact mathematical passage
A\cong\bigoplus_iM_{d_i}(k),\qquad k^n\cong\bigoplus_i(k^{d_i})^{\oplus m_i}, \qquad\sum_i d_im_i=n.
\sum_{i=0}^{d-1}Use in source · 1 passage
Search aliases: ∑_i=0^d-1 sum
Use in source · formal_appendix.tex, line 376
Let E be a complete normed ring with norm1=1, a submultiplicative norm, and the ultrametric inequality normx+y≤max(normx,normy). Assume norm A≤1. For a nonnegative integer d, define D_A,d(h) · =∑_i=0^d-1A^ihA^d-1-i, R_A,d(h) · =(A+h)^d-A^d-D_A,d(h). For d=0 the sum is empty and both maps are zero. The order of the factors in [source reference] matters. Replacing this operator by dA^d-1h requires A and h to commute. The replacement is invalid for general matrices.
Exact mathematical passage
D_{A,d}(h)&=\sum_{i=0}^{d-1}A^ihA^{d-1-i}, \label{eq:appendix-ordered-derivative}\\ R_{A,d}(h)&=(A+h)^d-A^d-D_{A,d}(h). \label{eq:appendix-ordered-remainder}
\sum_{j=0}^{e-1}Use in source · 1 passage
Search aliases: ∑_j=0^e-1 sum
Use in source · formal_appendix.tex, line 57
Suppose now that e is invertible in the coefficient ring. Define S=∑_j=0^e-1jT^j. A direct coefficient calculation gives (T-1)S=e-N. For x∈ker N, take y=e^-1Sx. Then (T-1)y=x, so every cocycle is a coboundary and H^1(C_e,M)=0. The averaging argument uses division by e at its decisive step. In residue characteristic dividing e, the same display remains an identity but no longer constructs y.
Exact mathematical passage
S=\sum_{j=0}^{e-1}jT^j
\sum_{j=0}^{r-1}Use in source · 1 passage
Search aliases: ∑_j=0^r-1 sum
Use in source · formal_appendix.tex, line 37
A one-cocycle is a function a:C_e→ M satisfying a(gh)=a(g)+g· a(h). Putting g=h=1 gives a(1)=0. If x=a(γ), repeated application of the cocycle equation gives a(γ^r)=∑_j=0^r-1T^j x (1≤ r≤ e). The group relation γ^e=1 therefore forces Nx=0. Conversely, if Nx=0, this formula defines a cocycle. Whenever exponents wrap around e, the discrepancy is a translate of Nx, which is zero. Thus evaluating at the chosen generator identifies cocycles with ker N.
Exact mathematical passage
a(\gamma^r)=\sum_{j=0}^{r-1}T^j x\qquad(1\le r\le e).
\sum_{k=0}^{n}Use in source · 1 passage
Search aliases: ∑_k=0^n sum
Use in source · body.tex, line 141
Let ξ be a generator of X, and identify its values with pth roots of unity. For a matrix A∈GL_n(CC), write det(TI-A) = ∑_k=0^n(-1)^k e_k(A)T^n-k.
Exact mathematical passage
\det(TI-A) = \sum_{k=0}^{n}(-1)^k e_k(A)T^{n-k}.
\sum_tUse in source · 1 passage
Search aliases: ∑_t sum
Use in source · formal_appendix.tex, line 448
For E=Mat_n( Q_p) use the entrywise supremum norm and normalize abs p_p=p^-1. Matrix multiplication satisfies abs(XY)_ij_p =|∑_tX_itY_tj|_p ≤max_tabsX_it_pabsY_tj_p ≤norm Xnorm Y. Taking the maximum over the finite set of entries proves submultiplicativity. The same ultrametric calculation removes a matrix dimension factor. Completeness is inherited from the finite product of copies of Q_p. These are the hypotheses required by the actual matrix application, rather than properties borrowed from a…
Exact mathematical passage
\abs{(XY)_{ij}}_p =\left|\sum_tX_{it}Y_{tj}\right|_p \le\max_t\abs{X_{it}}_p\abs{Y_{tj}}_p \le\norm X\norm Y.
\setminusUse in source · 1 passage
Search aliases: ∖ setminus
Use in source · body.tex, line 1433
[Trace separator for irreducible extensions] Let Htriangleleft G with G/H≅ C_p, and let ρ be an irreducible extension of an irreducible G invariant representation of H. Then there exists g∈ G∖ H such that Trρ(g)≠0. Consequently, the p quotient twists of ρ are distinguished by their traces at g.
Exact mathematical passage
g\in G\setminus H
\circUse in source · 1 passage
Search aliases: ∘ circ
Use in source · body.tex, line 263
Every algebraic character of GL_n is an integral power of the determinant. Thus det∘ r=det^a for one integer a. Evaluate both sides at zI_n. The left side is z^wN, while the right side is z^an.
Exact mathematical passage
\det\circ r=\det^a
\simeqUse in source · 2 passages
Search aliases: ≃ simeq
Use in source · formal_appendix.tex, line 47
For a zero-cochain y∈ M, its coboundary at γ is (T-1)y. Taking the quotient by these changes gives the precise presentation H^1(C_e,M)≃ ker N/im(T-1). If the norm operator is identically zero, this reduces to M/im(T-1). That reduction has a hypothesis. Dropping the kernel from the general formula introduces classes that cannot satisfy the cyclic relation.
Exact mathematical passage
H^1(C_e,M)\simeq \ker N/\operatorname{im}(T-1). \label{eq:appendix-cyclic-H1}
\neUse in source · 5 passages
Search aliases: ≠ ne
Use in source · formal_appendix.tex, line 194
For a proposed correction b, sum the equations c(γ^i,γ)=b(γ)-b(γ^i+1)+b(γ^i) for i=0,1,2. The right side telescopes to 3b(γ)=0, using b(1)=0 and the trivial action. The left side is 9≠0 in I. There is therefore no primitive. This explicit calculation recovers the nonzero obstruction class directly from its normalized representative.
Exact mathematical passage
9\ne0
\equivUse in source · 4 passages
Search aliases: ≡ equiv
Use in source · body.tex, line 1042
two extensions of the same absolutely irreducible representation of G_E. Their ratio is a character of G_F/G_E. Since the residue characteristic is different from p, the reduced quotient character has order p, and there is a unique index j_λ with overline r_π,λ ≅ overlineρ⊗overlineϵ^j_λ. At every good unramified place, P_π,v ≡ P_ρ⊗ϵ^j_λ,v pmodλ. Both polynomials belong to the finite alphabet calA. The prime is separating, so the congruence is equality in characteristic zero. Uniqueness follows f…
Exact mathematical passage
P_{\pi,v} \equiv P_{\rho\otimes\epsilon^{j_{\lambda}},v} \pmod\lambda.
\leUse in source · 36 passages
Search aliases: ≤ le
Use in source · body.tex, line 900
[Finite trace image bound] Let k be algebraically closed of characteristic zero or characteristic ℓ>n. Let Γ⊆GL_n(k) be a subgroup, and assume its defining module k^n is semisimple. Suppose its trace set is finite, with t=#{Tr(γ):γ∈Γ}. Then Γ is finite and absΓ≤ t^dim_k A≤ t^n^2, A=span_kΓ. In particular this bounds a finite semisimple residual matrix group in characteristic greater than its rank.
Exact mathematical passage
\abs\Gamma\le t^{\dim_k A}\le t^{n^2}, \qquad A=\operatorname{span}_k\Gamma.
\leqUse in source · 1 passage
Search aliases: ≤ leq
Use in source · body.tex, line 943
[Uniform bound for a residual Galois image] Let overline r:G_F⟶GL_n(overlineFF_ℓ) be continuous and semisimple, with ℓ>n. If its Frobenius traces belong to a set of size at most t on a Chebotarev dense set, then absimoverline r≤ t^n^2.
Exact mathematical passage
\boxed{ \abs{\im\overline r}\leq t^{n^2}. }
\geUse in source · 4 passages
Search aliases: ≥ ge
Use in source · body.tex, line 914
odule is the same thing as a Γ submodule. The given module is semisimple and faithful for A. The Jacobson radical annihilates every simple constituent, hence annihilates this faithful module. It is therefore zero. The split semisimple algebra has the form A≅⊕_iM_d_i(k), k^n≅⊕_i(k^d_i)^⊕ m_i, ∑_i d_im_i=n. Faithfulness gives m_i≥1 for every block. The trace pairing B(x,y)=Tr_k^n(xy) restricts on the ith block to m_iTr(XY). Matrix trace pairing is nondegenerate, as is seen by pairing matrix units …
Exact mathematical passage
m_i\ge1
\subseteqUse in source · 3 passages
Search aliases: ⊆ subseteq
Use in source · body.tex, line 895
[Finite trace image bound] Let k be algebraically closed of characteristic zero or characteristic ℓ>n. Let Γ⊆GL_n(k) be a subgroup, and assume its defining module k^n is semisimple. Suppose its trace set is finite, with t=#{Tr(γ):γ∈Γ}. Then Γ is finite and absΓ≤ t^dim_k A≤ t^n^2, A=span_kΓ. In particular this bounds a finite semisimple residual matrix group in characteristic greater than its rank.
Exact mathematical passage
\Gamma\subseteq\GL_n(k)
\bigoplus_iUse in source · 1 passage
Search aliases: ⊕_i bigoplus
Use in source · body.tex, line 909
oof Because products of group matrices are group matrices, A is a finite dimensional subalgebra of M_n(k). An A submodule is the same thing as a Γ submodule. The given module is semisimple and faithful for A. The Jacobson radical annihilates every simple constituent, hence annihilates this faithful module. It is therefore zero. The split semisimple algebra has the form A≅⊕_iM_d_i(k), k^n≅⊕_i(k^d_i)^⊕ m_i, ∑_i d_im_i=n. Faithfulness gives m_i≥1 for every block. The trace pairing B(x,y)=Tr_k^n(xy)…
Exact mathematical passage
A\cong\bigoplus_iM_{d_i}(k),\qquad k^n\cong\bigoplus_i(k^{d_i})^{\oplus m_i}, \qquad\sum_i d_im_i=n.
\bigoplus_{j=0}^{p-1}Use in source · 1 passage
Search aliases: ⊕_j=0^p-1 bigoplus
Use in source · body.tex, line 1568
The automorphic induction of the attached representation over E is already attached to Ind_G_E^G_Fτ ≅ ⊕_j=0^p-1 ρ⊗ϵ^j. It is also the isobaric sum of all descents. By [source reference], this total object is invariant under rotating either torsor. It proves the direct sum correspondence but cannot identify one constituent with the target extension.
Exact mathematical passage
\Ind_{G_E}^{G_F}\tau \cong \bigoplus_{j=0}^{p-1} \rho\otimes\epsilon^j.
\bigoplus_{\chi\in X}Use in source · 2 passages
Search aliases: ⊕_χ∈ X bigoplus chi in
Use in source · body.tex, line 317
[The total induced object is torsor blind] Let Htriangleleft G with cyclic quotient character group X, and let V be an extension of an H representation. Then ⊕_χ∈ X V⊗χ ≅ Ind_H^GRes_HV. The left side is unchanged when V is replaced by another member of its extension torsor. Likewise, for cyclic base change, mathopboxplus_χ∈ X π⊗χ ≅ AI_E/F(BC_E/Fπ) inside the established automorphic induction formalism. Therefore comparison of the total induced objects cannot determine the cyclic attachment offse…
Exact mathematical passage
\boxed{ \bigoplus_{\chi\in X} V\otimes\chi \cong \Ind_H^G\Res_HV. }
\bigcapUse in source · 2 passages
Search aliases: ⋂ bigcap
Use in source · body.tex, line 469
[Structure of the alignment set] The alignment set is empty or a coset of the common rotational stabilizer ⋂_τ {j:z^jP_τ,V=P_τ,V}. In particular, one stable type with aperiodic spectrum makes the alignment index unique whenever it exists.
Exact mathematical passage
\bigcap_\tau \{j:z^jP_{\tau,V}=P_{\tau,V}\}.
\bigcap_vUse in source · 1 passage
Search aliases: ⋂_v bigcap
Use in source · body.tex, line 650
[Global cyclic attachment criterion] With the preceding notation, the descent π_0⊗ϵ^j is attached to ρ if and only if j∈⋂_vcalJ_v. Consequently, the cyclic Strong Artin step succeeds exactly when this intersection is nonempty.
Exact mathematical passage
\boxed{ j\in\bigcap_v\calJ_v. }
\bigcap_\tauUse in source · 1 passage
Search aliases: ⋂_τ bigcap tau
Use in source · body.tex, line 469
[Structure of the alignment set] The alignment set is empty or a coset of the common rotational stabilizer ⋂_τ {j:z^jP_τ,V=P_τ,V}. In particular, one stable type with aperiodic spectrum makes the alignment index unique whenever it exists.
Exact mathematical passage
\bigcap_\tau \{j:z^jP_{\tau,V}=P_{\tau,V}\}.
\longmapstoUse in source · 2 passages
Search aliases: ⟼ longmapsto
Use in source · body.tex, line 1132
[Frobenius transform after reduction] Let P have integral coefficients at a place of residue characteristic p. Then overlineP^[p](T) = Frob(overline P(T)), where Frobenius raises every coefficient to its pth power. In particular, the map overline P⟼overlineP^[p] is injective over the residue field.
Exact mathematical passage
\overline P\longmapsto\overline{P^{[p]}}
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