Notation by work
Hilbert 12: analytic Fourier branch
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152 symbol spellings in 160 mathematical passages across 3 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 152 entries are available below.
+Use in source · 42 passages
Search aliases: +
Use in source · H12_Analytic_Fourier_Resolvent_Research_Note.md, line 11
For the number field K= Q(β), β^3=7, the previously constructed unramified cubic is generated by a root of f(X)=X^3-β X^2+1. Its cyclic Galois action extends to a globally defined order three Möbius transformation on P^1_K. The quotient coordinate d(X) presents a rational one parameter family of cyclic cubic polynomials whose discriminants are perfect squares. Its special fiber at d=0 is the original arithmetic cubic. The associated two primitive Fourier resolvents have exact, explicitly deforme…
Exact mathematical passage
f(X)=X^3-\beta X^2+1
=Use in source · 95 passages
Search aliases: =
Use in source · H12_Analytic_Fourier_Resolvent_Research_Note.md, line 11
For the number field K= Q(β), β^3=7, the previously constructed unramified cubic is generated by a root of f(X)=X^3-β X^2+1. Its cyclic Galois action extends to a globally defined order three Möbius transformation on P^1_K. The quotient coordinate d(X) presents a rational one parameter family of cyclic cubic polynomials whose discriminants are perfect squares. Its special fiber at d=0 is the original arithmetic cubic. The associated two primitive Fourier resolvents have exact, explicitly deforme…
Exact mathematical passage
K=\mathbb Q(\beta)
AUse in source · 8 passages
Search aliases: A
Use in source · H12_Analytic_Fourier_Resolvent_Research_Note.md, line 42
The previously constructed algebraic candidate satisfies Y_rm alg=Av^2+Bv+C where
Exact mathematical passage
Y_{\rm alg}=Av^2+Bv+C
aUse in source · 2 passages
Search aliases: a
Use in source · H12_Analytic_Fourier_Resolvent_Research_Note.md, line 42
The previously constructed algebraic candidate satisfies Y_rm alg=Av^2+Bv+C where
Exact mathematical passage
Y_{\rm alg}=Av^2+Bv+C
A^2Use in source · 2 passages
Search aliases: A^2
Use in source · H12_Analytic_Fourier_Resolvent_Research_Note.md, line 187
D_Y(Y) · =A(Y-C)+B^2+β AB, N_Y(Y) · =(Y-C)(B+β A)-A^2, c · =3(652β^2-1230β-33).
Exact mathematical passage
\begin{aligned} D_Y(Y)&=A(Y-C)+B^2+\beta AB,\\ N_Y(Y)&=(Y-C)(B+\beta A)-A^2,\\ c&=3(652\beta^2-1230\beta-33). \end{aligned}
\mathrm{alg}Use in source · 1 passage
Search aliases: alg mathrm
Use in source · H12_Analytic_Fourier_Resolvent_Research_Note.md, line 304
The orbit laws are **conjectural at the analytic gamma special values**, even though Ψ^3=1 is an exact algebraic theorem. If all three special-value identities hold, the established branch separation identifies U_Γ=U_alg. The nontrivial analytical content has not been proved by the algebraic construction.
Exact mathematical passage
U_\Gamma=U_{\mathrm{alg}}
BUse in source · 7 passages
Search aliases: B
Use in source · H12_Analytic_Fourier_Resolvent_Research_Note.md, line 42
The previously constructed algebraic candidate satisfies Y_rm alg=Av^2+Bv+C where
Exact mathematical passage
Y_{\rm alg}=Av^2+Bv+C
bUse in source · 1 passage
Search aliases: b
Use in source · H12_Analytic_Fourier_Resolvent_Research_Note.md, line 340
F_(t-a)/n,(s+b)/n(z) =∏_u=1-n^0∏_v=0^n-1 F_t,s(z+ufract-an+vfracs+bn).
Exact mathematical passage
F_{(t-a)/n,(s+b)/n}(z) =\prod_{u=1-n}^0\prod_{v=0}^{n-1} F_{t,s}\left(z+u\frac{t-a}{n}+v\frac{s+b}{n}\right).
B^2Use in source · 1 passage
Search aliases: B^2
Use in source · H12_Analytic_Fourier_Resolvent_Research_Note.md, line 187
D_Y(Y) · =A(Y-C)+B^2+β AB, N_Y(Y) · =(Y-C)(B+β A)-A^2, c · =3(652β^2-1230β-33).
Exact mathematical passage
\begin{aligned} D_Y(Y)&=A(Y-C)+B^2+\beta AB,\\ N_Y(Y)&=(Y-C)(B+\beta A)-A^2,\\ c&=3(652\beta^2-1230\beta-33). \end{aligned}
CUse in source · 8 passages
Search aliases: C
Use in source · H12_Analytic_Fourier_Resolvent_Research_Note.md, line 42
The previously constructed algebraic candidate satisfies Y_rm alg=Av^2+Bv+C where
Exact mathematical passage
Y_{\rm alg}=Av^2+Bv+C
cUse in source · 6 passages
Search aliases: c
Use in source · H12_Analytic_Fourier_Resolvent_Research_Note.md, line 187
D_Y(Y) · =A(Y-C)+B^2+β AB, N_Y(Y) · =(Y-C)(B+β A)-A^2, c · =3(652β^2-1230β-33).
Exact mathematical passage
\begin{aligned} D_Y(Y)&=A(Y-C)+B^2+\beta AB,\\ N_Y(Y)&=(Y-C)(B+\beta A)-A^2,\\ c&=3(652\beta^2-1230\beta-33). \end{aligned}
C_3Use in source · 1 passage
Search aliases: C_3
Use in source · H12_Analytic_Fourier_Resolvent_Research_Note.md, line 356
This identifies an obstruction rather than suppressing it. Every generic fiber d carries the same abstract C_3 symmetry. Symmetry, perfect-square discriminant, and Fourier diagonalization alone cannot force d=0. The missing input must be analytic and arithmetic normalization, not another abstract finite-group identity.
Exact mathematical passage
C_3
DUse in source · 10 passages
Search aliases: D
Use in source · H12_Analytic_Fourier_Resolvent_Research_Note.md, line 119
*Proof.* Work over K(X). Write D=(β^2X-4)(β^2X-5). Expanding the symmetric functions gives the three exact numerator identities
Exact mathematical passage
D=(\beta^2X-4)(\beta^2X-5)
dUse in source · 34 passages
Search aliases: d
Use in source · H12_Analytic_Fourier_Resolvent_Research_Note.md, line 11
For the number field K= Q(β), β^3=7, the previously constructed unramified cubic is generated by a root of f(X)=X^3-β X^2+1. Its cyclic Galois action extends to a globally defined order three Möbius transformation on P^1_K. The quotient coordinate d(X) presents a rational one parameter family of cyclic cubic polynomials whose discriminants are perfect squares. Its special fiber at d=0 is the original arithmetic cubic. The associated two primitive Fourier resolvents have exact, explicitly deforme…
Exact mathematical passage
d(X)
d^2Use in source · 2 passages
Search aliases: d^2
Use in source · H12_Analytic_Fourier_Resolvent_Research_Note.md, line 113
Disc_Z(P_d)=(49d^2-13d+1)^2.
Exact mathematical passage
\boxed{\operatorname{Disc}_Z(P_d)=(49d^2-13d+1)^2.}
D_YUse in source · 8 passages
Search aliases: D_Y
Use in source · H12_Analytic_Fourier_Resolvent_Research_Note.md, line 187
D_Y(Y) · =A(Y-C)+B^2+β AB, N_Y(Y) · =(Y-C)(B+β A)-A^2, c · =3(652β^2-1230β-33).
Exact mathematical passage
\begin{aligned} D_Y(Y)&=A(Y-C)+B^2+\beta AB,\\ N_Y(Y)&=(Y-C)(B+\beta A)-A^2,\\ c&=3(652\beta^2-1230\beta-33). \end{aligned}
D_Y'Use in source · 1 passage
Search aliases: D_Y'
Use in source · H12_Analytic_Fourier_Resolvent_Research_Note.md, line 205
N_Y'(Y)D_Y(Y)-N_Y(Y)D_Y'(Y)=c≠0.
Exact mathematical passage
\boxed{N_Y'(Y)D_Y(Y)-N_Y(Y)D_Y'(Y)=c\ne0.}
d_\GammaUse in source · 7 passages
Search aliases: d_Γ Gamma
Use in source · H12_Analytic_Fourier_Resolvent_Research_Note.md, line 11
ect squares. Its special fiber at d=0 is the original arithmetic cubic. The associated two primitive Fourier resolvents have exact, explicitly deformed norm and cube formulas. A rational inverse of the earlier quadratic arithmetic coordinate then turns the elliptic gamma function into a canonical meromorphic candidate for the cubic generator and supplies a single scalar analytic obstruction d_Γ(z). The three existing gamma charts agree numerically with the expected Möbius orbit at z=1/3, but not…
Exact mathematical passage
d_\Gamma(z)
\detUse in source · 1 passage
Search aliases: det
Use in source · H12_Analytic_Fourier_Resolvent_Research_Note.md, line 79
*Proof.* Direct calculation gives det G= -20+3β^3=1 and trG=1. Cayley–Hamilton proves projective order three. Multiplying each fractional formula by its denominator and reducing modulo v^3-β v^2+1 proves the polynomial expressions. The three roots are distinct because the discriminant is one. The nontrivial maps cyclically permute them. square
Exact mathematical passage
\det G= -20+3\beta^3=1
\operatorname{Disc}Use in source · 1 passage
Search aliases: Disc operatorname
Use in source · H12_Analytic_Fourier_Resolvent_Research_Note.md, line 113
Disc_Z(P_d)=(49d^2-13d+1)^2.
Exact mathematical passage
\boxed{\operatorname{Disc}_Z(P_d)=(49d^2-13d+1)^2.}
\operatorname{Disc}_ZUse in source · 1 passage
Search aliases: Disc_Z operatorname
Use in source · H12_Analytic_Fourier_Resolvent_Research_Note.md, line 113
Disc_Z(P_d)=(49d^2-13d+1)^2.
Exact mathematical passage
\boxed{\operatorname{Disc}_Z(P_d)=(49d^2-13d+1)^2.}
eUse in source · 7 passages
Search aliases: e
Use in source · H12_Analytic_Fourier_Resolvent_Research_Note.md, line 37
F(z)=fracΓ_rm ell(z;τ,s)^5Γ_rm ell(5z;5τ,5s), U_Γ=F(1/3), Y_Γ=U_Γ+U_Γ^-1.
Exact mathematical passage
F(z)=\frac{\Gamma_{\rm ell}(z;\tau,s)^5}{\Gamma_{\rm ell}(5z;5\tau,5s)}, \qquad U_\Gamma=F(1/3),\qquad Y_\Gamma=U_\Gamma+U_\Gamma^{-1}.
e^{2\pi in(s-z)}Use in source · 1 passage
Search aliases: e^2π in(s-z) pi
Use in source · H12_Analytic_Fourier_Resolvent_Research_Note.md, line 310
log F_t,s(z)=-5∑_substackn≥1 5nmid n frace^2π in(t+z)-e^2π in(s-z) n(1-e^2π int)(1-e^2π ins).
Exact mathematical passage
\log F_{t,s}(z)=-5\sum_{\substack{n\ge1\\5\nmid n}} \frac{e^{2\pi in(t+z)}-e^{2\pi in(s-z)}} {n(1-e^{2\pi int})(1-e^{2\pi ins})}.
e^{2\pi in(t+z)}Use in source · 1 passage
Search aliases: e^2π in(t+z) pi
Use in source · H12_Analytic_Fourier_Resolvent_Research_Note.md, line 310
log F_t,s(z)=-5∑_substackn≥1 5nmid n frace^2π in(t+z)-e^2π in(s-z) n(1-e^2π int)(1-e^2π ins).
Exact mathematical passage
\log F_{t,s}(z)=-5\sum_{\substack{n\ge1\\5\nmid n}} \frac{e^{2\pi in(t+z)}-e^{2\pi in(s-z)}} {n(1-e^{2\pi int})(1-e^{2\pi ins})}.
e^{2\pi ins}Use in source · 1 passage
Search aliases: e^2π ins pi
Use in source · H12_Analytic_Fourier_Resolvent_Research_Note.md, line 310
log F_t,s(z)=-5∑_substackn≥1 5nmid n frace^2π in(t+z)-e^2π in(s-z) n(1-e^2π int)(1-e^2π ins).
Exact mathematical passage
\log F_{t,s}(z)=-5\sum_{\substack{n\ge1\\5\nmid n}} \frac{e^{2\pi in(t+z)}-e^{2\pi in(s-z)}} {n(1-e^{2\pi int})(1-e^{2\pi ins})}.
e^{2\pi int}Use in source · 1 passage
Search aliases: e^2π int pi
Use in source · H12_Analytic_Fourier_Resolvent_Research_Note.md, line 310
log F_t,s(z)=-5∑_substackn≥1 5nmid n frace^2π in(t+z)-e^2π in(s-z) n(1-e^2π int)(1-e^2π ins).
Exact mathematical passage
\log F_{t,s}(z)=-5\sum_{\substack{n\ge1\\5\nmid n}} \frac{e^{2\pi in(t+z)}-e^{2\pi in(s-z)}} {n(1-e^{2\pi int})(1-e^{2\pi ins})}.
e^{2\pi is}Use in source · 1 passage
Search aliases: e^2π is pi
Use in source · H12_Analytic_Fourier_Resolvent_Research_Note.md, line 316
For real z, writing r=max(|e^2π it|,|e^2π is|) and q=min(…), the tail after N terms is bounded by
Exact mathematical passage
r=\max(|e^{2\pi it}|,|e^{2\pi is}|)
e^{2\pi it}Use in source · 1 passage
Search aliases: e^2π it pi
Use in source · H12_Analytic_Fourier_Resolvent_Research_Note.md, line 316
For real z, writing r=max(|e^2π it|,|e^2π is|) and q=min(…), the tail after N terms is bounded by
Exact mathematical passage
r=\max(|e^{2\pi it}|,|e^{2\pi is}|)
e_1Use in source · 3 passages
Search aliases: e_1
Use in source · H12_Analytic_Fourier_Resolvent_Research_Note.md, line 97
e_1 · =β(1+7d), e_2 · =9β^2d, e_3 · =-1+20d.
Exact mathematical passage
\begin{aligned} e_1&=\beta(1+7d),\\ e_2&=9\beta^2d,\\ e_3&=-1+20d. \end{aligned}
e_1^2Use in source · 1 passage
Search aliases: e_1^2
Use in source · H12_Analytic_Fourier_Resolvent_Research_Note.md, line 159
*Proof.* The norm identity follows from LL'=e_1^2-3e_2. The discriminant gives the square of the Vandermonde. Its orientation is fixed by the root ordering X,Φ(X),Φ^2(X), which at d=0 has Vandermonde +1. Hence
Exact mathematical passage
LL'=e_1^2-3e_2
e_1^3Use in source · 1 passage
Search aliases: e_1^3
Use in source · H12_Analytic_Fourier_Resolvent_Research_Note.md, line 167
L^3+(L')^3=27e_3-e_1^3+3e_1LL'=(98d-13)q(d).
Exact mathematical passage
L^3+(L')^3=27e_3-e_1^3+3e_1LL'=(98d-13)q(d).
e_2Use in source · 3 passages
Search aliases: e_2
Use in source · H12_Analytic_Fourier_Resolvent_Research_Note.md, line 97
e_1 · =β(1+7d), e_2 · =9β^2d, e_3 · =-1+20d.
Exact mathematical passage
\begin{aligned} e_1&=\beta(1+7d),\\ e_2&=9\beta^2d,\\ e_3&=-1+20d. \end{aligned}
e_3Use in source · 3 passages
Search aliases: e_3
Use in source · H12_Analytic_Fourier_Resolvent_Research_Note.md, line 97
e_1 · =β(1+7d), e_2 · =9β^2d, e_3 · =-1+20d.
Exact mathematical passage
\begin{aligned} e_1&=\beta(1+7d),\\ e_2&=9\beta^2d,\\ e_3&=-1+20d. \end{aligned}
FUse in source · 8 passages
Search aliases: F
Use in source · H12_Analytic_Fourier_Resolvent_Research_Note.md, line 37
F(z)=fracΓ_rm ell(z;τ,s)^5Γ_rm ell(5z;5τ,5s), U_Γ=F(1/3), Y_Γ=U_Γ+U_Γ^-1.
Exact mathematical passage
F(z)=\frac{\Gamma_{\rm ell}(z;\tau,s)^5}{\Gamma_{\rm ell}(5z;5\tau,5s)}, \qquad U_\Gamma=F(1/3),\qquad Y_\Gamma=U_\Gamma+U_\Gamma^{-1}.
fUse in source · 8 passages
Search aliases: f
Use in source · H12_Analytic_Fourier_Resolvent_Research_Note.md, line 11
For the number field K= Q(β), β^3=7, the previously constructed unramified cubic is generated by a root of f(X)=X^3-β X^2+1. Its cyclic Galois action extends to a globally defined order three Möbius transformation on P^1_K. The quotient coordinate d(X) presents a rational one parameter family of cyclic cubic polynomials whose discriminants are perfect squares. Its special fiber at d=0 is the original arithmetic cubic. The associated two primitive Fourier resolvents have exact, explicitly deforme…
Exact mathematical passage
f(X)=X^3-\beta X^2+1
F_{(t-a)/n,(s+b)/n}Use in source · 1 passage
Search aliases: F_(t-a)/n,(s+b)/n
Use in source · H12_Analytic_Fourier_Resolvent_Research_Note.md, line 340
F_(t-a)/n,(s+b)/n(z) =∏_u=1-n^0∏_v=0^n-1 F_t,s(z+ufract-an+vfracs+bn).
Exact mathematical passage
F_{(t-a)/n,(s+b)/n}(z) =\prod_{u=1-n}^0\prod_{v=0}^{n-1} F_{t,s}\left(z+u\frac{t-a}{n}+v\frac{s+b}{n}\right).
F_{t,s}Use in source · 2 passages
Search aliases: F_t,s
Use in source · H12_Analytic_Fourier_Resolvent_Research_Note.md, line 310
log F_t,s(z)=-5∑_substackn≥1 5nmid n frace^2π in(t+z)-e^2π in(s-z) n(1-e^2π int)(1-e^2π ins).
Exact mathematical passage
\log F_{t,s}(z)=-5\sum_{\substack{n\ge1\\5\nmid n}} \frac{e^{2\pi in(t+z)}-e^{2\pi in(s-z)}} {n(1-e^{2\pi int})(1-e^{2\pi ins})}.
F_{t_n,s_n}Use in source · 1 passage
Search aliases: F_t_n,s_n
Use in source · H12_Analytic_Fourier_Resolvent_Research_Note.md, line 292
Define U_n=F_t_n,s_n(1/3) and Y_n=U_n+U_n^-1. The exact sufficient analytic obligations now are
Exact mathematical passage
U_n=F_{t_n,s_n}(1/3)
GUse in source · 6 passages
Search aliases: G
Use in source · H12_Analytic_Fourier_Resolvent_Research_Note.md, line 61
G=5 · -3β β^2 · -4
Exact mathematical passage
G=\begin{pmatrix}5&-3\beta\\ \beta^2&-4\end{pmatrix}
gUse in source · 1 passage
Search aliases: g
Use in source · H12_Analytic_Fourier_Resolvent_Research_Note.md, line 42
The previously constructed algebraic candidate satisfies Y_rm alg=Av^2+Bv+C where
Exact mathematical passage
Y_{\rm alg}=Av^2+Bv+C
G^2Use in source · 1 passage
Search aliases: G^2
Use in source · H12_Analytic_Fourier_Resolvent_Research_Note.md, line 65
has determinant one and trace one. In particular G^2-G+I=0 and G^3=-I. Its action on P^1_K is
Exact mathematical passage
G^2-G+I=0
G^3Use in source · 1 passage
Search aliases: G^3
Use in source · H12_Analytic_Fourier_Resolvent_Research_Note.md, line 65
has determinant one and trace one. In particular G^2-G+I=0 and G^3=-I. Its action on P^1_K is
Exact mathematical passage
G^3=-I
hUse in source · 6 passages
Search aliases: h
Use in source · H12_Analytic_Fourier_Resolvent_Research_Note.md, line 209
Hence h(Y)=N_Y(Y)/D_Y(Y) is a Möbius transformation. On the algebraic cubic it inverts Y=Av^2+Bv+C.
Exact mathematical passage
h(Y)=N_Y(Y)/D_Y(Y)
h^{-1}Use in source · 1 passage
Search aliases: h^-1
Use in source · H12_Analytic_Fourier_Resolvent_Research_Note.md, line 252
Ψ=h^-1∘Φ∘ h,
Exact mathematical passage
\Psi=h^{-1}\circ\Phi\circ h,
IUse in source · 2 passages
Search aliases: I
Use in source · H12_Analytic_Fourier_Resolvent_Research_Note.md, line 65
has determinant one and trace one. In particular G^2-G+I=0 and G^3=-I. Its action on P^1_K is
Exact mathematical passage
G^2-G+I=0
iUse in source · 2 passages
Search aliases: i
Use in source · H12_Analytic_Fourier_Resolvent_Research_Note.md, line 310
log F_t,s(z)=-5∑_substackn≥1 5nmid n frace^2π in(t+z)-e^2π in(s-z) n(1-e^2π int)(1-e^2π ins).
Exact mathematical passage
\log F_{t,s}(z)=-5\sum_{\substack{n\ge1\\5\nmid n}} \frac{e^{2\pi in(t+z)}-e^{2\pi in(s-z)}} {n(1-e^{2\pi int})(1-e^{2\pi ins})}.
JUse in source · 4 passages
Search aliases: J
Use in source · H12_Analytic_Fourier_Resolvent_Research_Note.md, line 17
ω^2+ω+1=0, β=7^1/3ω^2, J=2+3ω, overline J=-1-3ω.
Exact mathematical passage
\omega^2+\omega+1=0,\qquad \beta=7^{1/3}\omega^2,\qquad J=2+3\omega, \qquad \overline J=-1-3\omega.
J^2Use in source · 3 passages
Search aliases: J^2
Use in source · H12_Analytic_Fourier_Resolvent_Research_Note.md, line 154
L^3=(J^2+49d)q(d), (L')^3=(overline J^ 2+49d)q(d).
Exact mathematical passage
\boxed{L^3=(J^2+49d)q(d),\qquad (L')^3=(\overline J^{\,2}+49d)q(d).}
KUse in source · 10 passages
Search aliases: K
Use in source · H12_Analytic_Fourier_Resolvent_Research_Note.md, line 11
For the number field K= Q(β), β^3=7, the previously constructed unramified cubic is generated by a root of f(X)=X^3-β X^2+1. Its cyclic Galois action extends to a globally defined order three Möbius transformation on P^1_K. The quotient coordinate d(X) presents a rational one parameter family of cyclic cubic polynomials whose discriminants are perfect squares. Its special fiber at d=0 is the original arithmetic cubic. The associated two primitive Fourier resolvents have exact, explicitly deforme…
Exact mathematical passage
K=\mathbb Q(\beta)
LUse in source · 7 passages
Search aliases: L
Use in source · H12_Analytic_Fourier_Resolvent_Research_Note.md, line 141
L=X_0+ω X_1+ω^2X_2, L'=X_0+ω^2X_1+ω X_2.
Exact mathematical passage
L=X_0+\omega X_1+\omega^2X_2,\qquad L'=X_0+\omega^2X_1+\omega X_2.
lUse in source · 2 passages
Search aliases: l
Use in source · H12_Analytic_Fourier_Resolvent_Research_Note.md, line 37
F(z)=fracΓ_rm ell(z;τ,s)^5Γ_rm ell(5z;5τ,5s), U_Γ=F(1/3), Y_Γ=U_Γ+U_Γ^-1.
Exact mathematical passage
F(z)=\frac{\Gamma_{\rm ell}(z;\tau,s)^5}{\Gamma_{\rm ell}(5z;5\tau,5s)}, \qquad U_\Gamma=F(1/3),\qquad Y_\Gamma=U_\Gamma+U_\Gamma^{-1}.
L'Use in source · 7 passages
Search aliases: L'
Use in source · H12_Analytic_Fourier_Resolvent_Research_Note.md, line 141
L=X_0+ω X_1+ω^2X_2, L'=X_0+ω^2X_1+ω X_2.
Exact mathematical passage
L=X_0+\omega X_1+\omega^2X_2,\qquad L'=X_0+\omega^2X_1+\omega X_2.
L^3Use in source · 4 passages
Search aliases: L^3
Use in source · H12_Analytic_Fourier_Resolvent_Research_Note.md, line 154
L^3=(J^2+49d)q(d), (L')^3=(overline J^ 2+49d)q(d).
Exact mathematical passage
\boxed{L^3=(J^2+49d)q(d),\qquad (L')^3=(\overline J^{\,2}+49d)q(d).}
\logUse in source · 1 passage
Search aliases: log
Use in source · H12_Analytic_Fourier_Resolvent_Research_Note.md, line 310
log F_t,s(z)=-5∑_substackn≥1 5nmid n frace^2π in(t+z)-e^2π in(s-z) n(1-e^2π int)(1-e^2π ins).
Exact mathematical passage
\log F_{t,s}(z)=-5\sum_{\substack{n\ge1\\5\nmid n}} \frac{e^{2\pi in(t+z)}-e^{2\pi in(s-z)}} {n(1-e^{2\pi int})(1-e^{2\pi ins})}.
MUse in source · 2 passages
Search aliases: M
Use in source · H12_Analytic_Fourier_Resolvent_Research_Note.md, line 256
where h(Y)=N_Y(Y)/D_Y(Y). If M_h is its 2×2 coefficient matrix, then the matrix for Ψ is M_h^-1GM_h. Its determinant is one and trace is one, so it also has order three.
Exact mathematical passage
M_h
M_hUse in source · 2 passages
Search aliases: M_h
Use in source · H12_Analytic_Fourier_Resolvent_Research_Note.md, line 256
where h(Y)=N_Y(Y)/D_Y(Y). If M_h is its 2×2 coefficient matrix, then the matrix for Ψ is M_h^-1GM_h. Its determinant is one and trace is one, so it also has order three.
Exact mathematical passage
M_h
M_h^{-1}Use in source · 1 passage
Search aliases: M_h^-1
Use in source · H12_Analytic_Fourier_Resolvent_Research_Note.md, line 256
where h(Y)=N_Y(Y)/D_Y(Y). If M_h is its 2×2 coefficient matrix, then the matrix for Ψ is M_h^-1GM_h. Its determinant is one and trace is one, so it also has order three.
Exact mathematical passage
M_h^{-1}GM_h
\maxUse in source · 1 passage
Search aliases: max
Use in source · H12_Analytic_Fourier_Resolvent_Research_Note.md, line 316
For real z, writing r=max(|e^2π it|,|e^2π is|) and q=min(…), the tail after N terms is bounded by
Exact mathematical passage
r=\max(|e^{2\pi it}|,|e^{2\pi is}|)
\minUse in source · 1 passage
Search aliases: min
Use in source · H12_Analytic_Fourier_Resolvent_Research_Note.md, line 316
For real z, writing r=max(|e^2π it|,|e^2π is|) and q=min(…), the tail after N terms is bounded by
Exact mathematical passage
q=\min(\ldots)
NUse in source · 11 passages
Search aliases: N
Use in source · H12_Analytic_Fourier_Resolvent_Research_Note.md, line 187
D_Y(Y) · =A(Y-C)+B^2+β AB, N_Y(Y) · =(Y-C)(B+β A)-A^2, c · =3(652β^2-1230β-33).
Exact mathematical passage
\begin{aligned} D_Y(Y)&=A(Y-C)+B^2+\beta AB,\\ N_Y(Y)&=(Y-C)(B+\beta A)-A^2,\\ c&=3(652\beta^2-1230\beta-33). \end{aligned}
nUse in source · 4 passages
Search aliases: n
Use in source · H12_Analytic_Fourier_Resolvent_Research_Note.md, line 292
Define U_n=F_t_n,s_n(1/3) and Y_n=U_n+U_n^-1. The exact sufficient analytic obligations now are
Exact mathematical passage
U_n=F_{t_n,s_n}(1/3)
N_YUse in source · 8 passages
Search aliases: N_Y
Use in source · H12_Analytic_Fourier_Resolvent_Research_Note.md, line 187
D_Y(Y) · =A(Y-C)+B^2+β AB, N_Y(Y) · =(Y-C)(B+β A)-A^2, c · =3(652β^2-1230β-33).
Exact mathematical passage
\begin{aligned} D_Y(Y)&=A(Y-C)+B^2+\beta AB,\\ N_Y(Y)&=(Y-C)(B+\beta A)-A^2,\\ c&=3(652\beta^2-1230\beta-33). \end{aligned}
N_Y'Use in source · 1 passage
Search aliases: N_Y'
Use in source · H12_Analytic_Fourier_Resolvent_Research_Note.md, line 205
N_Y'(Y)D_Y(Y)-N_Y(Y)D_Y'(Y)=c≠0.
Exact mathematical passage
\boxed{N_Y'(Y)D_Y(Y)-N_Y(Y)D_Y'(Y)=c\ne0.}
\nmidUse in source · 1 passage
Search aliases: nmid
Use in source · H12_Analytic_Fourier_Resolvent_Research_Note.md, line 310
log F_t,s(z)=-5∑_substackn≥1 5nmid n frace^2π in(t+z)-e^2π in(s-z) n(1-e^2π int)(1-e^2π ins).
Exact mathematical passage
\log F_{t,s}(z)=-5\sum_{\substack{n\ge1\\5\nmid n}} \frac{e^{2\pi in(t+z)}-e^{2\pi in(s-z)}} {n(1-e^{2\pi int})(1-e^{2\pi ins})}.
\overline JUse in source · 4 passages
Search aliases: overline J overline
Use in source · H12_Analytic_Fourier_Resolvent_Research_Note.md, line 17
ω^2+ω+1=0, β=7^1/3ω^2, J=2+3ω, overline J=-1-3ω.
Exact mathematical passage
\omega^2+\omega+1=0,\qquad \beta=7^{1/3}\omega^2,\qquad J=2+3\omega, \qquad \overline J=-1-3\omega.
\overline J^{\,2}Use in source · 3 passages
Search aliases: overline J^ 2 overline
Use in source · H12_Analytic_Fourier_Resolvent_Research_Note.md, line 154
L^3=(J^2+49d)q(d), (L')^3=(overline J^ 2+49d)q(d).
Exact mathematical passage
\boxed{L^3=(J^2+49d)q(d),\qquad (L')^3=(\overline J^{\,2}+49d)q(d).}
PUse in source · 2 passages
Search aliases: P
Use in source · H12_Analytic_Fourier_Resolvent_Research_Note.md, line 107
P_d(Z)=Z^3-β(1+7d)Z^2+9β^2dZ+1-20d.
Exact mathematical passage
\boxed{P_d(Z)=Z^3-\beta(1+7d)Z^2+9\beta^2dZ+1-20d.}
\mathbb PUse in source · 2 passages
Search aliases: P mathbb
Use in source · H12_Analytic_Fourier_Resolvent_Research_Note.md, line 11
For the number field K= Q(β), β^3=7, the previously constructed unramified cubic is generated by a root of f(X)=X^3-β X^2+1. Its cyclic Galois action extends to a globally defined order three Möbius transformation on P^1_K. The quotient coordinate d(X) presents a rational one parameter family of cyclic cubic polynomials whose discriminants are perfect squares. Its special fiber at d=0 is the original arithmetic cubic. The associated two primitive Fourier resolvents have exact, explicitly deforme…
Exact mathematical passage
\mathbb P^1_K
\mathbb P^1_KUse in source · 2 passages
Search aliases: P^1_K mathbb
Use in source · H12_Analytic_Fourier_Resolvent_Research_Note.md, line 11
For the number field K= Q(β), β^3=7, the previously constructed unramified cubic is generated by a root of f(X)=X^3-β X^2+1. Its cyclic Galois action extends to a globally defined order three Möbius transformation on P^1_K. The quotient coordinate d(X) presents a rational one parameter family of cyclic cubic polynomials whose discriminants are perfect squares. Its special fiber at d=0 is the original arithmetic cubic. The associated two primitive Fourier resolvents have exact, explicitly deforme…
Exact mathematical passage
\mathbb P^1_K
P_dUse in source · 2 passages
Search aliases: P_d
Use in source · H12_Analytic_Fourier_Resolvent_Research_Note.md, line 107
P_d(Z)=Z^3-β(1+7d)Z^2+9β^2dZ+1-20d.
Exact mathematical passage
\boxed{P_d(Z)=Z^3-\beta(1+7d)Z^2+9\beta^2dZ+1-20d.}
QUse in source · 1 passage
Search aliases: Q
Use in source · H12_Analytic_Fourier_Resolvent_Research_Note.md, line 246
Since Q(U)=U^3R(U+U^-1), this is exactly the original reciprocal-sextic membership target, now realized as an ambient cubic invariant.
Exact mathematical passage
Q(U)=U^3R(U+U^{-1})
\mathbb QUse in source · 3 passages
Search aliases: Q mathbb
Use in source · H12_Analytic_Fourier_Resolvent_Research_Note.md, line 11
For the number field K= Q(β), β^3=7, the previously constructed unramified cubic is generated by a root of f(X)=X^3-β X^2+1. Its cyclic Galois action extends to a globally defined order three Möbius transformation on P^1_K. The quotient coordinate d(X) presents a rational one parameter family of cyclic cubic polynomials whose discriminants are perfect squares. Its special fiber at d=0 is the original arithmetic cubic. The associated two primitive Fourier resolvents have exact, explicitly deforme…
Exact mathematical passage
K=\mathbb Q(\beta)
qUse in source · 7 passages
Search aliases: q
Use in source · H12_Analytic_Fourier_Resolvent_Research_Note.md, line 146
Define q(d)=1-13d+49d^2.
Exact mathematical passage
q(d)=1-13d+49d^2
RUse in source · 6 passages
Search aliases: R
Use in source · H12_Analytic_Fourier_Resolvent_Research_Note.md, line 50
R(Y)=Y^3+(6β^2-14β+2)Y^2 +(4β^2-6β-1)Y+94β^2-124β-107.
Exact mathematical passage
R(Y)=Y^3+(6\beta^2-14\beta+2)Y^2 +(4\beta^2-6\beta-1)Y+94\beta^2-124\beta-107.
rUse in source · 2 passages
Search aliases: r
Use in source · H12_Analytic_Fourier_Resolvent_Research_Note.md, line 316
For real z, writing r=max(|e^2π it|,|e^2π is|) and q=min(…), the tail after N terms is bounded by
Exact mathematical passage
r=\max(|e^{2\pi it}|,|e^{2\pi is}|)
r^{N+1}Use in source · 1 passage
Search aliases: r^N+1
Use in source · H12_Analytic_Fourier_Resolvent_Research_Note.md, line 318
|tail|≤frac10r^N+1(N+1)(1-r)^2(1-q).
Exact mathematical passage
|\text{tail}|\le\frac{10r^{N+1}}{(N+1)(1-r)^2(1-q)}.
\rmUse in source · 2 passages
Search aliases: rm
Use in source · H12_Analytic_Fourier_Resolvent_Research_Note.md, line 37
F(z)=fracΓ_rm ell(z;τ,s)^5Γ_rm ell(5z;5τ,5s), U_Γ=F(1/3), Y_Γ=U_Γ+U_Γ^-1.
Exact mathematical passage
F(z)=\frac{\Gamma_{\rm ell}(z;\tau,s)^5}{\Gamma_{\rm ell}(5z;5\tau,5s)}, \qquad U_\Gamma=F(1/3),\qquad Y_\Gamma=U_\Gamma+U_\Gamma^{-1}.
sUse in source · 7 passages
Search aliases: s
Use in source · H12_Analytic_Fourier_Resolvent_Research_Note.md, line 32
τ=fracβ^2+2β15, s=-fracβ+215, t=-τ,
Exact mathematical passage
\tau=\frac{\beta^2+2\beta}{15},\qquad s=-\frac{\beta+2}{15},\qquad t=-\tau,
s_0Use in source · 1 passage
Search aliases: s_0
Use in source · H12_Analytic_Fourier_Resolvent_Research_Note.md, line 286
(t_0,s_0)=(t,s), (t_10,s_10)=((t-1)/10,(s+3)/10), (t_23,s_23)=((t-5)/23,(s-2)/23).
Exact mathematical passage
(t_0,s_0)=(t,s),\quad (t_{10},s_{10})=((t-1)/10,(s+3)/10),\quad (t_{23},s_{23})=((t-5)/23,(s-2)/23).
s_{10}Use in source · 1 passage
Search aliases: s_10
Use in source · H12_Analytic_Fourier_Resolvent_Research_Note.md, line 286
(t_0,s_0)=(t,s), (t_10,s_10)=((t-1)/10,(s+3)/10), (t_23,s_23)=((t-5)/23,(s-2)/23).
Exact mathematical passage
(t_0,s_0)=(t,s),\quad (t_{10},s_{10})=((t-1)/10,(s+3)/10),\quad (t_{23},s_{23})=((t-5)/23,(s-2)/23).
s_{23}Use in source · 1 passage
Search aliases: s_23
Use in source · H12_Analytic_Fourier_Resolvent_Research_Note.md, line 286
(t_0,s_0)=(t,s), (t_10,s_10)=((t-1)/10,(s+3)/10), (t_23,s_23)=((t-5)/23,(s-2)/23).
Exact mathematical passage
(t_0,s_0)=(t,s),\quad (t_{10},s_{10})=((t-1)/10,(s+3)/10),\quad (t_{23},s_{23})=((t-5)/23,(s-2)/23).
s_nUse in source · 1 passage
Search aliases: s_n
Use in source · H12_Analytic_Fourier_Resolvent_Research_Note.md, line 292
Define U_n=F_t_n,s_n(1/3) and Y_n=U_n+U_n^-1. The exact sufficient analytic obligations now are
Exact mathematical passage
U_n=F_{t_n,s_n}(1/3)
\squareUse in source · 4 passages
Search aliases: square
Use in source · H12_Analytic_Fourier_Resolvent_Research_Note.md, line 79
*Proof.* Direct calculation gives det G= -20+3β^3=1 and trG=1. Cayley–Hamilton proves projective order three. Multiplying each fractional formula by its denominator and reducing modulo v^3-β v^2+1 proves the polynomial expressions. The three roots are distinct because the discriminant is one. The nontrivial maps cyclically permute them. square
Exact mathematical passage
\square
tUse in source · 6 passages
Search aliases: t
Use in source · H12_Analytic_Fourier_Resolvent_Research_Note.md, line 32
τ=fracβ^2+2β15, s=-fracβ+215, t=-τ,
Exact mathematical passage
\tau=\frac{\beta^2+2\beta}{15},\qquad s=-\frac{\beta+2}{15},\qquad t=-\tau,
t_0Use in source · 1 passage
Search aliases: t_0
Use in source · H12_Analytic_Fourier_Resolvent_Research_Note.md, line 286
(t_0,s_0)=(t,s), (t_10,s_10)=((t-1)/10,(s+3)/10), (t_23,s_23)=((t-5)/23,(s-2)/23).
Exact mathematical passage
(t_0,s_0)=(t,s),\quad (t_{10},s_{10})=((t-1)/10,(s+3)/10),\quad (t_{23},s_{23})=((t-5)/23,(s-2)/23).
t_{10}Use in source · 1 passage
Search aliases: t_10
Use in source · H12_Analytic_Fourier_Resolvent_Research_Note.md, line 286
(t_0,s_0)=(t,s), (t_10,s_10)=((t-1)/10,(s+3)/10), (t_23,s_23)=((t-5)/23,(s-2)/23).
Exact mathematical passage
(t_0,s_0)=(t,s),\quad (t_{10},s_{10})=((t-1)/10,(s+3)/10),\quad (t_{23},s_{23})=((t-5)/23,(s-2)/23).
t_{23}Use in source · 1 passage
Search aliases: t_23
Use in source · H12_Analytic_Fourier_Resolvent_Research_Note.md, line 286
(t_0,s_0)=(t,s), (t_10,s_10)=((t-1)/10,(s+3)/10), (t_23,s_23)=((t-5)/23,(s-2)/23).
Exact mathematical passage
(t_0,s_0)=(t,s),\quad (t_{10},s_{10})=((t-1)/10,(s+3)/10),\quad (t_{23},s_{23})=((t-5)/23,(s-2)/23).
t_nUse in source · 1 passage
Search aliases: t_n
Use in source · H12_Analytic_Fourier_Resolvent_Research_Note.md, line 292
Define U_n=F_t_n,s_n(1/3) and Y_n=U_n+U_n^-1. The exact sufficient analytic obligations now are
Exact mathematical passage
U_n=F_{t_n,s_n}(1/3)
\operatorname{tr}Use in source · 1 passage
Search aliases: tr operatorname
Use in source · H12_Analytic_Fourier_Resolvent_Research_Note.md, line 79
*Proof.* Direct calculation gives det G= -20+3β^3=1 and trG=1. Cayley–Hamilton proves projective order three. Multiplying each fractional formula by its denominator and reducing modulo v^3-β v^2+1 proves the polynomial expressions. The three roots are distinct because the discriminant is one. The nontrivial maps cyclically permute them. square
Exact mathematical passage
\operatorname{tr}G=1
UUse in source · 6 passages
Search aliases: U
Use in source · H12_Analytic_Fourier_Resolvent_Research_Note.md, line 37
F(z)=fracΓ_rm ell(z;τ,s)^5Γ_rm ell(5z;5τ,5s), U_Γ=F(1/3), Y_Γ=U_Γ+U_Γ^-1.
Exact mathematical passage
F(z)=\frac{\Gamma_{\rm ell}(z;\tau,s)^5}{\Gamma_{\rm ell}(5z;5\tau,5s)}, \qquad U_\Gamma=F(1/3),\qquad Y_\Gamma=U_\Gamma+U_\Gamma^{-1}.
uUse in source · 1 passage
Search aliases: u
Use in source · H12_Analytic_Fourier_Resolvent_Research_Note.md, line 340
F_(t-a)/n,(s+b)/n(z) =∏_u=1-n^0∏_v=0^n-1 F_t,s(z+ufract-an+vfracs+bn).
Exact mathematical passage
F_{(t-a)/n,(s+b)/n}(z) =\prod_{u=1-n}^0\prod_{v=0}^{n-1} F_{t,s}\left(z+u\frac{t-a}{n}+v\frac{s+b}{n}\right).
U^{-1}Use in source · 1 passage
Search aliases: U^-1
Use in source · H12_Analytic_Fourier_Resolvent_Research_Note.md, line 246
Since Q(U)=U^3R(U+U^-1), this is exactly the original reciprocal-sextic membership target, now realized as an ambient cubic invariant.
Exact mathematical passage
Q(U)=U^3R(U+U^{-1})
U^3Use in source · 1 passage
Search aliases: U^3
Use in source · H12_Analytic_Fourier_Resolvent_Research_Note.md, line 246
Since Q(U)=U^3R(U+U^-1), this is exactly the original reciprocal-sextic membership target, now realized as an ambient cubic invariant.
Exact mathematical passage
Q(U)=U^3R(U+U^{-1})
U_{\mathrm{alg}}Use in source · 1 passage
Search aliases: U_alg mathrm
Use in source · H12_Analytic_Fourier_Resolvent_Research_Note.md, line 304
The orbit laws are **conjectural at the analytic gamma special values**, even though Ψ^3=1 is an exact algebraic theorem. If all three special-value identities hold, the established branch separation identifies U_Γ=U_alg. The nontrivial analytical content has not been proved by the algebraic construction.
Exact mathematical passage
U_\Gamma=U_{\mathrm{alg}}
U_nUse in source · 2 passages
Search aliases: U_n
Use in source · H12_Analytic_Fourier_Resolvent_Research_Note.md, line 292
Define U_n=F_t_n,s_n(1/3) and Y_n=U_n+U_n^-1. The exact sufficient analytic obligations now are
Exact mathematical passage
U_n=F_{t_n,s_n}(1/3)
U_n^{-1}Use in source · 1 passage
Search aliases: U_n^-1
Use in source · H12_Analytic_Fourier_Resolvent_Research_Note.md, line 292
Define U_n=F_t_n,s_n(1/3) and Y_n=U_n+U_n^-1. The exact sufficient analytic obligations now are
Exact mathematical passage
Y_n=U_n+U_n^{-1}
U_\GammaUse in source · 3 passages
Search aliases: U_Γ Gamma
Use in source · H12_Analytic_Fourier_Resolvent_Research_Note.md, line 37
F(z)=fracΓ_rm ell(z;τ,s)^5Γ_rm ell(5z;5τ,5s), U_Γ=F(1/3), Y_Γ=U_Γ+U_Γ^-1.
Exact mathematical passage
F(z)=\frac{\Gamma_{\rm ell}(z;\tau,s)^5}{\Gamma_{\rm ell}(5z;5\tau,5s)}, \qquad U_\Gamma=F(1/3),\qquad Y_\Gamma=U_\Gamma+U_\Gamma^{-1}.
U_\Gamma^{-1}Use in source · 1 passage
Search aliases: U_Γ^-1 Gamma
Use in source · H12_Analytic_Fourier_Resolvent_Research_Note.md, line 37
F(z)=fracΓ_rm ell(z;τ,s)^5Γ_rm ell(5z;5τ,5s), U_Γ=F(1/3), Y_Γ=U_Γ+U_Γ^-1.
Exact mathematical passage
F(z)=\frac{\Gamma_{\rm ell}(z;\tau,s)^5}{\Gamma_{\rm ell}(5z;5\tau,5s)}, \qquad U_\Gamma=F(1/3),\qquad Y_\Gamma=U_\Gamma+U_\Gamma^{-1}.
vUse in source · 14 passages
Search aliases: v
Use in source · H12_Analytic_Fourier_Resolvent_Research_Note.md, line 28
The existing class field construction supplies an irreducible, everywhere unramified cyclic cubic model generated by a root v of f. Its field and ramification properties are inherited from the earlier archive and are not re-proved here. The polynomial discriminant is 1.
Exact mathematical passage
v
v^2Use in source · 8 passages
Search aliases: v^2
Use in source · H12_Analytic_Fourier_Resolvent_Research_Note.md, line 42
The previously constructed algebraic candidate satisfies Y_rm alg=Av^2+Bv+C where
Exact mathematical passage
Y_{\rm alg}=Av^2+Bv+C
v^3Use in source · 2 passages
Search aliases: v^3
Use in source · H12_Analytic_Fourier_Resolvent_Research_Note.md, line 79
*Proof.* Direct calculation gives det G= -20+3β^3=1 and trG=1. Cayley–Hamilton proves projective order three. Multiplying each fractional formula by its denominator and reducing modulo v^3-β v^2+1 proves the polynomial expressions. The three roots are distinct because the discriminant is one. The nontrivial maps cyclically permute them. square
Exact mathematical passage
v^3-\beta v^2+1
v_\GammaUse in source · 2 passages
Search aliases: v_Γ Gamma
Use in source · H12_Analytic_Fourier_Resolvent_Research_Note.md, line 221
v_Γ(z)=fracN_Y(F(z)+F(z)^-1)D_Y(F(z)+F(z)^-1).
Exact mathematical passage
\boxed{ v_\Gamma(z)=\frac{N_Y(F(z)+F(z)^{-1})}{D_Y(F(z)+F(z)^{-1})}. }
XUse in source · 17 passages
Search aliases: X
Use in source · H12_Analytic_Fourier_Resolvent_Research_Note.md, line 11
For the number field K= Q(β), β^3=7, the previously constructed unramified cubic is generated by a root of f(X)=X^3-β X^2+1. Its cyclic Galois action extends to a globally defined order three Möbius transformation on P^1_K. The quotient coordinate d(X) presents a rational one parameter family of cyclic cubic polynomials whose discriminants are perfect squares. Its special fiber at d=0 is the original arithmetic cubic. The associated two primitive Fourier resolvents have exact, explicitly deforme…
Exact mathematical passage
f(X)=X^3-\beta X^2+1
X^2Use in source · 2 passages
Search aliases: X^2
Use in source · H12_Analytic_Fourier_Resolvent_Research_Note.md, line 11
For the number field K= Q(β), β^3=7, the previously constructed unramified cubic is generated by a root of f(X)=X^3-β X^2+1. Its cyclic Galois action extends to a globally defined order three Möbius transformation on P^1_K. The quotient coordinate d(X) presents a rational one parameter family of cyclic cubic polynomials whose discriminants are perfect squares. Its special fiber at d=0 is the original arithmetic cubic. The associated two primitive Fourier resolvents have exact, explicitly deforme…
Exact mathematical passage
f(X)=X^3-\beta X^2+1
X^3Use in source · 2 passages
Search aliases: X^3
Use in source · H12_Analytic_Fourier_Resolvent_Research_Note.md, line 11
For the number field K= Q(β), β^3=7, the previously constructed unramified cubic is generated by a root of f(X)=X^3-β X^2+1. Its cyclic Galois action extends to a globally defined order three Möbius transformation on P^1_K. The quotient coordinate d(X) presents a rational one parameter family of cyclic cubic polynomials whose discriminants are perfect squares. Its special fiber at d=0 is the original arithmetic cubic. The associated two primitive Fourier resolvents have exact, explicitly deforme…
Exact mathematical passage
f(X)=X^3-\beta X^2+1
X_0Use in source · 2 passages
Search aliases: X_0
Use in source · H12_Analytic_Fourier_Resolvent_Research_Note.md, line 137
X_0=X, X_1=Φ(X), X_2=Φ^2(X),
Exact mathematical passage
X_0=X,\quad X_1=\Phi(X),\quad X_2=\Phi^2(X),
X_1Use in source · 2 passages
Search aliases: X_1
Use in source · H12_Analytic_Fourier_Resolvent_Research_Note.md, line 137
X_0=X, X_1=Φ(X), X_2=Φ^2(X),
Exact mathematical passage
X_0=X,\quad X_1=\Phi(X),\quad X_2=\Phi^2(X),
X_2Use in source · 2 passages
Search aliases: X_2
Use in source · H12_Analytic_Fourier_Resolvent_Research_Note.md, line 137
X_0=X, X_1=Φ(X), X_2=Φ^2(X),
Exact mathematical passage
X_0=X,\quad X_1=\Phi(X),\quad X_2=\Phi^2(X),
YUse in source · 32 passages
Search aliases: Y
Use in source · H12_Analytic_Fourier_Resolvent_Research_Note.md, line 37
F(z)=fracΓ_rm ell(z;τ,s)^5Γ_rm ell(5z;5τ,5s), U_Γ=F(1/3), Y_Γ=U_Γ+U_Γ^-1.
Exact mathematical passage
F(z)=\frac{\Gamma_{\rm ell}(z;\tau,s)^5}{\Gamma_{\rm ell}(5z;5\tau,5s)}, \qquad U_\Gamma=F(1/3),\qquad Y_\Gamma=U_\Gamma+U_\Gamma^{-1}.
Y^2Use in source · 1 passage
Search aliases: Y^2
Use in source · H12_Analytic_Fourier_Resolvent_Research_Note.md, line 50
R(Y)=Y^3+(6β^2-14β+2)Y^2 +(4β^2-6β-1)Y+94β^2-124β-107.
Exact mathematical passage
R(Y)=Y^3+(6\beta^2-14\beta+2)Y^2 +(4\beta^2-6\beta-1)Y+94\beta^2-124\beta-107.
Y^3Use in source · 1 passage
Search aliases: Y^3
Use in source · H12_Analytic_Fourier_Resolvent_Research_Note.md, line 50
R(Y)=Y^3+(6β^2-14β+2)Y^2 +(4β^2-6β-1)Y+94β^2-124β-107.
Exact mathematical passage
R(Y)=Y^3+(6\beta^2-14\beta+2)Y^2 +(4\beta^2-6\beta-1)Y+94\beta^2-124\beta-107.
Y_0Use in source · 7 passages
Search aliases: Y_0
Use in source · H12_Analytic_Fourier_Resolvent_Research_Note.md, line 258
Let Y_0=Y, Y_1=Ψ(Y), Y_2=Ψ^2(Y). Their trace is a rational function of the invariant d=d(h(Y)). Exact elimination gives
Exact mathematical passage
Y_0=Y
Y_1Use in source · 3 passages
Search aliases: Y_1
Use in source · H12_Analytic_Fourier_Resolvent_Research_Note.md, line 258
Let Y_0=Y, Y_1=Ψ(Y), Y_2=Ψ^2(Y). Their trace is a rational function of the invariant d=d(h(Y)). Exact elimination gives
Exact mathematical passage
Y_1=\Psi(Y)
Y_{10}Use in source · 3 passages
Search aliases: Y_10
Use in source · H12_Analytic_Fourier_Resolvent_Research_Note.md, line 294
Y_23=Ψ(Y_0), Y_10=Ψ^2(Y_0),
Exact mathematical passage
\boxed{Y_{23}=\Psi(Y_0),\qquad Y_{10}=\Psi^2(Y_0),}
Y_2Use in source · 3 passages
Search aliases: Y_2
Use in source · H12_Analytic_Fourier_Resolvent_Research_Note.md, line 258
Let Y_0=Y, Y_1=Ψ(Y), Y_2=Ψ^2(Y). Their trace is a rational function of the invariant d=d(h(Y)). Exact elimination gives
Exact mathematical passage
Y_2=\Psi^2(Y)
Y_{23}Use in source · 3 passages
Search aliases: Y_23
Use in source · H12_Analytic_Fourier_Resolvent_Research_Note.md, line 294
Y_23=Ψ(Y_0), Y_10=Ψ^2(Y_0),
Exact mathematical passage
\boxed{Y_{23}=\Psi(Y_0),\qquad Y_{10}=\Psi^2(Y_0),}
Y_nUse in source · 1 passage
Search aliases: Y_n
Use in source · H12_Analytic_Fourier_Resolvent_Research_Note.md, line 292
Define U_n=F_t_n,s_n(1/3) and Y_n=U_n+U_n^-1. The exact sufficient analytic obligations now are
Exact mathematical passage
Y_n=U_n+U_n^{-1}
Y_{\rm alg}Use in source · 1 passage
Search aliases: Y_rm alg rm
Use in source · H12_Analytic_Fourier_Resolvent_Research_Note.md, line 42
The previously constructed algebraic candidate satisfies Y_rm alg=Av^2+Bv+C where
Exact mathematical passage
Y_{\rm alg}=Av^2+Bv+C
Y_\GammaUse in source · 2 passages
Search aliases: Y_Γ Gamma
Use in source · H12_Analytic_Fourier_Resolvent_Research_Note.md, line 37
F(z)=fracΓ_rm ell(z;τ,s)^5Γ_rm ell(5z;5τ,5s), U_Γ=F(1/3), Y_Γ=U_Γ+U_Γ^-1.
Exact mathematical passage
F(z)=\frac{\Gamma_{\rm ell}(z;\tau,s)^5}{\Gamma_{\rm ell}(5z;5\tau,5s)}, \qquad U_\Gamma=F(1/3),\qquad Y_\Gamma=U_\Gamma+U_\Gamma^{-1}.
ZUse in source · 3 passages
Search aliases: Z
Use in source · H12_Analytic_Fourier_Resolvent_Research_Note.md, line 107
P_d(Z)=Z^3-β(1+7d)Z^2+9β^2dZ+1-20d.
Exact mathematical passage
\boxed{P_d(Z)=Z^3-\beta(1+7d)Z^2+9\beta^2dZ+1-20d.}
zUse in source · 18 passages
Search aliases: z
Use in source · H12_Analytic_Fourier_Resolvent_Research_Note.md, line 11
ect squares. Its special fiber at d=0 is the original arithmetic cubic. The associated two primitive Fourier resolvents have exact, explicitly deformed norm and cube formulas. A rational inverse of the earlier quadratic arithmetic coordinate then turns the elliptic gamma function into a canonical meromorphic candidate for the cubic generator and supplies a single scalar analytic obstruction d_Γ(z). The three existing gamma charts agree numerically with the expected Möbius orbit at z=1/3, but not…
Exact mathematical passage
d_\Gamma(z)
Z^2Use in source · 1 passage
Search aliases: Z^2
Use in source · H12_Analytic_Fourier_Resolvent_Research_Note.md, line 107
P_d(Z)=Z^3-β(1+7d)Z^2+9β^2dZ+1-20d.
Exact mathematical passage
\boxed{P_d(Z)=Z^3-\beta(1+7d)Z^2+9\beta^2dZ+1-20d.}
Z^3Use in source · 1 passage
Search aliases: Z^3
Use in source · H12_Analytic_Fourier_Resolvent_Research_Note.md, line 107
P_d(Z)=Z^3-β(1+7d)Z^2+9β^2dZ+1-20d.
Exact mathematical passage
\boxed{P_d(Z)=Z^3-\beta(1+7d)Z^2+9\beta^2dZ+1-20d.}
|Use in source · 3 passages
Search aliases: |
Use in source · H12_Analytic_Fourier_Resolvent_Research_Note.md, line 316
For real z, writing r=max(|e^2π it|,|e^2π is|) and q=min(…), the tail after N terms is bounded by
Exact mathematical passage
r=\max(|e^{2\pi it}|,|e^{2\pi is}|)
\timesUse in source · 7 passages
Search aliases: × times
Use in source · H12_Analytic_Fourier_Resolvent_Research_Note.md, line 256
where h(Y)=N_Y(Y)/D_Y(Y). If M_h is its 2×2 coefficient matrix, then the matrix for Ψ is M_h^-1GM_h. Its determinant is one and trace is one, so it also has order three.
Exact mathematical passage
2\times2
\betaUse in source · 34 passages
Search aliases: β beta
Use in source · H12_Analytic_Fourier_Resolvent_Research_Note.md, line 11
For the number field K= Q(β), β^3=7, the previously constructed unramified cubic is generated by a root of f(X)=X^3-β X^2+1. Its cyclic Galois action extends to a globally defined order three Möbius transformation on P^1_K. The quotient coordinate d(X) presents a rational one parameter family of cyclic cubic polynomials whose discriminants are perfect squares. Its special fiber at d=0 is the original arithmetic cubic. The associated two primitive Fourier resolvents have exact, explicitly deforme…
Exact mathematical passage
K=\mathbb Q(\beta)
\beta^2Use in source · 20 passages
Search aliases: β^2 beta
Use in source · H12_Analytic_Fourier_Resolvent_Research_Note.md, line 32
τ=fracβ^2+2β15, s=-fracβ+215, t=-τ,
Exact mathematical passage
\tau=\frac{\beta^2+2\beta}{15},\qquad s=-\frac{\beta+2}{15},\qquad t=-\tau,
\beta^3Use in source · 6 passages
Search aliases: β^3 beta
Use in source · H12_Analytic_Fourier_Resolvent_Research_Note.md, line 11
For the number field K= Q(β), β^3=7, the previously constructed unramified cubic is generated by a root of f(X)=X^3-β X^2+1. Its cyclic Galois action extends to a globally defined order three Möbius transformation on P^1_K. The quotient coordinate d(X) presents a rational one parameter family of cyclic cubic polynomials whose discriminants are perfect squares. Its special fiber at d=0 is the original arithmetic cubic. The associated two primitive Fourier resolvents have exact, explicitly deforme…
Exact mathematical passage
\beta^3=7
\GammaUse in source · 12 passages
Search aliases: Γ Gamma
Use in source · H12_Analytic_Fourier_Resolvent_Research_Note.md, line 11
ect squares. Its special fiber at d=0 is the original arithmetic cubic. The associated two primitive Fourier resolvents have exact, explicitly deformed norm and cube formulas. A rational inverse of the earlier quadratic arithmetic coordinate then turns the elliptic gamma function into a canonical meromorphic candidate for the cubic generator and supplies a single scalar analytic obstruction d_Γ(z). The three existing gamma charts agree numerically with the expected Möbius orbit at z=1/3, but not…
Exact mathematical passage
d_\Gamma(z)
\Gamma_{\rm ell}Use in source · 1 passage
Search aliases: Γ_rm ell Gamma rm
Use in source · H12_Analytic_Fourier_Resolvent_Research_Note.md, line 37
F(z)=fracΓ_rm ell(z;τ,s)^5Γ_rm ell(5z;5τ,5s), U_Γ=F(1/3), Y_Γ=U_Γ+U_Γ^-1.
Exact mathematical passage
F(z)=\frac{\Gamma_{\rm ell}(z;\tau,s)^5}{\Gamma_{\rm ell}(5z;5\tau,5s)}, \qquad U_\Gamma=F(1/3),\qquad Y_\Gamma=U_\Gamma+U_\Gamma^{-1}.
\thetaUse in source · 2 passages
Search aliases: θ theta
Use in source · H12_Analytic_Fourier_Resolvent_Research_Note.md, line 260
Y_0+Y_1+Y_2 =-6β^2+14β-2 +fracκ d1+θ d,
Exact mathematical passage
\boxed{ Y_0+Y_1+Y_2 =-6\beta^2+14\beta-2 +\frac{\kappa d}{1+\theta d}, }
\kappaUse in source · 3 passages
Search aliases: κ kappa
Use in source · H12_Analytic_Fourier_Resolvent_Research_Note.md, line 260
Y_0+Y_1+Y_2 =-6β^2+14β-2 +fracκ d1+θ d,
Exact mathematical passage
\boxed{ Y_0+Y_1+Y_2 =-6\beta^2+14\beta-2 +\frac{\kappa d}{1+\theta d}, }
\piUse in source · 2 passages
Search aliases: π pi
Use in source · H12_Analytic_Fourier_Resolvent_Research_Note.md, line 310
log F_t,s(z)=-5∑_substackn≥1 5nmid n frace^2π in(t+z)-e^2π in(s-z) n(1-e^2π int)(1-e^2π ins).
Exact mathematical passage
\log F_{t,s}(z)=-5\sum_{\substack{n\ge1\\5\nmid n}} \frac{e^{2\pi in(t+z)}-e^{2\pi in(s-z)}} {n(1-e^{2\pi int})(1-e^{2\pi ins})}.
\tauUse in source · 2 passages
Search aliases: τ tau
Use in source · H12_Analytic_Fourier_Resolvent_Research_Note.md, line 32
τ=fracβ^2+2β15, s=-fracβ+215, t=-τ,
Exact mathematical passage
\tau=\frac{\beta^2+2\beta}{15},\qquad s=-\frac{\beta+2}{15},\qquad t=-\tau,
\PhiUse in source · 10 passages
Search aliases: Φ Phi
Use in source · H12_Analytic_Fourier_Resolvent_Research_Note.md, line 67
Φ(X)=frac5X-3ββ^2X-4, Φ^2(X)=frac4X-3ββ^2X-5, Φ^3(X)=X.
Exact mathematical passage
\Phi(X)=\frac{5X-3\beta}{\beta^2X-4},\qquad \Phi^2(X)=\frac{4X-3\beta}{\beta^2X-5},\qquad \Phi^3(X)=X.
\Phi^2Use in source · 5 passages
Search aliases: Φ^2 Phi
Use in source · H12_Analytic_Fourier_Resolvent_Research_Note.md, line 67
Φ(X)=frac5X-3ββ^2X-4, Φ^2(X)=frac4X-3ββ^2X-5, Φ^3(X)=X.
Exact mathematical passage
\Phi(X)=\frac{5X-3\beta}{\beta^2X-4},\qquad \Phi^2(X)=\frac{4X-3\beta}{\beta^2X-5},\qquad \Phi^3(X)=X.
\Phi^3Use in source · 1 passage
Search aliases: Φ^3 Phi
Use in source · H12_Analytic_Fourier_Resolvent_Research_Note.md, line 67
Φ(X)=frac5X-3ββ^2X-4, Φ^2(X)=frac4X-3ββ^2X-5, Φ^3(X)=X.
Exact mathematical passage
\Phi(X)=\frac{5X-3\beta}{\beta^2X-4},\qquad \Phi^2(X)=\frac{4X-3\beta}{\beta^2X-5},\qquad \Phi^3(X)=X.
\PsiUse in source · 9 passages
Search aliases: Ψ Psi
Use in source · H12_Analytic_Fourier_Resolvent_Research_Note.md, line 252
Ψ=h^-1∘Φ∘ h,
Exact mathematical passage
\Psi=h^{-1}\circ\Phi\circ h,
\Psi^2Use in source · 3 passages
Search aliases: Ψ^2 Psi
Use in source · H12_Analytic_Fourier_Resolvent_Research_Note.md, line 258
Let Y_0=Y, Y_1=Ψ(Y), Y_2=Ψ^2(Y). Their trace is a rational function of the invariant d=d(h(Y)). Exact elimination gives
Exact mathematical passage
Y_2=\Psi^2(Y)
\Psi^3Use in source · 1 passage
Search aliases: Ψ^3 Psi
Use in source · H12_Analytic_Fourier_Resolvent_Research_Note.md, line 304
The orbit laws are **conjectural at the analytic gamma special values**, even though Ψ^3=1 is an exact algebraic theorem. If all three special-value identities hold, the established branch separation identifies U_Γ=U_alg. The nontrivial analytical content has not been proved by the algebraic construction.
Exact mathematical passage
\Psi^3=1
\omegaUse in source · 8 passages
Search aliases: ω omega
Use in source · H12_Analytic_Fourier_Resolvent_Research_Note.md, line 17
ω^2+ω+1=0, β=7^1/3ω^2, J=2+3ω, overline J=-1-3ω.
Exact mathematical passage
\omega^2+\omega+1=0,\qquad \beta=7^{1/3}\omega^2,\qquad J=2+3\omega, \qquad \overline J=-1-3\omega.
\omega^2Use in source · 6 passages
Search aliases: ω^2 omega
Use in source · H12_Analytic_Fourier_Resolvent_Research_Note.md, line 17
ω^2+ω+1=0, β=7^1/3ω^2, J=2+3ω, overline J=-1-3ω.
Exact mathematical passage
\omega^2+\omega+1=0,\qquad \beta=7^{1/3}\omega^2,\qquad J=2+3\omega, \qquad \overline J=-1-3\omega.
\ldotsUse in source · 1 passage
Search aliases: … ldots
Use in source · H12_Analytic_Fourier_Resolvent_Research_Note.md, line 316
For real z, writing r=max(|e^2π it|,|e^2π is|) and q=min(…), the tail after N terms is bounded by
Exact mathematical passage
q=\min(\ldots)
\prodUse in source · 1 passage
Search aliases: ∏ prod
Use in source · H12_Analytic_Fourier_Resolvent_Research_Note.md, line 340
F_(t-a)/n,(s+b)/n(z) =∏_u=1-n^0∏_v=0^n-1 F_t,s(z+ufract-an+vfracs+bn).
Exact mathematical passage
F_{(t-a)/n,(s+b)/n}(z) =\prod_{u=1-n}^0\prod_{v=0}^{n-1} F_{t,s}\left(z+u\frac{t-a}{n}+v\frac{s+b}{n}\right).
\prod_{u=1-n}^0Use in source · 1 passage
Search aliases: ∏_u=1-n^0 prod
Use in source · H12_Analytic_Fourier_Resolvent_Research_Note.md, line 340
F_(t-a)/n,(s+b)/n(z) =∏_u=1-n^0∏_v=0^n-1 F_t,s(z+ufract-an+vfracs+bn).
Exact mathematical passage
F_{(t-a)/n,(s+b)/n}(z) =\prod_{u=1-n}^0\prod_{v=0}^{n-1} F_{t,s}\left(z+u\frac{t-a}{n}+v\frac{s+b}{n}\right).
\prod_{v=0}^{n-1}Use in source · 1 passage
Search aliases: ∏_v=0^n-1 prod
Use in source · H12_Analytic_Fourier_Resolvent_Research_Note.md, line 340
F_(t-a)/n,(s+b)/n(z) =∏_u=1-n^0∏_v=0^n-1 F_t,s(z+ufract-an+vfracs+bn).
Exact mathematical passage
F_{(t-a)/n,(s+b)/n}(z) =\prod_{u=1-n}^0\prod_{v=0}^{n-1} F_{t,s}\left(z+u\frac{t-a}{n}+v\frac{s+b}{n}\right).
\sumUse in source · 1 passage
Search aliases: ∑ sum
Use in source · H12_Analytic_Fourier_Resolvent_Research_Note.md, line 310
log F_t,s(z)=-5∑_substackn≥1 5nmid n frace^2π in(t+z)-e^2π in(s-z) n(1-e^2π int)(1-e^2π ins).
Exact mathematical passage
\log F_{t,s}(z)=-5\sum_{\substack{n\ge1\\5\nmid n}} \frac{e^{2\pi in(t+z)}-e^{2\pi in(s-z)}} {n(1-e^{2\pi int})(1-e^{2\pi ins})}.
\sum_{\substack{n\ge1\\5\nmid n}}Use in source · 1 passage
Search aliases: ∑_substackn≥1 5nmid n sum substack ge nmid
Use in source · H12_Analytic_Fourier_Resolvent_Research_Note.md, line 310
log F_t,s(z)=-5∑_substackn≥1 5nmid n frace^2π in(t+z)-e^2π in(s-z) n(1-e^2π int)(1-e^2π ins).
Exact mathematical passage
\log F_{t,s}(z)=-5\sum_{\substack{n\ge1\\5\nmid n}} \frac{e^{2\pi in(t+z)}-e^{2\pi in(s-z)}} {n(1-e^{2\pi int})(1-e^{2\pi ins})}.
\circUse in source · 1 passage
Search aliases: ∘ circ
Use in source · H12_Analytic_Fourier_Resolvent_Research_Note.md, line 252
Ψ=h^-1∘Φ∘ h,
Exact mathematical passage
\Psi=h^{-1}\circ\Phi\circ h,
\neUse in source · 2 passages
Search aliases: ≠ ne
Use in source · H12_Analytic_Fourier_Resolvent_Research_Note.md, line 205
N_Y'(Y)D_Y(Y)-N_Y(Y)D_Y'(Y)=c≠0.
Exact mathematical passage
\boxed{N_Y'(Y)D_Y(Y)-N_Y(Y)D_Y'(Y)=c\ne0.}
\leUse in source · 1 passage
Search aliases: ≤ le
Use in source · H12_Analytic_Fourier_Resolvent_Research_Note.md, line 318
|tail|≤frac10r^N+1(N+1)(1-r)^2(1-q).
Exact mathematical passage
|\text{tail}|\le\frac{10r^{N+1}}{(N+1)(1-r)^2(1-q)}.
\geUse in source · 1 passage
Search aliases: ≥ ge
Use in source · H12_Analytic_Fourier_Resolvent_Research_Note.md, line 310
log F_t,s(z)=-5∑_substackn≥1 5nmid n frace^2π in(t+z)-e^2π in(s-z) n(1-e^2π int)(1-e^2π ins).
Exact mathematical passage
\log F_{t,s}(z)=-5\sum_{\substack{n\ge1\\5\nmid n}} \frac{e^{2\pi in(t+z)}-e^{2\pi in(s-z)}} {n(1-e^{2\pi int})(1-e^{2\pi ins})}.
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