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499 symbol spellings in 1142 mathematical passages across 29 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 499 entries are available below.
=Definition passage · 347 passages
Search aliases: =
Definition passage · 01_statement.tex, line 12
A positive solution of [source reference] is primitive when gcd(A,B,C)=1. Its ordered exponent triple (x,y,z) is its signature. The first two entries may be exchanged together with the first two bases. Primitivity becomes stronger on an actual solution than it is on an arbitrary integer triple. For a solution of [source reference], every prime dividing two bases divides the third. Consequently a primitive solution has pairwise coprime bases. If a prime p divides A and B, the equation gives pmid …
Exact mathematical passage
\gcd(A,B,C)=1
ADefinition passage · 99 passages
Search aliases: A
Definition passage · 01_statement.tex, line 12
A positive solution of [source reference] is primitive when gcd(A,B,C)=1. Its ordered exponent triple (x,y,z) is its signature. The first two entries may be exchanged together with the first two bases. Primitivity becomes stronger on an actual solution than it is on an arbitrary integer triple. For a solution of [source reference], every prime dividing two bases divides the third. Consequently a primitive solution has pairwise coprime bases. If a prime p divides A and B, the equation gives pmid …
Exact mathematical passage
\gcd(A,B,C)=1
BDefinition passage · 62 passages
Search aliases: B
Definition passage · 01_statement.tex, line 12
A positive solution of [source reference] is primitive when gcd(A,B,C)=1. Its ordered exponent triple (x,y,z) is its signature. The first two entries may be exchanged together with the first two bases. Primitivity becomes stronger on an actual solution than it is on an arbitrary integer triple. For a solution of [source reference], every prime dividing two bases divides the third. Consequently a primitive solution has pairwise coprime bases. If a prime p divides A and B, the equation gives pmid …
Exact mathematical passage
\gcd(A,B,C)=1
CDefinition passage · 77 passages
Search aliases: C
Definition passage · 01_statement.tex, line 12
A positive solution of [source reference] is primitive when gcd(A,B,C)=1. Its ordered exponent triple (x,y,z) is its signature. The first two entries may be exchanged together with the first two bases. Primitivity becomes stronger on an actual solution than it is on an arbitrary integer triple. For a solution of [source reference], every prime dividing two bases divides the third. Consequently a primitive solution has pairwise coprime bases. If a prime p divides A and B, the equation gives pmid …
Exact mathematical passage
\gcd(A,B,C)=1
\gcdDefinition passage · 5 passages
Search aliases: gcd
Definition passage · 01_statement.tex, line 12
A positive solution of [source reference] is primitive when gcd(A,B,C)=1. Its ordered exponent triple (x,y,z) is its signature. The first two entries may be exchanged together with the first two bases. Primitivity becomes stronger on an actual solution than it is on an arbitrary integer triple. For a solution of [source reference], every prime dividing two bases divides the third. Consequently a primitive solution has pairwise coprime bases. If a prime p divides A and B, the equation gives pmid …
Exact mathematical passage
\gcd(A,B,C)=1
xDefinition passage · 38 passages
Search aliases: x
Definition passage · 01_statement.tex, line 12
A positive solution of [source reference] is primitive when gcd(A,B,C)=1. Its ordered exponent triple (x,y,z) is its signature. The first two entries may be exchanged together with the first two bases. Primitivity becomes stronger on an actual solution than it is on an arbitrary integer triple. For a solution of [source reference], every prime dividing two bases divides the third. Consequently a primitive solution has pairwise coprime bases. If a prime p divides A and B, the equation gives pmid …
Exact mathematical passage
(x,y,z)
yDefinition passage · 30 passages
Search aliases: y
Definition passage · 01_statement.tex, line 12
A positive solution of [source reference] is primitive when gcd(A,B,C)=1. Its ordered exponent triple (x,y,z) is its signature. The first two entries may be exchanged together with the first two bases. Primitivity becomes stronger on an actual solution than it is on an arbitrary integer triple. For a solution of [source reference], every prime dividing two bases divides the third. Consequently a primitive solution has pairwise coprime bases. If a prime p divides A and B, the equation gives pmid …
Exact mathematical passage
(x,y,z)
zDefinition passage · 60 passages
Search aliases: z
Definition passage · 01_statement.tex, line 12
A positive solution of [source reference] is primitive when gcd(A,B,C)=1. Its ordered exponent triple (x,y,z) is its signature. The first two entries may be exchanged together with the first two bases. Primitivity becomes stronger on an actual solution than it is on an arbitrary integer triple. For a solution of [source reference], every prime dividing two bases divides the third. Consequently a primitive solution has pairwise coprime bases. If a prime p divides A and B, the equation gives pmid …
Exact mathematical passage
(x,y,z)
+Introduction passage · 139 passages
Search aliases: +
Introduction passage · 09_periods.tex, line 17
Put M=7^j and write [u] additively in F^×/F^× M for the local field F=K_v. Let X=[ϕ], Z=7[C]-3[A]-3[3]-5[sqrt5]. The entire marked period matrix becomes ([Π_ij])= Z+2X · -4X -4X · Z-2X =ZI+2XR, R=1 · -2 -2 · -1, R^2=5I. Since four is invertible modulo M, every marked period coordinate vanishes if and only if X=Z=0. The matrix R belongs to this marked local period calculation. It is not silently identified with a global endomorphism or with an integral semilinear descent operator.
Exact mathematical passage
\label{eq:periodmatrix} ([\Pi_{ij}])= \begin{pmatrix}Z+2X&-4X\\-4X&Z-2X\end{pmatrix} =ZI+2XR,\qquad R=\begin{pmatrix}1&-2\\-2&-1\end{pmatrix},\quad R^2=5I.
\bmodIntroduction passage · 2 passages
Search aliases: bmod
Introduction passage · 15_semilinear_image.tex, line 19
The depth one image theorem. Let K_∞ be the cyclotomic tower. Write B for the additive image of beta on its Galois group and d=dim_F_7B. The inherited group theorem at actual depth one identifies H=ρ(G_K_∞)= {M∈SL_2(Oloc): l Mbmod7=(1 · x 0 · 1), x∈ k, (M_21/7)bmod7∈ B }. In particular it contains the full kernel of reduction modulo forty nine. Its finite quotient orders are |Hbmod7^j|=7^6j-6+d (j≥2), |Hbmod7|=49. For the bounded patterns d=1, the orders at forty nine and 343 are 7^7 and 7^13. T…
Exact mathematical passage
\label{eq:imagegroup} H=\rho(G_{K_\infty})= \left\{M\in\mathrm{SL}_2(\Oloc): \begin{array}{l} M\bmod7=\left(\begin{smallmatrix}1&x\\0&1\end{smallmatrix}\right),\ x\in k,\\ (M_{21}/7)\bmod7\in B \end{array}\right\}.
DIntroduction passage · 20 passages
Search aliases: D
Introduction passage · 16_secondary.tex, line 8
Put s(g)=α(g)β(g). Expansion of the two cocycle laws gives ds=-(α∪β+β∪α). For a chosen primitive p, set D_12=-p and D_23=p-s. They form a defining system with differentials α∪β and β∪α. The associated twisted Massey type cocycle is μ_p=α∪(p-s)-p∪α. Explicitly it equals μ_p(g,h)=α(g)(p(h)-α(h)β(h)) -p(g)χ(g)α(h). Let R_p denote the right side of [source reference]. Direct expansion of f(g)=α(g)p(g) gives R_p+d_χ f=μ_p. Thus [R_p]=[μ_p] in H^2(K,k(χ)). For this fixed p, the upper equation is solub…
Exact mathematical passage
D_{12}=-p
dIntroduction passage · 31 passages
Search aliases: d
Introduction passage · 15_semilinear_image.tex, line 18
The depth one image theorem. Let K_∞ be the cyclotomic tower. Write B for the additive image of beta on its Galois group and d=dim_F_7B. The inherited group theorem at actual depth one identifies H=ρ(G_K_∞)= {M∈SL_2(Oloc): l Mbmod7=(1 · x
Exact mathematical passage
d=\dim_{\F_7}B
D_{12}Introduction passage · 1 passage
Search aliases: D_12
Introduction passage · 16_secondary.tex, line 8
Put s(g)=α(g)β(g). Expansion of the two cocycle laws gives ds=-(α∪β+β∪α). For a chosen primitive p, set D_12=-p and D_23=p-s. They form a defining system with differentials α∪β and β∪α. The associated twisted Massey type cocycle is μ_p=α∪(p-s)-p∪α. Explicitly it equals μ_p(g,h)=α(g)(p(h)-α(h)β(h)) -p(g)χ(g)α(h). Let R_p denote the right side of [source reference]. Direct expansion of f(g)=α(g)p(g) gives R_p+d_χ f=μ_p. Thus [R_p]=[μ_p] in H^2(K,k(χ)). For this fixed p, the upper equation is solub…
Exact mathematical passage
D_{12}=-p
D_{23}Introduction passage · 1 passage
Search aliases: D_23
Introduction passage · 16_secondary.tex, line 8
Put s(g)=α(g)β(g). Expansion of the two cocycle laws gives ds=-(α∪β+β∪α). For a chosen primitive p, set D_12=-p and D_23=p-s. They form a defining system with differentials α∪β and β∪α. The associated twisted Massey type cocycle is μ_p=α∪(p-s)-p∪α. Explicitly it equals μ_p(g,h)=α(g)(p(h)-α(h)β(h)) -p(g)χ(g)α(h). Let R_p denote the right side of [source reference]. Direct expansion of f(g)=α(g)p(g) gives R_p+d_χ f=μ_p. Thus [R_p]=[μ_p] in H^2(K,k(χ)). For this fixed p, the upper equation is solub…
Exact mathematical passage
D_{23}=p-s
d_\chiIntroduction passage · 4 passages
Search aliases: d_χ chi
Introduction passage · 16_secondary.tex, line 18
th differentials α∪β and β∪α. The associated twisted Massey type cocycle is μ_p=α∪(p-s)-p∪α. Explicitly it equals μ_p(g,h)=α(g)(p(h)-α(h)β(h)) -p(g)χ(g)α(h). Let R_p denote the right side of [source reference]. Direct expansion of f(g)=α(g)p(g) gives R_p+d_χ f=μ_p. Thus [R_p]=[μ_p] in H^2(K,k(χ)). For this fixed p, the upper equation is soluble if and only if this class is zero.
Exact mathematical passage
\label{eq:secondaryidentity} R_p+d_\chi f=\mu_p.
\dimIntroduction passage · 4 passages
Search aliases: dim
Introduction passage · 15_semilinear_image.tex, line 18
The depth one image theorem. Let K_∞ be the cyclotomic tower. Write B for the additive image of beta on its Galois group and d=dim_F_7B. The inherited group theorem at actual depth one identifies H=ρ(G_K_∞)= {M∈SL_2(Oloc): l Mbmod7=(1 · x
Exact mathematical passage
d=\dim_{\F_7}B
\dim_{\F_7}Introduction passage · 1 passage
Search aliases: dim_F_7 dim F
Introduction passage · 15_semilinear_image.tex, line 18
The depth one image theorem. Let K_∞ be the cyclotomic tower. Write B for the additive image of beta on its Galois group and d=dim_F_7B. The inherited group theorem at actual depth one identifies H=ρ(G_K_∞)= {M∈SL_2(Oloc): l Mbmod7=(1 · x
Exact mathematical passage
d=\dim_{\F_7}B
FIntroduction passage · 51 passages
Search aliases: F
Introduction passage · 09_periods.tex, line 12
Put M=7^j and write [u] additively in F^×/F^× M for the local field F=K_v. Let X=[ϕ], Z=7[C]-3[A]-3[3]-5[sqrt5]. The entire marked period matrix becomes ([Π_ij])= Z+2X · -4X
Exact mathematical passage
F^\times/F^{\times M}
\FIntroduction passage · 26 passages
Search aliases: F
Introduction passage · 15_semilinear_image.tex, line 18
The depth one image theorem. Let K_∞ be the cyclotomic tower. Write B for the additive image of beta on its Galois group and d=dim_F_7B. The inherited group theorem at actual depth one identifies H=ρ(G_K_∞)= {M∈SL_2(Oloc): l Mbmod7=(1 · x
Exact mathematical passage
d=\dim_{\F_7}B
fIntroduction passage · 13 passages
Search aliases: f
Introduction passage · 16_secondary.tex, line 17
_23=p-s. They form a defining system with differentials α∪β and β∪α. The associated twisted Massey type cocycle is μ_p=α∪(p-s)-p∪α. Explicitly it equals μ_p(g,h)=α(g)(p(h)-α(h)β(h)) -p(g)χ(g)α(h). Let R_p denote the right side of [source reference]. Direct expansion of f(g)=α(g)p(g) gives R_p+d_χ f=μ_p. Thus [R_p]=[μ_p] in H^2(K,k(χ)). For this fixed p, the upper equation is soluble if and only if this class is zero.
Exact mathematical passage
f(g)=\alpha(g)p(g)
F^\timesIntroduction passage · 2 passages
Search aliases: F^× times
Introduction passage · 09_periods.tex, line 12
Put M=7^j and write [u] additively in F^×/F^× M for the local field F=K_v. Let X=[ϕ], Z=7[C]-3[A]-3[3]-5[sqrt5]. The entire marked period matrix becomes ([Π_ij])= Z+2X · -4X
Exact mathematical passage
F^\times/F^{\times M}
F^{\times M}Introduction passage · 1 passage
Search aliases: F^× M times
Introduction passage · 09_periods.tex, line 12
Put M=7^j and write [u] additively in F^×/F^× M for the local field F=K_v. Let X=[ϕ], Z=7[C]-3[A]-3[3]-5[sqrt5]. The entire marked period matrix becomes ([Π_ij])= Z+2X · -4X
Exact mathematical passage
F^\times/F^{\times M}
\F_7Introduction passage · 17 passages
Search aliases: F_7 F
Introduction passage · 15_semilinear_image.tex, line 18
The depth one image theorem. Let K_∞ be the cyclotomic tower. Write B for the additive image of beta on its Galois group and d=dim_F_7B. The inherited group theorem at actual depth one identifies H=ρ(G_K_∞)= {M∈SL_2(Oloc): l Mbmod7=(1 · x
Exact mathematical passage
d=\dim_{\F_7}B
GIntroduction passage · 9 passages
Search aliases: G
Introduction passage · 15_semilinear_image.tex, line 19
The depth one image theorem. Let K_∞ be the cyclotomic tower. Write B for the additive image of beta on its Galois group and d=dim_F_7B. The inherited group theorem at actual depth one identifies H=ρ(G_K_∞)= {M∈SL_2(Oloc): l Mbmod7=(1 · x 0 · 1), x∈ k, (M_21/7)bmod7∈ B }. In particular it contains the full kernel of reduction modulo forty nine. Its finite quotient orders are |Hbmod7^j|=7^6j-6+d (j≥2), |Hbmod7|=49. For the bounded patterns d=1, the orders at forty nine and 343 are 7^7 and 7^13. T…
Exact mathematical passage
\label{eq:imagegroup} H=\rho(G_{K_\infty})= \left\{M\in\mathrm{SL}_2(\Oloc): \begin{array}{l} M\bmod7=\left(\begin{smallmatrix}1&x\\0&1\end{smallmatrix}\right),\ x\in k,\\ (M_{21}/7)\bmod7\in B \end{array}\right\}.
gIntroduction passage · 60 passages
Search aliases: g
Introduction passage · 16_secondary.tex, line 4
Put s(g)=α(g)β(g). Expansion of the two cocycle laws gives ds=-(α∪β+β∪α). For a chosen primitive p, set D_12=-p and D_23=p-s. They form a defining system with differentials α∪β and β∪α. The associated twisted Massey type cocycle is μ_p=α∪(p-s)-p∪α. Explicitly it equals μ_p(g,h)=α(g)(p(h)-α(h)β(h)) -p(g)χ(g)α(h). Let R_p denote the right side of [source reference]. Direct expansion of f(g)=α(g)p(g) gives R_p+d_χ f=μ_p. Thus [R_p]=[μ_
Exact mathematical passage
s(g)=\alpha(g)\beta(g)
G_{K_\infty}Introduction passage · 1 passage
Search aliases: G_K_∞ infty
Introduction passage · 15_semilinear_image.tex, line 19
The depth one image theorem. Let K_∞ be the cyclotomic tower. Write B for the additive image of beta on its Galois group and d=dim_F_7B. The inherited group theorem at actual depth one identifies H=ρ(G_K_∞)= {M∈SL_2(Oloc): l Mbmod7=(1 · x 0 · 1), x∈ k, (M_21/7)bmod7∈ B }. In particular it contains the full kernel of reduction modulo forty nine. Its finite quotient orders are |Hbmod7^j|=7^6j-6+d (j≥2), |Hbmod7|=49. For the bounded patterns d=1, the orders at forty nine and 343 are 7^7 and 7^13. T…
Exact mathematical passage
\label{eq:imagegroup} H=\rho(G_{K_\infty})= \left\{M\in\mathrm{SL}_2(\Oloc): \begin{array}{l} M\bmod7=\left(\begin{smallmatrix}1&x\\0&1\end{smallmatrix}\right),\ x\in k,\\ (M_{21}/7)\bmod7\in B \end{array}\right\}.
HIntroduction passage · 35 passages
Search aliases: H
Introduction passage · 15_semilinear_image.tex, line 19
The depth one image theorem. Let K_∞ be the cyclotomic tower. Write B for the additive image of beta on its Galois group and d=dim_F_7B. The inherited group theorem at actual depth one identifies H=ρ(G_K_∞)= {M∈SL_2(Oloc): l Mbmod7=(1 · x 0 · 1), x∈ k, (M_21/7)bmod7∈ B }. In particular it contains the full kernel of reduction modulo forty nine. Its finite quotient orders are |Hbmod7^j|=7^6j-6+d (j≥2), |Hbmod7|=49. For the bounded patterns d=1, the orders at forty nine and 343 are 7^7 and 7^13. T…
Exact mathematical passage
\label{eq:imagegroup} H=\rho(G_{K_\infty})= \left\{M\in\mathrm{SL}_2(\Oloc): \begin{array}{l} M\bmod7=\left(\begin{smallmatrix}1&x\\0&1\end{smallmatrix}\right),\ x\in k,\\ (M_{21}/7)\bmod7\in B \end{array}\right\}.
hIntroduction passage · 43 passages
Search aliases: h
Introduction passage · 16_secondary.tex, line 13
cycle laws gives ds=-(α∪β+β∪α). For a chosen primitive p, set D_12=-p and D_23=p-s. They form a defining system with differentials α∪β and β∪α. The associated twisted Massey type cocycle is μ_p=α∪(p-s)-p∪α. Explicitly it equals μ_p(g,h)=α(g)(p(h)-α(h)β(h)) -p(g)χ(g)α(h). Let R_p denote the right side of [source reference]. Direct expansion of f(g)=α(g)p(g) gives R_p+d_χ f=μ_p. Thus [R_p]=[μ_p] in H^2(K,k(χ)). For this fixed p, the upper equation is soluble if and only if this class is zero.
Exact mathematical passage
\mu_p(g,h)=\alpha(g)(p(h)-\alpha(h)\beta(h)) -p(g)\chi(g)\alpha(h).
H^2Introduction passage · 8 passages
Search aliases: H^2
Introduction passage · 16_secondary.tex, line 21
ype cocycle is μ_p=α∪(p-s)-p∪α. Explicitly it equals μ_p(g,h)=α(g)(p(h)-α(h)β(h)) -p(g)χ(g)α(h). Let R_p denote the right side of [source reference]. Direct expansion of f(g)=α(g)p(g) gives R_p+d_χ f=μ_p. Thus [R_p]=[μ_p] in H^2(K,k(χ)). For this fixed p, the upper equation is soluble if and only if this class is zero.
Exact mathematical passage
H^2(K,k(\chi))
IIntroduction passage · 24 passages
Search aliases: I
Introduction passage · 09_periods.tex, line 17
Put M=7^j and write [u] additively in F^×/F^× M for the local field F=K_v. Let X=[ϕ], Z=7[C]-3[A]-3[3]-5[sqrt5]. The entire marked period matrix becomes ([Π_ij])= Z+2X · -4X -4X · Z-2X =ZI+2XR, R=1 · -2 -2 · -1, R^2=5I. Since four is invertible modulo M, every marked period coordinate vanishes if and only if X=Z=0. The matrix R belongs to this marked local period calculation. It is not silently identified with a global endomorphism or with an integral semilinear descent operator.
Exact mathematical passage
\label{eq:periodmatrix} ([\Pi_{ij}])= \begin{pmatrix}Z+2X&-4X\\-4X&Z-2X\end{pmatrix} =ZI+2XR,\qquad R=\begin{pmatrix}1&-2\\-2&-1\end{pmatrix},\quad R^2=5I.
iIntroduction passage · 9 passages
Search aliases: i
Introduction passage · 09_periods.tex, line 17
Put M=7^j and write [u] additively in F^×/F^× M for the local field F=K_v. Let X=[ϕ], Z=7[C]-3[A]-3[3]-5[sqrt5]. The entire marked period matrix becomes ([Π_ij])= Z+2X · -4X -4X · Z-2X =ZI+2XR, R=1 · -2 -2 · -1, R^2=5I. Since four is invertible modulo M, every marked period coordinate vanishes if and only if X=Z=0. The matrix R belongs to this marked local period calculation. It is not silently identified with a global endomorphism or with an integral semilinear descent operator.
Exact mathematical passage
\label{eq:periodmatrix} ([\Pi_{ij}])= \begin{pmatrix}Z+2X&-4X\\-4X&Z-2X\end{pmatrix} =ZI+2XR,\qquad R=\begin{pmatrix}1&-2\\-2&-1\end{pmatrix},\quad R^2=5I.
jIntroduction passage · 36 passages
Search aliases: j
Introduction passage · 09_periods.tex, line 12
Put M=7^j and write [u] additively in F^×/F^× M for the local field F=K_v. Let X=[ϕ], Z=7[C]-3[A]-3[3]-5[sqrt5]. The entire marked period matrix becomes ([Π_ij])= Z+2X · -4X
Exact mathematical passage
M=7^j
KIntroduction passage · 45 passages
Search aliases: K
Introduction passage · 09_periods.tex, line 12
Put M=7^j and write [u] additively in F^×/F^× M for the local field F=K_v. Let X=[ϕ], Z=7[C]-3[A]-3[3]-5[sqrt5]. The entire marked period matrix becomes ([Π_ij])= Z+2X · -4X
Exact mathematical passage
F=K_v
kIntroduction passage · 48 passages
Search aliases: k
Introduction passage · 15_semilinear_image.tex, line 19
The depth one image theorem. Let K_∞ be the cyclotomic tower. Write B for the additive image of beta on its Galois group and d=dim_F_7B. The inherited group theorem at actual depth one identifies H=ρ(G_K_∞)= {M∈SL_2(Oloc): l Mbmod7=(1 · x 0 · 1), x∈ k, (M_21/7)bmod7∈ B }. In particular it contains the full kernel of reduction modulo forty nine. Its finite quotient orders are |Hbmod7^j|=7^6j-6+d (j≥2), |Hbmod7|=49. For the bounded patterns d=1, the orders at forty nine and 343 are 7^7 and 7^13. T…
Exact mathematical passage
\label{eq:imagegroup} H=\rho(G_{K_\infty})= \left\{M\in\mathrm{SL}_2(\Oloc): \begin{array}{l} M\bmod7=\left(\begin{smallmatrix}1&x\\0&1\end{smallmatrix}\right),\ x\in k,\\ (M_{21}/7)\bmod7\in B \end{array}\right\}.
K_vIntroduction passage · 5 passages
Search aliases: K_v
Introduction passage · 09_periods.tex, line 12
Put M=7^j and write [u] additively in F^×/F^× M for the local field F=K_v. Let X=[ϕ], Z=7[C]-3[A]-3[3]-5[sqrt5]. The entire marked period matrix becomes ([Π_ij])= Z+2X · -4X
Exact mathematical passage
F=K_v
K_\inftyIntroduction passage · 3 passages
Search aliases: K_∞ infty
Introduction passage · 15_semilinear_image.tex, line 18
The depth one image theorem. Let K_∞ be the cyclotomic tower. Write B for the additive image of beta on its Galois group and d=dim_F_7B. The inherited group theorem at actual depth one identifies H=ρ(G_K_∞)= {M∈SL_2(Oloc): l Mbmod7=(1 · x
Exact mathematical passage
K_\infty
lIntroduction passage · 3 passages
Search aliases: l
Introduction passage · 15_semilinear_image.tex, line 19
The depth one image theorem. Let K_∞ be the cyclotomic tower. Write B for the additive image of beta on its Galois group and d=dim_F_7B. The inherited group theorem at actual depth one identifies H=ρ(G_K_∞)= {M∈SL_2(Oloc): l Mbmod7=(1 · x 0 · 1), x∈ k, (M_21/7)bmod7∈ B }. In particular it contains the full kernel of reduction modulo forty nine. Its finite quotient orders are |Hbmod7^j|=7^6j-6+d (j≥2), |Hbmod7|=49. For the bounded patterns d=1, the orders at forty nine and 343 are 7^7 and 7^13. T…
Exact mathematical passage
\label{eq:imagegroup} H=\rho(G_{K_\infty})= \left\{M\in\mathrm{SL}_2(\Oloc): \begin{array}{l} M\bmod7=\left(\begin{smallmatrix}1&x\\0&1\end{smallmatrix}\right),\ x\in k,\\ (M_{21}/7)\bmod7\in B \end{array}\right\}.
MIntroduction passage · 32 passages
Search aliases: M
Introduction passage · 09_periods.tex, line 12
Put M=7^j and write [u] additively in F^×/F^× M for the local field F=K_v. Let X=[ϕ], Z=7[C]-3[A]-3[3]-5[sqrt5]. The entire marked period matrix becomes ([Π_ij])= Z+2X · -4X
Exact mathematical passage
M=7^j
M_{21}Introduction passage · 1 passage
Search aliases: M_21
Introduction passage · 15_semilinear_image.tex, line 19
The depth one image theorem. Let K_∞ be the cyclotomic tower. Write B for the additive image of beta on its Galois group and d=dim_F_7B. The inherited group theorem at actual depth one identifies H=ρ(G_K_∞)= {M∈SL_2(Oloc): l Mbmod7=(1 · x 0 · 1), x∈ k, (M_21/7)bmod7∈ B }. In particular it contains the full kernel of reduction modulo forty nine. Its finite quotient orders are |Hbmod7^j|=7^6j-6+d (j≥2), |Hbmod7|=49. For the bounded patterns d=1, the orders at forty nine and 343 are 7^7 and 7^13. T…
Exact mathematical passage
\label{eq:imagegroup} H=\rho(G_{K_\infty})= \left\{M\in\mathrm{SL}_2(\Oloc): \begin{array}{l} M\bmod7=\left(\begin{smallmatrix}1&x\\0&1\end{smallmatrix}\right),\ x\in k,\\ (M_{21}/7)\bmod7\in B \end{array}\right\}.
\OlocIntroduction passage · 27 passages
Search aliases: Oloc
Introduction passage · 15_semilinear_image.tex, line 19
The depth one image theorem. Let K_∞ be the cyclotomic tower. Write B for the additive image of beta on its Galois group and d=dim_F_7B. The inherited group theorem at actual depth one identifies H=ρ(G_K_∞)= {M∈SL_2(Oloc): l Mbmod7=(1 · x 0 · 1), x∈ k, (M_21/7)bmod7∈ B }. In particular it contains the full kernel of reduction modulo forty nine. Its finite quotient orders are |Hbmod7^j|=7^6j-6+d (j≥2), |Hbmod7|=49. For the bounded patterns d=1, the orders at forty nine and 343 are 7^7 and 7^13. T…
Exact mathematical passage
\label{eq:imagegroup} H=\rho(G_{K_\infty})= \left\{M\in\mathrm{SL}_2(\Oloc): \begin{array}{l} M\bmod7=\left(\begin{smallmatrix}1&x\\0&1\end{smallmatrix}\right),\ x\in k,\\ (M_{21}/7)\bmod7\in B \end{array}\right\}.
pIntroduction passage · 122 passages
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Introduction passage · 16_secondary.tex, line 8
Put s(g)=α(g)β(g). Expansion of the two cocycle laws gives ds=-(α∪β+β∪α). For a chosen primitive p, set D_12=-p and D_23=p-s. They form a defining system with differentials α∪β and β∪α. The associated twisted Massey type cocycle is μ_p=α∪(p-s)-p∪α. Explicitly it equals μ_p(g,h)=α(g)(p(h)-α(h)β(h)) -p(g)χ(g)α(h). Let R_p denote the right side of [source reference]. Direct expansion of f(g)=α(g)p(g) gives R_p+d_χ f=μ_p. Thus [R_p]=[μ_p] in H^2(K,k(χ)). For this fixed p, the upper equation is solub…
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p
RIntroduction passage · 18 passages
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Introduction passage · 09_periods.tex, line 17
Put M=7^j and write [u] additively in F^×/F^× M for the local field F=K_v. Let X=[ϕ], Z=7[C]-3[A]-3[3]-5[sqrt5]. The entire marked period matrix becomes ([Π_ij])= Z+2X · -4X -4X · Z-2X =ZI+2XR, R=1 · -2 -2 · -1, R^2=5I. Since four is invertible modulo M, every marked period coordinate vanishes if and only if X=Z=0. The matrix R belongs to this marked local period calculation. It is not silently identified with a global endomorphism or with an integral semilinear descent operator.
Exact mathematical passage
\label{eq:periodmatrix} ([\Pi_{ij}])= \begin{pmatrix}Z+2X&-4X\\-4X&Z-2X\end{pmatrix} =ZI+2XR,\qquad R=\begin{pmatrix}1&-2\\-2&-1\end{pmatrix},\quad R^2=5I.
R^2Introduction passage · 2 passages
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Introduction passage · 09_periods.tex, line 17
Put M=7^j and write [u] additively in F^×/F^× M for the local field F=K_v. Let X=[ϕ], Z=7[C]-3[A]-3[3]-5[sqrt5]. The entire marked period matrix becomes ([Π_ij])= Z+2X · -4X -4X · Z-2X =ZI+2XR, R=1 · -2 -2 · -1, R^2=5I. Since four is invertible modulo M, every marked period coordinate vanishes if and only if X=Z=0. The matrix R belongs to this marked local period calculation. It is not silently identified with a global endomorphism or with an integral semilinear descent operator.
Exact mathematical passage
\label{eq:periodmatrix} ([\Pi_{ij}])= \begin{pmatrix}Z+2X&-4X\\-4X&Z-2X\end{pmatrix} =ZI+2XR,\qquad R=\begin{pmatrix}1&-2\\-2&-1\end{pmatrix},\quad R^2=5I.
R_pIntroduction passage · 5 passages
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Introduction passage · 16_secondary.tex, line 17
α). For a chosen primitive p, set D_12=-p and D_23=p-s. They form a defining system with differentials α∪β and β∪α. The associated twisted Massey type cocycle is μ_p=α∪(p-s)-p∪α. Explicitly it equals μ_p(g,h)=α(g)(p(h)-α(h)β(h)) -p(g)χ(g)α(h). Let R_p denote the right side of [source reference]. Direct expansion of f(g)=α(g)p(g) gives R_p+d_χ f=μ_p. Thus [R_p]=[μ_p] in H^2(K,k(χ)). For this fixed p, the upper equation is soluble if and only if this class is zero.
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R_p
sIntroduction passage · 36 passages
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Introduction passage · 16_secondary.tex, line 4
Put s(g)=α(g)β(g). Expansion of the two cocycle laws gives ds=-(α∪β+β∪α). For a chosen primitive p, set D_12=-p and D_23=p-s. They form a defining system with differentials α∪β and β∪α. The associated twisted Massey type cocycle is μ_p=α∪(p-s)-p∪α. Explicitly it equals μ_p(g,h)=α(g)(p(h)-α(h)β(h)) -p(g)χ(g)α(h). Let R_p denote the right side of [source reference]. Direct expansion of f(g)=α(g)p(g) gives R_p+d_χ f=μ_p. Thus [R_p]=[μ_
Exact mathematical passage
s(g)=\alpha(g)\beta(g)
\mathrm{SL}Introduction passage · 1 passage
Search aliases: SL mathrm
Introduction passage · 15_semilinear_image.tex, line 19
The depth one image theorem. Let K_∞ be the cyclotomic tower. Write B for the additive image of beta on its Galois group and d=dim_F_7B. The inherited group theorem at actual depth one identifies H=ρ(G_K_∞)= {M∈SL_2(Oloc): l Mbmod7=(1 · x 0 · 1), x∈ k, (M_21/7)bmod7∈ B }. In particular it contains the full kernel of reduction modulo forty nine. Its finite quotient orders are |Hbmod7^j|=7^6j-6+d (j≥2), |Hbmod7|=49. For the bounded patterns d=1, the orders at forty nine and 343 are 7^7 and 7^13. T…
Exact mathematical passage
\label{eq:imagegroup} H=\rho(G_{K_\infty})= \left\{M\in\mathrm{SL}_2(\Oloc): \begin{array}{l} M\bmod7=\left(\begin{smallmatrix}1&x\\0&1\end{smallmatrix}\right),\ x\in k,\\ (M_{21}/7)\bmod7\in B \end{array}\right\}.
\mathrm{SL}_2Introduction passage · 1 passage
Search aliases: SL_2 mathrm
Introduction passage · 15_semilinear_image.tex, line 19
The depth one image theorem. Let K_∞ be the cyclotomic tower. Write B for the additive image of beta on its Galois group and d=dim_F_7B. The inherited group theorem at actual depth one identifies H=ρ(G_K_∞)= {M∈SL_2(Oloc): l Mbmod7=(1 · x 0 · 1), x∈ k, (M_21/7)bmod7∈ B }. In particular it contains the full kernel of reduction modulo forty nine. Its finite quotient orders are |Hbmod7^j|=7^6j-6+d (j≥2), |Hbmod7|=49. For the bounded patterns d=1, the orders at forty nine and 343 are 7^7 and 7^13. T…
Exact mathematical passage
\label{eq:imagegroup} H=\rho(G_{K_\infty})= \left\{M\in\mathrm{SL}_2(\Oloc): \begin{array}{l} M\bmod7=\left(\begin{smallmatrix}1&x\\0&1\end{smallmatrix}\right),\ x\in k,\\ (M_{21}/7)\bmod7\in B \end{array}\right\}.
uIntroduction passage · 34 passages
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Introduction passage · 09_periods.tex, line 12
Put M=7^j and write [u] additively in F^×/F^× M for the local field F=K_v. Let X=[ϕ], Z=7[C]-3[A]-3[3]-5[sqrt5]. The entire marked period matrix becomes ([Π_ij])= Z+2X · -4X
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[u]
vIntroduction passage · 60 passages
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Introduction passage · 09_periods.tex, line 12
Put M=7^j and write [u] additively in F^×/F^× M for the local field F=K_v. Let X=[ϕ], Z=7[C]-3[A]-3[3]-5[sqrt5]. The entire marked period matrix becomes ([Π_ij])= Z+2X · -4X
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F=K_v
XIntroduction passage · 31 passages
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Introduction passage · 09_periods.tex, line 13
Put M=7^j and write [u] additively in F^×/F^× M for the local field F=K_v. Let X=[ϕ], Z=7[C]-3[A]-3[3]-5[sqrt5]. The entire marked period matrix becomes ([Π_ij])= Z+2X · -4X
Exact mathematical passage
X=[\phi],\qquad Z=7[C]-3[A]-3[3]-5[\sqrt5].
ZIntroduction passage · 10 passages
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Introduction passage · 09_periods.tex, line 13
Put M=7^j and write [u] additively in F^×/F^× M for the local field F=K_v. Let X=[ϕ], Z=7[C]-3[A]-3[3]-5[sqrt5]. The entire marked period matrix becomes ([Π_ij])= Z+2X · -4X
Exact mathematical passage
X=[\phi],\qquad Z=7[C]-3[A]-3[3]-5[\sqrt5].
\timesIntroduction passage · 9 passages
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Introduction passage · 09_periods.tex, line 12
Put M=7^j and write [u] additively in F^×/F^× M for the local field F=K_v. Let X=[ϕ], Z=7[C]-3[A]-3[3]-5[sqrt5]. The entire marked period matrix becomes ([Π_ij])= Z+2X · -4X
Exact mathematical passage
F^\times/F^{\times M}
\alphaIntroduction passage · 44 passages
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Introduction passage · 16_secondary.tex, line 4
Put s(g)=α(g)β(g). Expansion of the two cocycle laws gives ds=-(α∪β+β∪α). For a chosen primitive p, set D_12=-p and D_23=p-s. They form a defining system with differentials α∪β and β∪α. The associated twisted Massey type cocycle is μ_p=α∪(p-s)-p∪α. Explicitly it equals μ_p(g,h)=α(g)(p(h)-α(h)β(h)) -p(g)χ(g)α(h). Let R_p denote the right side of [source reference]. Direct expansion of f(g)=α(g)p(g) gives R_p+d_χ f=μ_p. Thus [R_p]=[μ_
Exact mathematical passage
s(g)=\alpha(g)\beta(g)
\betaIntroduction passage · 41 passages
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Introduction passage · 16_secondary.tex, line 4
Put s(g)=α(g)β(g). Expansion of the two cocycle laws gives ds=-(α∪β+β∪α). For a chosen primitive p, set D_12=-p and D_23=p-s. They form a defining system with differentials α∪β and β∪α. The associated twisted Massey type cocycle is μ_p=α∪(p-s)-p∪α. Explicitly it equals μ_p(g,h)=α(g)(p(h)-α(h)β(h)) -p(g)χ(g)α(h). Let R_p denote the right side of [source reference]. Direct expansion of f(g)=α(g)p(g) gives R_p+d_χ f=μ_p. Thus [R_p]=[μ_
Exact mathematical passage
s(g)=\alpha(g)\beta(g)
\muIntroduction passage · 22 passages
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Introduction passage · 16_secondary.tex, line 9
Put s(g)=α(g)β(g). Expansion of the two cocycle laws gives ds=-(α∪β+β∪α). For a chosen primitive p, set D_12=-p and D_23=p-s. They form a defining system with differentials α∪β and β∪α. The associated twisted Massey type cocycle is μ_p=α∪(p-s)-p∪α. Explicitly it equals μ_p(g,h)=α(g)(p(h)-α(h)β(h)) -p(g)χ(g)α(h). Let R_p denote the right side of [source reference]. Direct expansion of f(g)=α(g)p(g) gives R_p+d_χ f=μ_p. Thus [R_p]=[μ_p] in H^2(K,k(χ)). For this fixed p, the upper equation is solub…
Exact mathematical passage
\label{eq:mu} \mu_p=\alpha\cup(p-s)-p\cup\alpha.
\mu_pIntroduction passage · 6 passages
Search aliases: μ_p mu
Introduction passage · 16_secondary.tex, line 9
Put s(g)=α(g)β(g). Expansion of the two cocycle laws gives ds=-(α∪β+β∪α). For a chosen primitive p, set D_12=-p and D_23=p-s. They form a defining system with differentials α∪β and β∪α. The associated twisted Massey type cocycle is μ_p=α∪(p-s)-p∪α. Explicitly it equals μ_p(g,h)=α(g)(p(h)-α(h)β(h)) -p(g)χ(g)α(h). Let R_p denote the right side of [source reference]. Direct expansion of f(g)=α(g)p(g) gives R_p+d_χ f=μ_p. Thus [R_p]=[μ_p] in H^2(K,k(χ)). For this fixed p, the upper equation is solub…
Exact mathematical passage
\label{eq:mu} \mu_p=\alpha\cup(p-s)-p\cup\alpha.
\PiIntroduction passage · 2 passages
Search aliases: Π Pi
Introduction passage · 09_periods.tex, line 17
Put M=7^j and write [u] additively in F^×/F^× M for the local field F=K_v. Let X=[ϕ], Z=7[C]-3[A]-3[3]-5[sqrt5]. The entire marked period matrix becomes ([Π_ij])= Z+2X · -4X -4X · Z-2X =ZI+2XR, R=1 · -2 -2 · -1, R^2=5I. Since four is invertible modulo M, every marked period coordinate vanishes if and only if X=Z=0. The matrix R belongs to this marked local period calculation. It is not silently identified with a global endomorphism or with an integral semilinear descent operator.
Exact mathematical passage
\label{eq:periodmatrix} ([\Pi_{ij}])= \begin{pmatrix}Z+2X&-4X\\-4X&Z-2X\end{pmatrix} =ZI+2XR,\qquad R=\begin{pmatrix}1&-2\\-2&-1\end{pmatrix},\quad R^2=5I.
\Pi_{ij}Introduction passage · 1 passage
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Introduction passage · 09_periods.tex, line 17
Put M=7^j and write [u] additively in F^×/F^× M for the local field F=K_v. Let X=[ϕ], Z=7[C]-3[A]-3[3]-5[sqrt5]. The entire marked period matrix becomes ([Π_ij])= Z+2X · -4X -4X · Z-2X =ZI+2XR, R=1 · -2 -2 · -1, R^2=5I. Since four is invertible modulo M, every marked period coordinate vanishes if and only if X=Z=0. The matrix R belongs to this marked local period calculation. It is not silently identified with a global endomorphism or with an integral semilinear descent operator.
Exact mathematical passage
\label{eq:periodmatrix} ([\Pi_{ij}])= \begin{pmatrix}Z+2X&-4X\\-4X&Z-2X\end{pmatrix} =ZI+2XR,\qquad R=\begin{pmatrix}1&-2\\-2&-1\end{pmatrix},\quad R^2=5I.
\rhoIntroduction passage · 16 passages
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Introduction passage · 15_semilinear_image.tex, line 19
The depth one image theorem. Let K_∞ be the cyclotomic tower. Write B for the additive image of beta on its Galois group and d=dim_F_7B. The inherited group theorem at actual depth one identifies H=ρ(G_K_∞)= {M∈SL_2(Oloc): l Mbmod7=(1 · x 0 · 1), x∈ k, (M_21/7)bmod7∈ B }. In particular it contains the full kernel of reduction modulo forty nine. Its finite quotient orders are |Hbmod7^j|=7^6j-6+d (j≥2), |Hbmod7|=49. For the bounded patterns d=1, the orders at forty nine and 343 are 7^7 and 7^13. T…
Exact mathematical passage
\label{eq:imagegroup} H=\rho(G_{K_\infty})= \left\{M\in\mathrm{SL}_2(\Oloc): \begin{array}{l} M\bmod7=\left(\begin{smallmatrix}1&x\\0&1\end{smallmatrix}\right),\ x\in k,\\ (M_{21}/7)\bmod7\in B \end{array}\right\}.
\phiIntroduction passage · 33 passages
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Introduction passage · 09_periods.tex, line 13
Put M=7^j and write [u] additively in F^×/F^× M for the local field F=K_v. Let X=[ϕ], Z=7[C]-3[A]-3[3]-5[sqrt5]. The entire marked period matrix becomes ([Π_ij])= Z+2X · -4X
Exact mathematical passage
X=[\phi],\qquad Z=7[C]-3[A]-3[3]-5[\sqrt5].
\chiIntroduction passage · 57 passages
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Introduction passage · 16_secondary.tex, line 13
cycle laws gives ds=-(α∪β+β∪α). For a chosen primitive p, set D_12=-p and D_23=p-s. They form a defining system with differentials α∪β and β∪α. The associated twisted Massey type cocycle is μ_p=α∪(p-s)-p∪α. Explicitly it equals μ_p(g,h)=α(g)(p(h)-α(h)β(h)) -p(g)χ(g)α(h). Let R_p denote the right side of [source reference]. Direct expansion of f(g)=α(g)p(g) gives R_p+d_χ f=μ_p. Thus [R_p]=[μ_p] in H^2(K,k(χ)). For this fixed p, the upper equation is soluble if and only if this class is zero.
Exact mathematical passage
\mu_p(g,h)=\alpha(g)(p(h)-\alpha(h)\beta(h)) -p(g)\chi(g)\alpha(h).
\inIntroduction passage · 39 passages
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Introduction passage · 15_semilinear_image.tex, line 19
The depth one image theorem. Let K_∞ be the cyclotomic tower. Write B for the additive image of beta on its Galois group and d=dim_F_7B. The inherited group theorem at actual depth one identifies H=ρ(G_K_∞)= {M∈SL_2(Oloc): l Mbmod7=(1 · x 0 · 1), x∈ k, (M_21/7)bmod7∈ B }. In particular it contains the full kernel of reduction modulo forty nine. Its finite quotient orders are |Hbmod7^j|=7^6j-6+d (j≥2), |Hbmod7|=49. For the bounded patterns d=1, the orders at forty nine and 343 are 7^7 and 7^13. T…
Exact mathematical passage
\label{eq:imagegroup} H=\rho(G_{K_\infty})= \left\{M\in\mathrm{SL}_2(\Oloc): \begin{array}{l} M\bmod7=\left(\begin{smallmatrix}1&x\\0&1\end{smallmatrix}\right),\ x\in k,\\ (M_{21}/7)\bmod7\in B \end{array}\right\}.
\inftyIntroduction passage · 3 passages
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Introduction passage · 15_semilinear_image.tex, line 18
The depth one image theorem. Let K_∞ be the cyclotomic tower. Write B for the additive image of beta on its Galois group and d=dim_F_7B. The inherited group theorem at actual depth one identifies H=ρ(G_K_∞)= {M∈SL_2(Oloc): l Mbmod7=(1 · x
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K_\infty
\cupIntroduction passage · 16 passages
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Introduction passage · 16_secondary.tex, line 5
Put s(g)=α(g)β(g). Expansion of the two cocycle laws gives ds=-(α∪β+β∪α). For a chosen primitive p, set D_12=-p and D_23=p-s. They form a defining system with differentials α∪β and β∪α. The associated twisted Massey type cocycle is μ_p=α∪(p-s)-p∪α. Explicitly it equals μ_p(g,h)=α(g)(p(h)-α(h)β(h)) -p(g)χ(g)α(h). Let R_p denote the right side of [source reference]. Direct expansion of f(g)=α(g)p(g) gives R_p+d_χ f=μ_p. Thus [R_p]=[μ_p] in H^2(K,k(χ)). For this fixed p, the upper equation is solub…
Exact mathematical passage
\label{eq:ds} ds=-(\alpha\cup\beta+\beta\cup\alpha).
*Use in source · 3 passages
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Use in source · 22_affine.tex, line 77
[Affine fibers] If Stage P has one solution with parameter z_*, its complete solution set for these linear conditions is z_*+ker A. It has 49^3,49^3,49^2,49^3 choices in patterns A through D, respectively. If the target is not in the image, the set is empty. Az=t is equivalent to A(z-z_*)=0. The exact kernel dimensions give the counts over k. An additional common pointwise semilinear j, frame, and reference condition makes the relevant minus conductor coefficients lie in wF_7. Under that extra h…
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z_*
aUse in source · 64 passages
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Use in source · 02_signatures.tex, line 13
(D^n/4)^4. Apply this operation to each term separately. The new bases have exactly the same prime supports as the old bases, so primitivity is preserved. The converse follows by reducing any hypothetical primitive solution in this way. The proposition retains infinitely many signatures. It does not reduce them all to (3,5,7). A theorem for that signature also handles signatures (3a,5b,7c) after absorbing the multipliers into the bases, and the symmetric version with the first two terms exchange…
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(3a,5b,7c)
A'Use in source · 4 passages
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Use in source · 01_statement.tex, line 27
he common prime is 3, so this equality illustrates the conjecture rather than contradicting it. Its powered terms are 27,216,243. figplaceholder01_beal_termsThe equation in Example~[source reference] has a common prime in its bases. Exact equality alone does not supply primitivity. An ordinary division by gcd(A,B,C) is not a valid reduction to a primitive solution. If A=dA', B=dB', C=dC', the three terms carry factors d^x,d^y,d^z. Unless the exponents agree, division by one of these powers chang…
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A=dA'
a^{-1}Use in source · 3 passages
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Use in source · 10_reciprocity.tex, line 5
In the descended all inert regime, let M=Q(ζ_7) and let γ∈ H^1(Q,F_7(χ^-1)) restrict to the relevant beta class. Over M, the restriction is additive and satisfies γ(δ_a gδ_a^-1)=a^-1γ(g). At a conductor prime q≡1pmod7, there are six degree one primes in M. At q≡-1pmod7, there are three degree two primes. If z_q is a chosen seventh root of unity and ℓ_q is the corresponding seventh power logarithm, the local functional at the embedding ζ_7↦ z_q^b is b s_qℓ_q.
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\gamma(\delta_a g\delta_a^{-1})=a^{-1}\gamma(g).
A^2Use in source · 2 passages
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Use in source · 05_frey.tex, line 38
A characteristic seven cancellation. In characteristic seven, the Cartier coefficient calculation comes from f^3. Its relevant coefficients are [X^6]f^3 · =1575B^4, · [X^5]f^3 · =4320A^2B, [X^13]f^3 · =36A, · [X^12]f^3 · =315B^2. The diagonal lanes vanish modulo seven and the off diagonal lanes become M=0 · A^2B
Exact mathematical passage
[X^6]f^3&=1575B^4,& [X^5]f^3&=4320A^2B,\\ [X^{13}]f^3&=36A,& [X^{12}]f^3&=315B^2.
a^2Use in source · 1 passage
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Use in source · 13_deformation.tex, line 18
0 · 1), direct conjugation sends (p,q,b) to (p+ax^-1b, xq-2ap-a^2x^-1b, x^-1b). Thus the three coupled equations are β(gh) · =β(g)+χ(g)^-1β(h),
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(p+ax^{-1}b,\;xq-2ap-a^2x^{-1}b,\;x^{-1}b).
A^3Use in source · 12 passages
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Use in source · 03_scaling.tex, line 19
For a nonzero rational solution, define t=A^3/C^7 and λ=C^8/(AB^4). Substitution recovers [source reference]. If the original bases are positive, then 0<t<1 and λ>0. The rational parametrization therefore captures the entire positive rational problem. Its missing condition is exactly primitive integrality.
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t=A^3/C^7
a^3Use in source · 2 passages
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Use in source · 04_local_strata.tex, line 7
The complete local squareclass test. Write a hypothetical primitive solution as A=au^2, B=bv^2, C=cw^2, where a,b,c are positive pairwise coprime squarefree integers. The equation becomes a^3u^6+b^5v^10=c^7w^14. At a prime p, a local solution is called primitive when at most one base is divisible by p. Roots and bases are required to be nonzero in Z_p. The roots may depend on p. This last freedom is exactly why local existence can be weaker than an integer solution.
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\label{eq:strata} a^3u^6+b^5v^{10}=c^7w^{14}.
A^6Use in source · 5 passages
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Use in source · 05_frey.tex, line 24
ion this becomes 2^123^65^5C^14. Give X,A,B weights 1,5,3. Then f_λ^5A,λ^3B(λ X)=λ^6f_A,B(X). The six roots scale by λ, so their fifteen squared differences give discriminant weight thirty. The only nonnegative integer solutions of 5i+3j=30 are (i,j)=(6,0),(3,5),(0,10). The discriminant therefore has the form uA^6+vA^3B^5+wB^10.
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uA^6+vA^3B^5+wB^{10}
a^6Use in source · 1 passage
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Use in source · 12_fixed_seven.tex, line 45
Use the three coordinate basis X_j=[ϕ], H_j=[3], T_j=[7]. The third coordinate is valuation. The actual local scalar class in the window is κ_j=2ϵ wX_j+Λ_jH_j+7cT_j, where Λ_j=7ℓ_j(C_0)-3ℓ_j(A)-3-5d_j, ℓ_j(a)=fraclog(a^6)log(3^6)pmod7^j, d_j=fracℓ_j(5)2. At level 49, with Q_A=(A^6-1)/7 modulo 49 and q_0=(C_0^6-1)/7 modulo seven, Λ_2=47+38Q_A+14Q_A^2-7q_0pmod49. Both C_0 and the coordinate 7c[7] disappeared modulo seven. Neither may be omitted modulo forty nine.
Exact mathematical passage
\Lambda_j=7\ell_j(C_0)-3\ell_j(A)-3-5d_j, \quad \ell_j(a)=\frac{\log(a^6)}{\log(3^6)}\pmod{7^j}, \quad d_j=\frac{\ell_j(5)}2.
A^xUse in source · 3 passages
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Use in source · 01_statement.tex, line 4
Three powers and one common prime. [Beal] If positive integers A,B,C and integers x,y,z>2 satisfy A^x+B^y=C^z, then A,B,C have a common prime divisor. The exponents are allowed to differ. That is the essential extra freedom beyond the equal exponent Fermat equation. The common prime belongs to all three bases, not merely to the product of two of them. The historical statement and its generalized Fermat context are discussed in Mauldin's account and its authenticated published translation [source…
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\label{eq:beal} A^x+B^y=C^z,
a_{\mathfrak q}Use in source · 1 passage
Search aliases: a_ q mathfrak
Use in source · 06_modular_labels.tex, line 4
Suppose a Hecke eigenvalue a_ q is replaced by its minimal polynomial. A resultant against a possible curve trace polynomial asks whether some conjugate can match some allowable trace. Repeating the test at a different prime can choose a different conjugate. If the original representation requires one residue embedding for every prime, this relaxed sequence of tests may preserve candidates that the labelled comparison would exclude. An exclusion obtained from the relaxation remains valid. Surviv…
Exact mathematical passage
a_{\mathfrak q}
A_0Use in source · 2 passages
Search aliases: A_0
Use in source · 03_scaling.tex, line 5
A rational parametrization and its integral defect. For the signature (3,5,7), rational points are easy to manufacture. Primitive integer points are not. The following polynomial identity makes the difference exact. [Universal power identity] In any commutative ring, put s=1-t and A_0=t^12s^28, B_0=t^7s^17, C_0=t^5s^12. Then A_0^3+B_0^5=C_0^7. Weighted scaling (A,B,C)=(λ^35A_0,λ^21B_0,λ^15C_0) preserves this equality. The two terms on the left are t^36s^84 and t^35s^85. Their sum is t^35s^84(t+s…
Exact mathematical passage
A_0=t^{12}s^{28},\qquad B_0=t^7s^{17},\qquad C_0=t^5s^{12}.
A_0^3Use in source · 2 passages
Search aliases: A_0^3
Use in source · 03_scaling.tex, line 8
rization and its integral defect For the signature (3,5,7), rational points are easy to manufacture. Primitive integer points are not. The following polynomial identity makes the difference exact. [Universal power identity] In any commutative ring, put s=1-t and A_0=t^12s^28, B_0=t^7s^17, C_0=t^5s^12. Then A_0^3+B_0^5=C_0^7. Weighted scaling (A,B,C)=(λ^35A_0,λ^21B_0,λ^15C_0) preserves this equality. The two terms on the left are t^36s^84 and t^35s^85. Their sum is t^35s^84(t+s)=t^35s^84, which i…
Exact mathematical passage
A_0^3+B_0^5=C_0^7
a_3Use in source · 3 passages
Search aliases: a_3
Use in source · 08_global_coefficients.tex, line 3
Global coefficients, diagonal characters, and Frobenius precision. The natural class has a fixed global expansion α=a_ϕ[ϕ]+a_3[3]+a_5[sqrt5], a_3a_5≠0. This lies in a Kummer group tensored with k. It does not mean that an element of F_49 is an exponent of a base field unit. The three coefficients are normalized together, up to one common scalar. Using a different scalar at each completion would discard the global information that makes the reciprocity tests effective.
Exact mathematical passage
\label{eq:globalalpha} \alpha=a_\phi[\phi]+a_3[3]+a_5[\sqrt5],\qquad a_3a_5\ne0.
a_5Use in source · 3 passages
Search aliases: a_5
Use in source · 08_global_coefficients.tex, line 3
Global coefficients, diagonal characters, and Frobenius precision. The natural class has a fixed global expansion α=a_ϕ[ϕ]+a_3[3]+a_5[sqrt5], a_3a_5≠0. This lies in a Kummer group tensored with k. It does not mean that an element of F_49 is an exponent of a base field unit. The three coefficients are normalized together, up to one common scalar. Using a different scalar at each completion would discard the global information that makes the reciprocity tests effective.
Exact mathematical passage
\label{eq:globalalpha} \alpha=a_\phi[\phi]+a_3[3]+a_5[\sqrt5],\qquad a_3a_5\ne0.
a_qUse in source · 1 passage
Search aliases: a_q
Use in source · 20_localization.tex, line 13
, the two independent pro seven unit coordinates are ϕ,3, whose normalized logarithmic first digits are 5sqrt5,6, and the third coordinate is the uniformizer seven. Each of the nine inert support primes has one seventh power residue coordinate and one unramified coordinate. Real odd primary characters vanish. The total local dimension is 3+3+2·9=24 over k. The chosen g_q=ϕ+a_q generating the residue seventh power quotient supplies I_q. It need not generate the whole multiplicative group. Only it…
Exact mathematical passage
g_q=\phi+a_q
a_\phiUse in source · 4 passages
Search aliases: a_ϕ phi
Use in source · 08_global_coefficients.tex, line 3
Global coefficients, diagonal characters, and Frobenius precision. The natural class has a fixed global expansion α=a_ϕ[ϕ]+a_3[3]+a_5[sqrt5], a_3a_5≠0. This lies in a Kummer group tensored with k. It does not mean that an element of F_49 is an exponent of a base field unit. The three coefficients are normalized together, up to one common scalar. Using a different scalar at each completion would discard the global information that makes the reciprocity tests effective.
Exact mathematical passage
\label{eq:globalalpha} \alpha=a_\phi[\phi]+a_3[3]+a_5[\sqrt5],\qquad a_3a_5\ne0.
a_\ellUse in source · 2 passages
Search aliases: a_ℓ ell
Use in source · 03_scaling.tex, line 36
v_ℓ(t)<0 · Longrightarrowquad7mid -v_ℓ(t). When it exists, λ is unique. Write m=v_ℓ(t), n=v_ℓ(1-t) and h=v_ℓ(λ). Because t+(1-t)=1, the pair (m,n) belongs to one of four sectors: (m>0,n=0), (m=0,n>0), (m=n<0), (m=n=0). The output valuations are a_ℓ · =35h+12m+28n, b_ℓ · =21h+7m+17n, c_ℓ · =15h+5m+12n. They must be nonnegative, and their minimum must be zero. Equivalently, the minimum of 3a_ℓ,5b_ℓ,7c_ℓ must be zero.
Exact mathematical passage
a_\ell&=35h+12m+28n,\\ b_\ell&=21h+7m+17n,\\ c_\ell&=15h+5m+12n.
\AdUse in source · 2 passages
Search aliases: Ad
Use in source · 13_deformation.tex, line 14
beta(g) · -p(g)∈sl_2(k). The determinant condition makes X traceless. Multiplication shows that ρ is a representation if and only if X(gh)=X(g)+Ad(overlineρ_0(g))X(h). For overlineρ_0(g)=(x · a
Exact mathematical passage
X(gh)=X(g)+\Ad(\overline\rho_0(g))X(h).
\mathrm{an}Use in source · 1 passage
Search aliases: an mathrm
Use in source · 09_periods.tex, line 25
[Full torsion and period powers] For the stipulated split uniformization J^an= G_m^2/Λ with rational period lattice, full J[M] is F rational if and only if μ_M⊂ F and every coordinate of a basis of Λ is an Mth power in F. The uniformization gives 0⟶μ_M^2⟶ J[M] ⟶Λ/MΛ⟶0. If all torsion points are rational, so are μ_M^2. For λ∈Λ, choose an Mth root u in the torus. Rationality of its torsion class gives g(u)/u∈Λ for every g∈ G_F. Since λ is rational, this quotient also lies in μ_M^2. The lattice is …
Exact mathematical passage
J^{\mathrm{an}}=\mathbb G_m^2/\Lambda
bUse in source · 69 passages
Search aliases: b
Use in source · 02_signatures.tex, line 13
(D^n/4)^4. Apply this operation to each term separately. The new bases have exactly the same prime supports as the old bases, so primitivity is preserved. The converse follows by reducing any hypothetical primitive solution in this way. The proposition retains infinitely many signatures. It does not reduce them all to (3,5,7). A theorem for that signature also handles signatures (3a,5b,7c) after absorbing the multipliers into the bases, and the symmetric version with the first two terms exchange…
Exact mathematical passage
(3a,5b,7c)
B'Use in source · 1 passage
Search aliases: B'
Use in source · 01_statement.tex, line 27
prime is 3, so this equality illustrates the conjecture rather than contradicting it. Its powered terms are 27,216,243. figplaceholder01_beal_termsThe equation in Example~[source reference] has a common prime in its bases. Exact equality alone does not supply primitivity. An ordinary division by gcd(A,B,C) is not a valid reduction to a primitive solution. If A=dA', B=dB', C=dC', the three terms carry factors d^x,d^y,d^z. Unless the exponents agree, division by one of these powers changes the oth…
Exact mathematical passage
B=dB'
B^{10}Use in source · 1 passage
Search aliases: B^10
Use in source · 05_frey.tex, line 24
ion this becomes 2^123^65^5C^14. Give X,A,B weights 1,5,3. Then f_λ^5A,λ^3B(λ X)=λ^6f_A,B(X). The six roots scale by λ, so their fifteen squared differences give discriminant weight thirty. The only nonnegative integer solutions of 5i+3j=30 are (i,j)=(6,0),(3,5),(0,10). The discriminant therefore has the form uA^6+vA^3B^5+wB^10.
Exact mathematical passage
uA^6+vA^3B^5+wB^{10}
B^2Use in source · 5 passages
Search aliases: B^2
Use in source · 05_frey.tex, line 3
A curve that retains the exponent seven. The signature route uses the integral curve y^2+(X^3+B)y=2BX^3+3AX+B^2. Completing the square with Z=2y+X^3+B gives Z^2=f_A,B(X)=X^6+10BX^3+12AX+5B^2. This is the genus two family used in the recovered source. The comparison with the family of Pacetti and Villagra Torcomian is a rational change of coordinates, with their variables translated as (a,b,c,p)=(B,-C,A,7) [source reference]. Explicitly, put t=-B^5/A^3, d=-B^2/A, and substitute x_PV=d/X, y_PV=tZ/…
Exact mathematical passage
\label{eq:integralcurve} y^2+(X^3+B)y=2BX^3+3AX+B^2.
B^4Use in source · 2 passages
Search aliases: B^4
Use in source · 03_scaling.tex, line 19
For a nonzero rational solution, define t=A^3/C^7 and λ=C^8/(AB^4). Substitution recovers [source reference]. If the original bases are positive, then 0<t<1 and λ>0. The rational parametrization therefore captures the entire positive rational problem. Its missing condition is exactly primitive integrality.
Exact mathematical passage
\lambda=C^8/(AB^4)
B^5Use in source · 5 passages
Search aliases: B^5
Use in source · 05_frey.tex, line 10
n Completing the square with Z=2y+X^3+B gives Z^2=f_A,B(X)=X^6+10BX^3+12AX+5B^2. This is the genus two family used in the recovered source. The comparison with the family of Pacetti and Villagra Torcomian is a rational change of coordinates, with their variables translated as (a,b,c,p)=(B,-C,A,7) [source reference]. Explicitly, put t=-B^5/A^3, d=-B^2/A, and substitute x_PV=d/X, y_PV=tZ/X^3 into their equation y^2=5x^6-12x^5+10tx^3+t^2. Multiplication by X^6/t^2 gives [source reference]. The map …
Exact mathematical passage
t=-B^5/A^3
b^5Use in source · 3 passages
Search aliases: b^5
Use in source · 04_local_strata.tex, line 7
The complete local squareclass test. Write a hypothetical primitive solution as A=au^2, B=bv^2, C=cw^2, where a,b,c are positive pairwise coprime squarefree integers. The equation becomes a^3u^6+b^5v^10=c^7w^14. At a prime p, a local solution is called primitive when at most one base is divisible by p. Roots and bases are required to be nonzero in Z_p. The roots may depend on p. This last freedom is exactly why local existence can be weaker than an integer solution.
Exact mathematical passage
\label{eq:strata} a^3u^6+b^5v^{10}=c^7w^{14}.
B^yUse in source · 5 passages
Search aliases: B^y
Use in source · 01_statement.tex, line 4
Three powers and one common prime. [Beal] If positive integers A,B,C and integers x,y,z>2 satisfy A^x+B^y=C^z, then A,B,C have a common prime divisor. The exponents are allowed to differ. That is the essential extra freedom beyond the equal exponent Fermat equation. The common prime belongs to all three bases, not merely to the product of two of them. The historical statement and its generalized Fermat context are discussed in Mauldin's account and its authenticated published translation [source…
Exact mathematical passage
\label{eq:beal} A^x+B^y=C^z,
B_0Use in source · 4 passages
Search aliases: B_0
Use in source · 03_scaling.tex, line 5
A rational parametrization and its integral defect. For the signature (3,5,7), rational points are easy to manufacture. Primitive integer points are not. The following polynomial identity makes the difference exact. [Universal power identity] In any commutative ring, put s=1-t and A_0=t^12s^28, B_0=t^7s^17, C_0=t^5s^12. Then A_0^3+B_0^5=C_0^7. Weighted scaling (A,B,C)=(λ^35A_0,λ^21B_0,λ^15C_0) preserves this equality. The two terms on the left are t^36s^84 and t^35s^85. Their sum is t^35s^84(t+s…
Exact mathematical passage
A_0=t^{12}s^{28},\qquad B_0=t^7s^{17},\qquad C_0=t^5s^{12}.
b_0Use in source · 4 passages
Search aliases: b_0
Use in source · 13_deformation.tex, line 3
The coupled adjoint equations. Choose one reference Kummer representation over Oloc/49, ρ_0(g)=χ(g) · b_0(g) 0 · 1, whose reduction has the same residual α as the actual natural representation. The reference is built by lifting the coefficients of the same global classes [ϕ],[3],[sqrt5]. It has no seven valuation coordinate at the fixed place. It is an auxiliary representation, not the actual one.
Exact mathematical passage
\label{eq:rho0} \rho_0(g)=\begin{pmatrix}\chi(g)&b_0(g)\\0&1\end{pmatrix},
B_0^5Use in source · 1 passage
Search aliases: B_0^5
Use in source · 03_scaling.tex, line 8
rization and its integral defect For the signature (3,5,7), rational points are easy to manufacture. Primitive integer points are not. The following polynomial identity makes the difference exact. [Universal power identity] In any commutative ring, put s=1-t and A_0=t^12s^28, B_0=t^7s^17, C_0=t^5s^12. Then A_0^3+B_0^5=C_0^7. Weighted scaling (A,B,C)=(λ^35A_0,λ^21B_0,λ^15C_0) preserves this equality. The two terms on the left are t^36s^84 and t^35s^85. Their sum is t^35s^84(t+s)=t^35s^84, which i…
Exact mathematical passage
A_0^3+B_0^5=C_0^7
b_7Use in source · 1 passage
Search aliases: b_7
Use in source · 14_integral_trace.tex, line 14
ntative of X has p=β=0 and upper entry q a k(χ) Kummer cocycle. Define ν_7(X) to be its [7] valuation coordinate. Remaining aligned strict gauges change q by a coboundary and a scalar multiple of residual α, both of valuation zero. Thus the coordinate is well defined under these strict equivalences. The reference b_0 has valuation zero and the actual difference is b_7-b_0=7q, giving the fixed value below. This local alignment never erases beta from the global deformation.
Exact mathematical passage
b_7-b_0=7q
b_qUse in source · 7 passages
Search aliases: b_q
Use in source · 21_ramified_frobenius.tex, line 31
Because each support prime is inert in K/Q, restriction at a rational q≡1pmod7 multiplies the compatible rational Frobenius value by two: b_q=2∑_rD_q,rs_r. For q≡-1pmod7, the compatible squared rational Frobenius has beta value zero, from the cocycle factor 1+χ(q)^-1=0. This does not say beta is unramified there.
Exact mathematical passage
\label{eq:betaF} b_q=2\sum_rD_{q,r}s_r.
b_\ellUse in source · 2 passages
Search aliases: b_ℓ ell
Use in source · 03_scaling.tex, line 36
v_ℓ(t)<0 · Longrightarrowquad7mid -v_ℓ(t). When it exists, λ is unique. Write m=v_ℓ(t), n=v_ℓ(1-t) and h=v_ℓ(λ). Because t+(1-t)=1, the pair (m,n) belongs to one of four sectors: (m>0,n=0), (m=0,n>0), (m=n<0), (m=n=0). The output valuations are a_ℓ · =35h+12m+28n, b_ℓ · =21h+7m+17n, c_ℓ · =15h+5m+12n. They must be nonnegative, and their minimum must be zero. Equivalently, the minimum of 3a_ℓ,5b_ℓ,7c_ℓ must be zero.
Exact mathematical passage
a_\ell&=35h+12m+28n,\\ b_\ell&=21h+7m+17n,\\ c_\ell&=15h+5m+12n.
\operatorname{Br}Use in source · 1 passage
Search aliases: Br operatorname
Use in source · 16_secondary.tex, line 45
For K=Q(sqrt5), localization on H^2(K,k(χ)) is injective. To see the coefficient translation, restrict to L=K(ζ_7), whose degree over K is six. Restriction followed by corestriction multiplies by six, which is invertible on seven primary cohomology. Over L, the module is two trivial F_7 copies and μ_7 is present. Kummer theory identifies each copy of H^2(L,F_7) with Br(L)[7]. Brauer--Hasse--Noether gives local to global injectivity for the Brauer group. A class zero at every completion over K is…
Exact mathematical passage
\operatorname{Br}(L)[7]
cUse in source · 66 passages
Search aliases: c
Use in source · 02_signatures.tex, line 13
(D^n/4)^4. Apply this operation to each term separately. The new bases have exactly the same prime supports as the old bases, so primitivity is preserved. The converse follows by reducing any hypothetical primitive solution in this way. The proposition retains infinitely many signatures. It does not reduce them all to (3,5,7). A theorem for that signature also handles signatures (3a,5b,7c) after absorbing the multipliers into the bases, and the symmetric version with the first two terms exchange…
Exact mathematical passage
(3a,5b,7c)
C'Use in source · 1 passage
Search aliases: C'
Use in source · 01_statement.tex, line 27
3, so this equality illustrates the conjecture rather than contradicting it. Its powered terms are 27,216,243. figplaceholder01_beal_termsThe equation in Example~[source reference] has a common prime in its bases. Exact equality alone does not supply primitivity. An ordinary division by gcd(A,B,C) is not a valid reduction to a primitive solution. If A=dA', B=dB', C=dC', the three terms carry factors d^x,d^y,d^z. Unless the exponents agree, division by one of these powers changes the other coeffi…
Exact mathematical passage
C=dC'
C^{-1}Use in source · 1 passage
Search aliases: C^-1
Use in source · 15_semilinear_image.tex, line 3
Integral semilinearity and the unlabelled image. The integral semilinear theorem supplies more than equality of residual cohomology classes. Under the retained geometric inputs, an actual map between the conjugate curves induces an integral operator U=Cσ on the Tate lattice, with ρ(jgj^-1)=Cσ(ρ(g))C^-1, Cσ(C)=-ρ(j^2), C_21∈7^nOloc. It preserves the natural endpoint, the opposite endpoint, and the stable lattice segment. The minus sign in the square comes from the actual conjugate curve map. It c…
Exact mathematical passage
\rho(jgj^{-1})=C\sigma(\rho(g))C^{-1},\qquad C\sigma(C)=-\rho(j^2),\qquad C_{21}\in7^n\Oloc.
C^{14}Use in source · 1 passage
Search aliases: C^14
Use in source · 05_frey.tex, line 17
[Polynomial discriminant] The sextic in [source reference] satisfies disc(f_A,B)=2^123^65^5(A^3+B^5)^2. For a hypothetical (3,5,7) solution this becomes 2^123^65^5C^14. Give X,A,B weights 1,5,3. Then f_λ^5A,λ^3B(λ X)=λ^6f_A,B(X). The six roots scale by λ, so their fifteen squared differences give discriminant weight thirty. The only nonnegative integer solutions of 5i+3j=30 are (i,j)=(6,0),(3,5),(0,10). The discriminant therefore has the form uA^6+vA^3B^5+wB^10.
Exact mathematical passage
2^{12}3^65^5C^{14}
C^7Use in source · 3 passages
Search aliases: C^7
Use in source · 03_scaling.tex, line 19
For a nonzero rational solution, define t=A^3/C^7 and λ=C^8/(AB^4). Substitution recovers [source reference]. If the original bases are positive, then 0<t<1 and λ>0. The rational parametrization therefore captures the entire positive rational problem. Its missing condition is exactly primitive integrality.
Exact mathematical passage
t=A^3/C^7
c^7Use in source · 3 passages
Search aliases: c^7
Use in source · 04_local_strata.tex, line 7
The complete local squareclass test. Write a hypothetical primitive solution as A=au^2, B=bv^2, C=cw^2, where a,b,c are positive pairwise coprime squarefree integers. The equation becomes a^3u^6+b^5v^10=c^7w^14. At a prime p, a local solution is called primitive when at most one base is divisible by p. Roots and bases are required to be nonzero in Z_p. The roots may depend on p. This last freedom is exactly why local existence can be weaker than an integer solution.
Exact mathematical passage
\label{eq:strata} a^3u^6+b^5v^{10}=c^7w^{14}.
C^8Use in source · 1 passage
Search aliases: C^8
Use in source · 03_scaling.tex, line 19
For a nonzero rational solution, define t=A^3/C^7 and λ=C^8/(AB^4). Substitution recovers [source reference]. If the original bases are positive, then 0<t<1 and λ>0. The rational parametrization therefore captures the entire positive rational problem. Its missing condition is exactly primitive integrality.
Exact mathematical passage
\lambda=C^8/(AB^4)
C^zUse in source · 6 passages
Search aliases: C^z
Use in source · 01_statement.tex, line 4
Three powers and one common prime. [Beal] If positive integers A,B,C and integers x,y,z>2 satisfy A^x+B^y=C^z, then A,B,C have a common prime divisor. The exponents are allowed to differ. That is the essential extra freedom beyond the equal exponent Fermat equation. The common prime belongs to all three bases, not merely to the product of two of them. The historical statement and its generalized Fermat context are discussed in Mauldin's account and its authenticated published translation [source…
Exact mathematical passage
\label{eq:beal} A^x+B^y=C^z,
C_0Use in source · 7 passages
Search aliases: C_0
Use in source · 03_scaling.tex, line 5
A rational parametrization and its integral defect. For the signature (3,5,7), rational points are easy to manufacture. Primitive integer points are not. The following polynomial identity makes the difference exact. [Universal power identity] In any commutative ring, put s=1-t and A_0=t^12s^28, B_0=t^7s^17, C_0=t^5s^12. Then A_0^3+B_0^5=C_0^7. Weighted scaling (A,B,C)=(λ^35A_0,λ^21B_0,λ^15C_0) preserves this equality. The two terms on the left are t^36s^84 and t^35s^85. Their sum is t^35s^84(t+s…
Exact mathematical passage
A_0=t^{12}s^{28},\qquad B_0=t^7s^{17},\qquad C_0=t^5s^{12}.
C_0^{35}Use in source · 1 passage
Search aliases: C_0^35
Use in source · 09_periods.tex, line 48
At level seven, the last unit condition reduces to 75A being a seventh power. At level forty nine, exact multiplicative identities reduce it to 3·5^9A/C_0^35∈(κ_v^×)^49. Thus full forty nine torsion is rational if and only if 49mid Q-1, 7mid c, ϕ is a forty ninth power residue, and the displayed rational unit is as well. These are two way local criteria for the stipulated local Jacobian.
Exact mathematical passage
3\cdot5^9A/C_0^{35}\in(\kappa_v^\times)^{49}.
C_0^6Use in source · 1 passage
Search aliases: C_0^6
Use in source · 12_fixed_seven.tex, line 50
d H_j=[3], T_j=[7]. The third coordinate is valuation. The actual local scalar class in the window is κ_j=2ϵ wX_j+Λ_jH_j+7cT_j, where Λ_j=7ℓ_j(C_0)-3ℓ_j(A)-3-5d_j, ℓ_j(a)=fraclog(a^6)log(3^6)pmod7^j, d_j=fracℓ_j(5)2. At level 49, with Q_A=(A^6-1)/7 modulo 49 and q_0=(C_0^6-1)/7 modulo seven, Λ_2=47+38Q_A+14Q_A^2-7q_0pmod49. Both C_0 and the coordinate 7c[7] disappeared modulo seven. Neither may be omitted modulo forty nine.
Exact mathematical passage
q_0=(C_0^6-1)/7
C_0^7Use in source · 4 passages
Search aliases: C_0^7
Use in source · 03_scaling.tex, line 8
rization and its integral defect For the signature (3,5,7), rational points are easy to manufacture. Primitive integer points are not. The following polynomial identity makes the difference exact. [Universal power identity] In any commutative ring, put s=1-t and A_0=t^12s^28, B_0=t^7s^17, C_0=t^5s^12. Then A_0^3+B_0^5=C_0^7. Weighted scaling (A,B,C)=(λ^35A_0,λ^21B_0,λ^15C_0) preserves this equality. The two terms on the left are t^36s^84 and t^35s^85. Their sum is t^35s^84(t+s)=t^35s^84, which i…
Exact mathematical passage
A_0^3+B_0^5=C_0^7
C_2Use in source · 4 passages
Search aliases: C_2
Use in source · 08_global_coefficients.tex, line 24
[Discriminant collapse] Suppose a good Frobenius polynomial has integral coefficients and factorization P(T)=(T^2-τ T+Q)(T^2-σ(τ)T+Q) =T^4-ST^3+C_2T^2-QST+Q^2, where Oloc/Z_7 is unramified quadratic, σ is its nontrivial automorphism, all complex roots have absolute value sqrt Q, and τ≡ Q+1pmod7^rOloc. Then the integer D=S^2-4(C_2-2Q) satisfies 7^2rmid D and 0≤ D≤16Q. If 7^2r>16Q, then P(T)=(T^2-aT+Q)^2, a∈Z, |a|≤2sqrt Q, a≡ Q+1pmod7^r. Expansion gives S=τ+σ(τ), C_2-2Q=τσ(τ), and D=(τ-σ(τ))^2. Th…
Exact mathematical passage
P(T)=(T^2-\tau T+Q)(T^2-\sigma(\tau)T+Q) =T^4-ST^3+C_2T^2-QST+Q^2,
C_{21}Use in source · 1 passage
Search aliases: C_21
Use in source · 15_semilinear_image.tex, line 3
Integral semilinearity and the unlabelled image. The integral semilinear theorem supplies more than equality of residual cohomology classes. Under the retained geometric inputs, an actual map between the conjugate curves induces an integral operator U=Cσ on the Tate lattice, with ρ(jgj^-1)=Cσ(ρ(g))C^-1, Cσ(C)=-ρ(j^2), C_21∈7^nOloc. It preserves the natural endpoint, the opposite endpoint, and the stable lattice segment. The minus sign in the square comes from the actual conjugate curve map. It c…
Exact mathematical passage
\rho(jgj^{-1})=C\sigma(\rho(g))C^{-1},\qquad C\sigma(C)=-\rho(j^2),\qquad C_{21}\in7^n\Oloc.
C_6Use in source · 1 passage
Search aliases: C_6
Use in source · 17_local_flexibility.tex, line 4
For a finite module A, let A'=Hom(A,μ_7) and let G' be the finite group of the field cut out by that dual. The cited sufficient criterion is H^1(G',A')=0. For A=k with trivial action, A'≃ k(χ) and G'=C_6. Averaging kills its positive cohomology because six is invertible modulo seven. For A=k(χ), the source action cancels the cyclotomic target action, so the dual is trivial and G'=1. Both satisfy the criterion.
Exact mathematical passage
G'=C_6
c_qUse in source · 8 passages
Search aliases: c_q
Use in source · 21_ramified_frobenius.tex, line 37
The ordinary inertia generator differs from the frozen twisted residue convention. Its beta value is c_q=s_qℓ_q(Nm_F_q^2/F_qg_q) (q≡1pmod7), and c_q=s_qℓ_q(g_q) for q≡-1pmod7. The conversion factors are rrrrr 197:5 · 307:2 · 337:5 · 547:4 · 673:1
Exact mathematical passage
c_q=s_q\ell_q(\operatorname{Nm}_{\F_{q^2}/\F_q}g_q) \quad(q\equiv1\pmod7),
c_\ellUse in source · 2 passages
Search aliases: c_ℓ ell
Use in source · 03_scaling.tex, line 36
v_ℓ(t)<0 · Longrightarrowquad7mid -v_ℓ(t). When it exists, λ is unique. Write m=v_ℓ(t), n=v_ℓ(1-t) and h=v_ℓ(λ). Because t+(1-t)=1, the pair (m,n) belongs to one of four sectors: (m>0,n=0), (m=0,n>0), (m=n<0), (m=n=0). The output valuations are a_ℓ · =35h+12m+28n, b_ℓ · =21h+7m+17n, c_ℓ · =15h+5m+12n. They must be nonnegative, and their minimum must be zero. Equivalently, the minimum of 3a_ℓ,5b_ℓ,7c_ℓ must be zero.
Exact mathematical passage
a_\ell&=35h+12m+28n,\\ b_\ell&=21h+7m+17n,\\ c_\ell&=15h+5m+12n.
\mathrm{char}Use in source · 4 passages
Search aliases: char mathrm
Use in source · 22_affine.tex, line 56
A candidate p=p_0+z_char meets the fixed local condition precisely when the selected local character coordinates equal those of p_v^fix-p_0|_v. At support, substitution in [source reference] gives c_qz_char(F_q)-b_qz_char(T_q)=t_q(p_0). Together, these fifteen equations are exactly Az=t(p_0). Thus Stage P is soluble if and only if t(p_0)∈im A.
Exact mathematical passage
p=p_0+z_{\mathrm{char}}
\ClUse in source · 1 passage
Search aliases: Cl
Use in source · 19_bare_primitive.tex, line 21
ma[The dual Kummer kernel] For every finite Σ containing the prime above seven and all real places, Sha^1_Σ(K,μ_7) =ker(H^1(Gsig,μ_7) ⟶∏_v∈ΣH^1(K_v,μ_7))=0. The discriminant of K is five. Minkowski's ideal bound is sqrt5/2<2, so every ideal class has a representative of norm one. Thus Cl(K)=1 [source reference].
Exact mathematical passage
\Cl(K)=1
\cokerUse in source · 3 passages
Search aliases: coker
Use in source · 22_affine.tex, line 18
The homogeneous results. The complete exact matrices, row reductions, null bases, and rank minors give crrrl Pattern · rankA · dim_kker A · dim_kcoker A · Remaining difference support
Exact mathematical passage
\dim_k\coker A
D^2Use in source · 1 passage
Search aliases: D^2
Use in source · 10_reciprocity.tex, line 27
7c · d, det F=Q. Put d=1+7D and tr(F)=Q+1+7T. Then a=Q+7(T-D), and the determinant equation gives bc=T+(Q-1)D+7TD-7D^2. Modulo seven this becomes fractr(F)-Q-17=α(F)β(F) in k. This is the coefficient field rank two trace of the actual representation. It is not half the rational four dimensional trace. The hypothesis Q≡1pmod7 makes the extra diagonal term disappear. At other residue norms, [source reference] must be retained.
Exact mathematical passage
\label{eq:trace-product-exact} bc=T+(Q-1)D+7TD-7D^2.
D^nUse in source · 2 passages
Search aliases: D^n
Use in source · 02_signatures.tex, line 11
on[Prime or four reduction] If a primitive solution of [source reference] exists, there is a primitive solution whose three exponents each belong to {4}∪{p:p is an odd prime}. Conversely, absence of primitive solutions for every ordered signature in this set implies Beal. Let n>2. If n has an odd prime divisor p, write D^n=(D^n/p)^p. If it has no odd prime divisor, it is a power of two at least four, so write D^n=(D^n/4)^4. Apply this operation to each term separately. The new bases have exactly…
Exact mathematical passage
D^n=(D^{n/p})^p
D^{n/4}Use in source · 1 passage
Search aliases: D^n/4
Use in source · 02_signatures.tex, line 11
ists, there is a primitive solution whose three exponents each belong to {4}∪{p:p is an odd prime}. Conversely, absence of primitive solutions for every ordered signature in this set implies Beal. Let n>2. If n has an odd prime divisor p, write D^n=(D^n/p)^p. If it has no odd prime divisor, it is a power of two at least four, so write D^n=(D^n/4)^4. Apply this operation to each term separately. The new bases have exactly the same prime supports as the old bases, so primitivity is preserved. The …
Exact mathematical passage
D^n=(D^{n/4})^4
D^{n/p}Use in source · 1 passage
Search aliases: D^n/p
Use in source · 02_signatures.tex, line 11
on[Prime or four reduction] If a primitive solution of [source reference] exists, there is a primitive solution whose three exponents each belong to {4}∪{p:p is an odd prime}. Conversely, absence of primitive solutions for every ordered signature in this set implies Beal. Let n>2. If n has an odd prime divisor p, write D^n=(D^n/p)^p. If it has no odd prime divisor, it is a power of two at least four, so write D^n=(D^n/4)^4. Apply this operation to each term separately. The new bases have exactly…
Exact mathematical passage
D^n=(D^{n/p})^p
d^xUse in source · 1 passage
Search aliases: d^x
Use in source · 01_statement.tex, line 27
conjecture rather than contradicting it. Its powered terms are 27,216,243. figplaceholder01_beal_termsThe equation in Example~[source reference] has a common prime in its bases. Exact equality alone does not supply primitivity. An ordinary division by gcd(A,B,C) is not a valid reduction to a primitive solution. If A=dA', B=dB', C=dC', the three terms carry factors d^x,d^y,d^z. Unless the exponents agree, division by one of these powers changes the other coefficients. Later, a weighted scaling wi…
Exact mathematical passage
d^x,d^y,d^z
d^yUse in source · 1 passage
Search aliases: d^y
Use in source · 01_statement.tex, line 27
conjecture rather than contradicting it. Its powered terms are 27,216,243. figplaceholder01_beal_termsThe equation in Example~[source reference] has a common prime in its bases. Exact equality alone does not supply primitivity. An ordinary division by gcd(A,B,C) is not a valid reduction to a primitive solution. If A=dA', B=dB', C=dC', the three terms carry factors d^x,d^y,d^z. Unless the exponents agree, division by one of these powers changes the other coefficients. Later, a weighted scaling wi…
Exact mathematical passage
d^x,d^y,d^z
d^zUse in source · 1 passage
Search aliases: d^z
Use in source · 01_statement.tex, line 27
conjecture rather than contradicting it. Its powered terms are 27,216,243. figplaceholder01_beal_termsThe equation in Example~[source reference] has a common prime in its bases. Exact equality alone does not supply primitivity. An ordinary division by gcd(A,B,C) is not a valid reduction to a primitive solution. If A=dA', B=dB', C=dC', the three terms carry factors d^x,d^y,d^z. Unless the exponents agree, division by one of these powers changes the other coefficients. Later, a weighted scaling wi…
Exact mathematical passage
d^x,d^y,d^z
d_jUse in source · 1 passage
Search aliases: d_j
Use in source · 12_fixed_seven.tex, line 45
Use the three coordinate basis X_j=[ϕ], H_j=[3], T_j=[7]. The third coordinate is valuation. The actual local scalar class in the window is κ_j=2ϵ wX_j+Λ_jH_j+7cT_j, where Λ_j=7ℓ_j(C_0)-3ℓ_j(A)-3-5d_j, ℓ_j(a)=fraclog(a^6)log(3^6)pmod7^j, d_j=fracℓ_j(5)2. At level 49, with Q_A=(A^6-1)/7 modulo 49 and q_0=(C_0^6-1)/7 modulo seven, Λ_2=47+38Q_A+14Q_A^2-7q_0pmod49. Both C_0 and the coordinate 7c[7] disappeared modulo seven. Neither may be omitted modulo forty nine.
Exact mathematical passage
\Lambda_j=7\ell_j(C_0)-3\ell_j(A)-3-5d_j, \quad \ell_j(a)=\frac{\log(a^6)}{\log(3^6)}\pmod{7^j}, \quad d_j=\frac{\ell_j(5)}2.
D_{q,q}Use in source · 1 passage
Search aliases: D_q,q
Use in source · 21_ramified_frobenius.tex, line 19
At the chosen q place, write π_q=q u_q with u_q a unit. The same norm identity gives ℓ_q(u_q)=-∑_b=2^6ℓ_q(π_q(z_q^b)). Principal reciprocity on π_q therefore supplies the own conductor column D_q,q=∑_b=2^6(1-b)ℓ_q(π_q(z_q^b)). For r≠ q, the column is D_q,r=-∑_b∈reps_rbℓ_r(π_q(z_r^b)). The degree two representative convention is the one proved in [source reference]. These formulas use the same arithmetic Frobenius principal idele convention as the ordinary localization rows.
Exact mathematical passage
\label{eq:selfcolumn} D_{q,q}=\sum_{b=2}^6(1-b)\ell_q(\pi_q(z_q^b)).
D_{q,r}Use in source · 2 passages
Search aliases: D_q,r
Use in source · 21_ramified_frobenius.tex, line 23
n q place, write π_q=q u_q with u_q a unit. The same norm identity gives ℓ_q(u_q)=-∑_b=2^6ℓ_q(π_q(z_q^b)). Principal reciprocity on π_q therefore supplies the own conductor column D_q,q=∑_b=2^6(1-b)ℓ_q(π_q(z_q^b)). For r≠ q, the column is D_q,r=-∑_b∈reps_rbℓ_r(π_q(z_r^b)). The degree two representative convention is the one proved in [source reference]. These formulas use the same arithmetic Frobenius principal idele convention as the ordinary localization rows.
Exact mathematical passage
\label{eq:othercolumn} D_{q,r}=-\sum_{b\in\mathrm{reps}_r}b\ell_r(\pi_q(z_r^b)).
\detUse in source · 2 passages
Search aliases: det
Use in source · 10_reciprocity.tex, line 23
The same representation trace product. At depth one and an unramified place of residue norm Q≡1pmod7, write the actual Frobenius in the fixed natural frame as F=a · b 7c · d, det F=Q. Put d=1+7D and tr(F)=Q+1+7T. Then a=Q+7(T-D), and the determinant equation gives bc=T+(Q-1)D+7TD-7D^2. Modulo seven this becomes fractr(F)-Q-17=α(F)β(F) in k. This is the coefficient field rank two trace of the actual representation. It is not half the rational four dimensional trace. The hypothesis Q≡1pmod7 makes …
Exact mathematical passage
F=\begin{pmatrix}a&b\\7c&d\end{pmatrix},\qquad\det F=Q.
\operatorname{diag}Use in source · 1 passage
Search aliases: diag operatorname
Use in source · 07_natural_depth.tex, line 36
An actual finite segment of stable lattices. The inherited split toric calculation at three supplies Frobenius eigenvalues nine and one. Their difference eight is a seven adic unit. The projectors E_9=fracρ(g_0)-I8, E_1=frac9I-ρ(g_0)8 are complementary integral idempotents. Choose generators of their images. In this basis, ρ(g_0)=diag(9,1) and every lower entry is divisible by seven. Natural upper monodromy at three has a unit entry. Absolute irreducibility in characteristic zero forces some low…
Exact mathematical passage
\rho(g_0)=\operatorname{diag}(9,1)
\dim_kUse in source · 2 passages
Search aliases: dim_k dim
Use in source · 22_affine.tex, line 18
The homogeneous results. The complete exact matrices, row reductions, null bases, and rank minors give crrrl Pattern · rankA · dim_kker A · dim_kcoker A · Remaining difference support
Exact mathematical passage
\dim_k\ker A
\operatorname{disc}Use in source · 1 passage
Search aliases: disc operatorname
Use in source · 05_frey.tex, line 14
[Polynomial discriminant] The sextic in [source reference] satisfies disc(f_A,B)=2^123^65^5(A^3+B^5)^2. For a hypothetical (3,5,7) solution this becomes 2^123^65^5C^14. Give X,A,B weights 1,5,3. Then f_λ^5A,λ^3B(λ X)=λ^6f_A,B(X). The six roots scale by λ, so their fifteen squared differences give discriminant weight thirty. The only nonnegative integer solutions of 5i+3j=30 are (i,j)=(6,0),(3,5),(0,10). The discriminant therefore has the form uA^6+vA^3B^5+wB^10.
Exact mathematical passage
\operatorname{disc}(f_{A,B})=2^{12}3^65^5(A^3+B^5)^2.
EUse in source · 5 passages
Search aliases: E
Use in source · 07_natural_depth.tex, line 33
An actual finite segment of stable lattices. The inherited split toric calculation at three supplies Frobenius eigenvalues nine and one. Their difference eight is a seven adic unit. The projectors E_9=fracρ(g_0)-I8, E_1=frac9I-ρ(g_0)8 are complementary integral idempotents. Choose generators of their images. In this basis, ρ(g_0)=diag(9,1) and every lower entry is divisible by seven. Natural upper monodromy at three has a unit entry. Absolute irreducibility in characteristic zero forces some low…
Exact mathematical passage
E_9=\frac{\rho(g_0)-I}{8},\qquad E_1=\frac{9I-\rho(g_0)}8
eUse in source · 10 passages
Search aliases: e
Use in source · 04_local_strata.tex, line 45
Odd primes away from the coefficients. Let α,β,γ be the quadratic characters of a,b,c, and let e=(-1/p). A primitive residue solution with one zero base exists if β=γ, α=γ, α=eβ, respectively. The same cyclic group argument proves this, and the lifting construction replaces the zero root by a nonzero multiple of p.
Exact mathematical passage
e=(-1/p)
E_1Use in source · 1 passage
Search aliases: E_1
Use in source · 07_natural_depth.tex, line 33
An actual finite segment of stable lattices. The inherited split toric calculation at three supplies Frobenius eigenvalues nine and one. Their difference eight is a seven adic unit. The projectors E_9=fracρ(g_0)-I8, E_1=frac9I-ρ(g_0)8 are complementary integral idempotents. Choose generators of their images. In this basis, ρ(g_0)=diag(9,1) and every lower entry is divisible by seven. Natural upper monodromy at three has a unit entry. Absolute irreducibility in characteristic zero forces some low…
Exact mathematical passage
E_9=\frac{\rho(g_0)-I}{8},\qquad E_1=\frac{9I-\rho(g_0)}8
e_1Use in source · 1 passage
Search aliases: e_1
Use in source · 07_natural_depth.tex, line 43
Define n=min_g:ρ(g)_21≠0v_7(ρ(g)_21). This positive integer is attained. It is an invariant of the fixed natural frame, not a truncation selected for a computation. Every Oloc lattice stable under ρ is preserved by the two projectors, so up to homothety it has the form Λ_j=Oloc e_1+Oloc 7^j e_2. The unit upper monodromy forces j≥0. The attained lower entry minimum forces j≤ n. Conversely every j in that interval is stable. Thus the stable lattices form a finite segment with the natural and oppos…
Exact mathematical passage
\Lambda_j=\Oloc e_1+\Oloc 7^j e_2.
E_{12}Use in source · 2 passages
Search aliases: E_12
Use in source · 14_integral_trace.tex, line 17
The actual period valuation matrix is 7cI, so this probe has ρ(τ_j)=I+7cPE_12P^-1pmod7^j, where the same local toric line has the required alignment with the natural line. At actual depth one and a detecting g in the residual kernel, [source reference] gives D(τ,g)=49ν_7(X)β(g)pmod343. The actual fixed seven period comparison requires ν_7(X)=cfracv2ϵ in F_7. If β(g)≠0, the mixed trace determines this value.
Exact mathematical passage
\rho(\tau_j)=I+7cPE_{12}P^{-1}\pmod{7^j},
e_2Use in source · 1 passage
Search aliases: e_2
Use in source · 07_natural_depth.tex, line 43
Define n=min_g:ρ(g)_21≠0v_7(ρ(g)_21). This positive integer is attained. It is an invariant of the fixed natural frame, not a truncation selected for a computation. Every Oloc lattice stable under ρ is preserved by the two projectors, so up to homothety it has the form Λ_j=Oloc e_1+Oloc 7^j e_2. The unit upper monodromy forces j≥0. The attained lower entry minimum forces j≤ n. Conversely every j in that interval is stable. Thus the stable lattices form a finite segment with the natural and oppos…
Exact mathematical passage
\Lambda_j=\Oloc e_1+\Oloc 7^j e_2.
E_9Use in source · 1 passage
Search aliases: E_9
Use in source · 07_natural_depth.tex, line 33
An actual finite segment of stable lattices. The inherited split toric calculation at three supplies Frobenius eigenvalues nine and one. Their difference eight is a seven adic unit. The projectors E_9=fracρ(g_0)-I8, E_1=frac9I-ρ(g_0)8 are complementary integral idempotents. Choose generators of their images. In this basis, ρ(g_0)=diag(9,1) and every lower entry is divisible by seven. Natural upper monodromy at three has a unit entry. Absolute irreducibility in characteristic zero forces some low…
Exact mathematical passage
E_9=\frac{\rho(g_0)-I}{8},\qquad E_1=\frac{9I-\rho(g_0)}8
\epsUse in source · 13 passages
Search aliases: eps
Use in source · 02_signatures.tex, line 29
The conditional abc height bound. An abc estimate makes the role of the signature explicit. Let H=C^z, R=rad(ABC), s=frac1x+frac1y+frac1z. For a primitive solution, the powered terms are pairwise coprime and their radical is R. Positivity gives ABC<H^s. Assume the all epsilon abc estimate H≤ K_eps R^1+eps for every eps>0, with one constant K_eps independent of the triple. Then H<K_eps H^(1+eps)s. Whenever (1+eps)s<1, it follows that H<K_eps^1/(1-(1+eps)s). This is a conditional bound, not an unc…
Exact mathematical passage
\label{eq:abc-assumed} H\le K_\eps R^{1+\eps}
F^{-1}Use in source · 1 passage
Search aliases: F^-1
Use in source · 07_natural_depth.tex, line 90
If α∪β were a coboundary in the endpoint case, the two adjacent entries would extend to a representation in the three by three upper unitriangular group over k. That group has exponent seven. The tame relation FTF^-1=T^Q with Q≡1pmod7 would force the images to commute. Their central commutator is α(F)β(T)-α(T)β(F)=α(F)β(T)≠0. This contradiction proves local nonvanishing at the endpoint. The zero below the endpoint and nonzero at it are statements about the same labelled local representation.
Exact mathematical passage
FTF^{-1}=T^Q
f^3Use in source · 2 passages
Search aliases: f^3
Use in source · 05_frey.tex, line 37
A characteristic seven cancellation. In characteristic seven, the Cartier coefficient calculation comes from f^3. Its relevant coefficients are [X^6]f^3 · =1575B^4, · [X^5]f^3 · =4320A^2B,
Exact mathematical passage
f^3
F^{\times7^j}Use in source · 1 passage
Search aliases: F^×7^j times
Use in source · 12_fixed_seven.tex, line 34
Higher precision retains a valuation coordinate. For 1≤ j≤7c-1, the deep period errors vanish in F^×/F^×7^j. This follows because seventh powering takes 1+7^sOloc_F bijectively onto 1+7^s+1Oloc_F for s≥1. The guaranteed window does not assert that actual errors become nontrivial at the next precision.
Exact mathematical passage
F^\times/F^{\times7^j}
F_2Use in source · 2 passages
Search aliases: F_2
Use in source · 20_localization.tex, line 3
The complete ordinary localization map. Let z be an ordinary k valued character of Gsig. It is a difference of bare primitives, not beta and not either primitive itself. Local class field theory gives the following coordinate order: F_2,F_3,F_5,quad7:ϕ,3,7, I_q_1,F_q_1,…,I_q_9,F_q_9. We use the local Artin convention sending a selected uniformizer to arithmetic Frobenius. At five the uniformizer is sqrt5. At support primes it is the rational prime q. Every sign in the reciprocity rows and every …
Exact mathematical passage
\label{eq:localcoords} F_2,F_3,F_5,\quad7:\phi,3,7,\quad I_{q_1},F_{q_1},\ldots,I_{q_9},F_{q_9}.
F_{21}Use in source · 1 passage
Search aliases: F_21
Use in source · 07_natural_depth.tex, line 56
-y^2 · xy, where μ∈Oloc∖{0}, (x,y)∈Oloc^2 is primitive, t=v(μ)≥0, and FN=QN, NF=N, F_21∈π^nOloc, v(N_21)=n. Set h=v(Q-1). Then n=t+2s, s=v(y),qquad0≤ s≤ h. The lower nilpotent entry is -μ y^2, so its valuation gives n=t+2s. The common image and kernel line of N is generated by (x,y)^T. The two identities for F show that F-I has image in this line and kills no extra quotient direction. Write F=I+x
Exact mathematical passage
FN=QN,\qquad NF=N,\qquad F_{21}\in\pi^n\Oloc,\qquad v(N_{21})=n.
F_3Use in source · 2 passages
Search aliases: F_3
Use in source · 20_localization.tex, line 3
The complete ordinary localization map. Let z be an ordinary k valued character of Gsig. It is a difference of bare primitives, not beta and not either primitive itself. Local class field theory gives the following coordinate order: F_2,F_3,F_5,quad7:ϕ,3,7, I_q_1,F_q_1,…,I_q_9,F_q_9. We use the local Artin convention sending a selected uniformizer to arithmetic Frobenius. At five the uniformizer is sqrt5. At support primes it is the rational prime q. Every sign in the reciprocity rows and every …
Exact mathematical passage
\label{eq:localcoords} F_2,F_3,F_5,\quad7:\phi,3,7,\quad I_{q_1},F_{q_1},\ldots,I_{q_9},F_{q_9}.
\F_{49}Use in source · 3 passages
Search aliases: F_49 F
Use in source · 07_natural_depth.tex, line 7
The natural lattice and the depth of an opposite extension. The reducible branch uses two different kinds of radical. The base field is K=Q(sqrt5), ϕ=(1+sqrt5)/2. The coefficient ring and field are Oloc=Z_7[w]/(w^2-5), k=F_7[w]/(w^2-5)=F_49. The base unit ϕ defines Kummer classes. The coefficient w acts on a rank two representation. Although both involve a square root of five, they have different roles and are never identified by an implicit residue substitution.
Exact mathematical passage
\Oloc=\Z_7[w]/(w^2-5),\qquad k=\F_7[w]/(w^2-5)=\F_{49}.
\F_{49}^\timesUse in source · 1 passage
Search aliases: F_49^× F times
Use in source · 05_frey.tex, line 55
The global unit step uses ϕ=(1+sqrt5)/2, whose residue has order sixteen in F_49^×. The local weight possibilities at seven are 0,1,7,8, up to a common reciprocity sign. Applying reciprocity to the eighth power of each constituent gives overlineϕ^8e=1 and rules out the mixed weights 1,7. The determinant norm relation δ^8=χ^2 then leaves one constituent unramified at seven, without assuming an unproved rank two determinant normalization in this argument. Its eighth power is everywhere unramified,…
Exact mathematical passage
\F_{49}^\times
F_5Use in source · 2 passages
Search aliases: F_5
Use in source · 20_localization.tex, line 3
The complete ordinary localization map. Let z be an ordinary k valued character of Gsig. It is a difference of bare primitives, not beta and not either primitive itself. Local class field theory gives the following coordinate order: F_2,F_3,F_5,quad7:ϕ,3,7, I_q_1,F_q_1,…,I_q_9,F_q_9. We use the local Artin convention sending a selected uniformizer to arithmetic Frobenius. At five the uniformizer is sqrt5. At support primes it is the rational prime q. Every sign in the reciprocity rows and every …
Exact mathematical passage
\label{eq:localcoords} F_2,F_3,F_5,\quad7:\phi,3,7,\quad I_{q_1},F_{q_1},\ldots,I_{q_9},F_{q_9}.
\F_7^\timesUse in source · 1 passage
Search aliases: F_7^× F times
Use in source · 10_reciprocity.tex, line 51
Coefficient rigidity without radical identification. The recovered coefficient theorem, initially in the descended depth one regime and later in the all depth semilinear formulation, normalizes the natural class as α=vw[ϕ]+r[3]+[sqrt5], v,r∈F_7^×. Its mechanism retains the actual joint finite quotient. The three Kummer classes are independent after cyclotomic restriction. Distinct cyclotomic eigenspaces separate the natural and opposite images. Chebotarev then realizes joint vectors at good spli…
Exact mathematical passage
\label{eq:rigidalpha} \alpha=vw[\phi]+r[3]+[\sqrt5],\qquad v,r\in\F_7^\times.
f_{A,B}Use in source · 4 passages
Search aliases: f_A,B
Use in source · 05_frey.tex, line 7
A curve that retains the exponent seven. The signature route uses the integral curve y^2+(X^3+B)y=2BX^3+3AX+B^2. Completing the square with Z=2y+X^3+B gives Z^2=f_A,B(X)=X^6+10BX^3+12AX+5B^2. This is the genus two family used in the recovered source. The comparison with the family of Pacetti and Villagra Torcomian is a rational change of coordinates, with their variables translated as (a,b,c,p)=(B,-C,A,7) [source reference]. Explicitly, put t=-B^5/A^3, d=-B^2/A, and substitute x_PV=d/X, y_PV=tZ/…
Exact mathematical passage
\label{eq:sextic} Z^2=f_{A,B}(X)=X^6+10BX^3+12AX+5B^2.
\F_pUse in source · 1 passage
Search aliases: F_p F
Use in source · 16_secondary.tex, line 55
General Massey vanishing theorems do not erase these qualifications. The all prime theorem of Min avc and T an concerns ordinary products over absolute Galois groups with trivial F_p coefficients [source reference]. The JEMS theorem cited in some older discussion has a mod two statement [source reference]. The 2026 strong vanishing theorem likewise concerns its stated ordinary coefficient setting and may introduce additional ramification [source reference]. None is a theorem that the one twisted…
Exact mathematical passage
\F_p
\F_p^\timesUse in source · 1 passage
Search aliases: F_p^× F times
Use in source · 04_local_strata.tex, line 42
If pmid a, primitivity forces the other two bases to be units. Reduction gives b^5v^10=c^7w^14. The quadratic character of this equality proves the necessary symbol condition. Conversely, v^10w^-14 ranges over the squares of F_p^×, because gcd(10,14,p-1)=2. The symbol condition lets us choose the two unit roots, then lift. The other two cases use gcd(6,14,p-1)=2 and gcd(6,10,p-1)=2. Moving a term across the equality gives the minus sign when pmid c.
Exact mathematical passage
\F_p^\times
F_qUse in source · 7 passages
Search aliases: F_q
Use in source · 20_localization.tex, line 35
An explicit image parametrization. Put a=z_7(3) and x_i=z(I_q_i). Let ℓ_i be the seventh power logarithm normalized by ℓ_i(g_q_i)=1. The supporting symbols of ϕ vanish. For rational seven adic units define ℓ_7(t)=fraclog(t^6)log(3^6)pmod7 =6fract^6-17pmod7. Principal reciprocity determines the remaining coordinates: z_7(ϕ) · =0, z(F_2) · =-5a-∑_iℓ_i(2)x_i, z(F_3) · =-a-∑_iℓ_i(3)x_i, z(F_5) · =-4a-∑_iℓ_i(sqrt5)x_i, z_7(7) · =-∑_iℓ_i(7)x_i, z(F_q) · =-ℓ_7(q)a-∑_i:q_i≠ qℓ_i(q)x_i. At its own conduc…
Exact mathematical passage
z_7(\phi)&=0,\label{eq:Jphi}\\ z(F_2)&=-5a-\sum_i\ell_i(2)x_i,\label{eq:J2}\\ z(F_3)&=-a-\sum_i\ell_i(3)x_i,\label{eq:J3}\\ z(F_5)&=-4a-\sum_i\ell_i(\sqrt5)x_i,\label{eq:J5}\\ z_7(7)&=-\sum_i\ell_i(7)x_i,\label{eq:J7}\\ z(F_q)&=-\ell_7(q)a-\sum_{i:q_i\ne q}\ell_i(q)x_i.\label{eq:Jq}
\F_qUse in source · 3 passages
Search aliases: F_q F
Use in source · 06_modular_labels.tex, line 15
From point counts to a common polynomial. For a smooth genus two curve over F_q, let N_1,N_2 count its points over F_q,F_q^2, including infinity. Write s_1=q+1-N_1, s_2=fracs_1^2-(q^2+1-N_2)2. Its Frobenius polynomial is T^4-s_1T^3+s_2T^2-qs_1T+q^2. In the RM setting, the rank two traces are the roots of X^2-s_1X+(s_2-2q). The two roots belong to one common quartic. Choosing one root from one fiber and another root from an unrelated fiber can enlarge a necessary locus, but does not give a compat…
Exact mathematical passage
\F_q
f_qUse in source · 1 passage
Search aliases: f_q
Use in source · 10_reciprocity.tex, line 19
_q(x^q)=-ℓ_q(x). Therefore (-b)ℓ_q(x^q)=bℓ_q(x). The functional is well defined on prime ideals, not just on the six embeddings. Principal reciprocity gives, for a test prime generator π_l outside the conductor, R_l,q=-∑_b∈reps_qbℓ_q(π_l(z_q^b)) =-f_q^-1∑_b=1^6bℓ_q(π_l(z_q^b)), where f_q=1 or two. Using all six terms without dividing by two in the second case would double count each prime. Constant weights or inverse b weights give different modules.
Exact mathematical passage
f_q=1
f_q^{-1}Use in source · 1 passage
Search aliases: f_q^-1
Use in source · 10_reciprocity.tex, line 15
For degree two, the indices b and -b represent the same prime. Frobenius sends a residue x to x^q, and ℓ_q(x^q)=-ℓ_q(x). Therefore (-b)ℓ_q(x^q)=bℓ_q(x). The functional is well defined on prime ideals, not just on the six embeddings. Principal reciprocity gives, for a test prime generator π_l outside the conductor, R_l,q=-∑_b∈reps_qbℓ_q(π_l(z_q^b)) =-f_q^-1∑_b=1^6bℓ_q(π_l(z_q^b)), where f_q=1 or two. Using all six terms without dividing by two in the second case would double count each prime. Con…
Exact mathematical passage
\label{eq:orbitreciprocity} R_{l,q}=-\sum_{b\in\mathrm{reps}_q}b\ell_q(\pi_l(z_q^b)) =-f_q^{-1}\sum_{b=1}^6b\ell_q(\pi_l(z_q^b)),
\F_{q^2}Use in source · 2 passages
Search aliases: F_q^2 F
Use in source · 06_modular_labels.tex, line 15
From point counts to a common polynomial. For a smooth genus two curve over F_q, let N_1,N_2 count its points over F_q,F_q^2, including infinity. Write s_1=q+1-N_1, s_2=fracs_1^2-(q^2+1-N_2)2. Its Frobenius polynomial is T^4-s_1T^3+s_2T^2-qs_1T+q^2. In the RM setting, the rank two traces are the roots of X^2-s_1X+(s_2-2q). The two roots belong to one common quartic. Choosing one root from one fiber and another root from an unrelated fiber can enlarge a necessary locus, but does not give a compat…
Exact mathematical passage
\F_q,\F_{q^2}
F_{q_1}Use in source · 1 passage
Search aliases: F_q_1
Use in source · 20_localization.tex, line 3
The complete ordinary localization map. Let z be an ordinary k valued character of Gsig. It is a difference of bare primitives, not beta and not either primitive itself. Local class field theory gives the following coordinate order: F_2,F_3,F_5,quad7:ϕ,3,7, I_q_1,F_q_1,…,I_q_9,F_q_9. We use the local Artin convention sending a selected uniformizer to arithmetic Frobenius. At five the uniformizer is sqrt5. At support primes it is the rational prime q. Every sign in the reciprocity rows and every …
Exact mathematical passage
\label{eq:localcoords} F_2,F_3,F_5,\quad7:\phi,3,7,\quad I_{q_1},F_{q_1},\ldots,I_{q_9},F_{q_9}.
F_{q_9}Use in source · 1 passage
Search aliases: F_q_9
Use in source · 20_localization.tex, line 3
The complete ordinary localization map. Let z be an ordinary k valued character of Gsig. It is a difference of bare primitives, not beta and not either primitive itself. Local class field theory gives the following coordinate order: F_2,F_3,F_5,quad7:ϕ,3,7, I_q_1,F_q_1,…,I_q_9,F_q_9. We use the local Artin convention sending a selected uniformizer to arithmetic Frobenius. At five the uniformizer is sqrt5. At support primes it is the rational prime q. Every sign in the reciprocity rows and every …
Exact mathematical passage
\label{eq:localcoords} F_2,F_3,F_5,\quad7:\phi,3,7,\quad I_{q_1},F_{q_1},\ldots,I_{q_9},F_{q_9}.
f_{\lambda^5A,\lambda^3B}Use in source · 1 passage
Search aliases: f_λ^5A,λ^3B lambda lambda
Use in source · 05_frey.tex, line 21
[Polynomial discriminant] The sextic in [source reference] satisfies disc(f_A,B)=2^123^65^5(A^3+B^5)^2. For a hypothetical (3,5,7) solution this becomes 2^123^65^5C^14. Give X,A,B weights 1,5,3. Then f_λ^5A,λ^3B(λ X)=λ^6f_A,B(X). The six roots scale by λ, so their fifteen squared differences give discriminant weight thirty. The only nonnegative integer solutions of 5i+3j=30 are (i,j)=(6,0),(3,5),(0,10). The discriminant therefore has the form uA^6+vA^3B^5+wB^10.
Exact mathematical passage
f_{\lambda^5A,\lambda^3B}(\lambda X)=\lambda^6f_{A,B}(X).
\mathrm{fix}Use in source · 10 passages
Search aliases: fix mathrm
Use in source · FORMAL_STATUS.md, line 118
β(g)=(χ(g)^-1-1)λ, p^fix(g)=λχ(g)^-1α(g).
Exact mathematical passage
\beta(g)=(\chi(g)^{-1}-1)\lambda, \qquad p^{\mathrm{fix}}(g)=\lambda\chi(g)^{-1}\alpha(g).
\mathbb GUse in source · 1 passage
Search aliases: G mathbb
Use in source · 09_periods.tex, line 25
[Full torsion and period powers] For the stipulated split uniformization J^an= G_m^2/Λ with rational period lattice, full J[M] is F rational if and only if μ_M⊂ F and every coordinate of a basis of Λ is an Mth power in F. The uniformization gives 0⟶μ_M^2⟶ J[M] ⟶Λ/MΛ⟶0. If all torsion points are rational, so are μ_M^2. For λ∈Λ, choose an Mth root u in the torus. Rationality of its torsion class gives g(u)/u∈Λ for every g∈ G_F. Since λ is rational, this quotient also lies in μ_M^2. The lattice is …
Exact mathematical passage
J^{\mathrm{an}}=\mathbb G_m^2/\Lambda
G'Use in source · 4 passages
Search aliases: G'
Use in source · 17_local_flexibility.tex, line 4
For a finite module A, let A'=Hom(A,μ_7) and let G' be the finite group of the field cut out by that dual. The cited sufficient criterion is H^1(G',A')=0. For A=k with trivial action, A'≃ k(χ) and G'=C_6. Averaging kills its positive cohomology because six is invertible modulo seven. For A=k(χ), the source action cancels the cyclotomic target action, so the dual is trivial and G'=1. Both satisfy the criterion.
Exact mathematical passage
G'
g_0Use in source · 3 passages
Search aliases: g_0
Use in source · 07_natural_depth.tex, line 33
An actual finite segment of stable lattices. The inherited split toric calculation at three supplies Frobenius eigenvalues nine and one. Their difference eight is a seven adic unit. The projectors E_9=fracρ(g_0)-I8, E_1=frac9I-ρ(g_0)8 are complementary integral idempotents. Choose generators of their images. In this basis, ρ(g_0)=diag(9,1) and every lower entry is divisible by seven. Natural upper monodromy at three has a unit entry. Absolute irreducibility in characteristic zero forces some low…
Exact mathematical passage
E_9=\frac{\rho(g_0)-I}{8},\qquad E_1=\frac{9I-\rho(g_0)}8
G_bUse in source · 2 passages
Search aliases: G_b
Use in source · 15_semilinear_image.tex, line 35
The image group contains upper and lower elements T_a,G_b with tr(T_aG_b)-tr(T_a)-tr(G_b)+2=49abpmod343. All leading residue values occur as the elements vary. This flexibility does not allow a fixed local uniformizer probe to be replaced by any convenient upper element. Preimages in the Galois group may have different local membership and different arithmetic labels. Knowledge of the unlabelled image is therefore weaker than the quotient map and compatible local embeddings required by the mixed…
Exact mathematical passage
T_a,G_b
G_FUse in source · 1 passage
Search aliases: G_F
Use in source · 09_periods.tex, line 33
dinate of a basis of Λ is an Mth power in F. The uniformization gives 0⟶μ_M^2⟶ J[M] ⟶Λ/MΛ⟶0. If all torsion points are rational, so are μ_M^2. For λ∈Λ, choose an Mth root u in the torus. Rationality of its torsion class gives g(u)/u∈Λ for every g∈ G_F. Since λ is rational, this quotient also lies in μ_M^2. The lattice is torsion free, so the intersection is trivial. Thus u is rational. Every period coordinate is an Mth power.
Exact mathematical passage
g\in G_F
g_jUse in source · 4 passages
Search aliases: g_j
Use in source · 12_fixed_seven.tex, line 56
At higher actual lattice depth, put m=lfloor n/2rfloor+1. The inherited diagonal theorem supplies an actual global upper extension α_j for 1≤ j≤ m. In the common range 1≤ j≤min(m,7c-1), its local coordinates satisfy U_j=2ϵ wg_j, V_j=Λ_jg_j, W_j=7cg_j for one unit g_j. The exact valuation matrix preserves the valuation identity at every legitimate triangular precision 1≤ j≤ m, even beyond the proved unit period window. Consequently v_7^trunc(W_j)=min(j,1+v_7(c)). At depth one, only the residual g…
Exact mathematical passage
U_j=2\epsilon wg_j
G_KUse in source · 1 passage
Search aliases: G_K
Use in source · 15_semilinear_image.tex, line 15
of j, one has ρ(j^2)=I. The coefficient unit (1+w)/2 has norm minus one, so dividing U by it gives a semilinear involution. An adapted basis of its plus lattice produces a global endpoint preserving integral frame with ρ(jgj^-1)=σ(ρ(g)). This is an existential normal form. The unknown candidate frame has not been numerically constructed. Its plus lattice need not be G_K stable, so the statement is not ordinary rank two Q_7 descent.
Exact mathematical passage
G_K
\mathbb G_m^2Use in source · 1 passage
Search aliases: G_m^2 mathbb
Use in source · 09_periods.tex, line 25
[Full torsion and period powers] For the stipulated split uniformization J^an= G_m^2/Λ with rational period lattice, full J[M] is F rational if and only if μ_M⊂ F and every coordinate of a basis of Λ is an Mth power in F. The uniformization gives 0⟶μ_M^2⟶ J[M] ⟶Λ/MΛ⟶0. If all torsion points are rational, so are μ_M^2. For λ∈Λ, choose an Mth root u in the torus. Rationality of its torsion class gives g(u)/u∈Λ for every g∈ G_F. Since λ is rational, this quotient also lies in μ_M^2. The lattice is …
Exact mathematical passage
J^{\mathrm{an}}=\mathbb G_m^2/\Lambda
g_qUse in source · 3 passages
Search aliases: g_q
Use in source · 20_localization.tex, line 13
, the two independent pro seven unit coordinates are ϕ,3, whose normalized logarithmic first digits are 5sqrt5,6, and the third coordinate is the uniformizer seven. Each of the nine inert support primes has one seventh power residue coordinate and one unramified coordinate. Real odd primary characters vanish. The total local dimension is 3+3+2·9=24 over k. The chosen g_q=ϕ+a_q generating the residue seventh power quotient supplies I_q. It need not generate the whole multiplicative group. Only it…
Exact mathematical passage
g_q=\phi+a_q
g_{q_i}Use in source · 1 passage
Search aliases: g_q_i
Use in source · 20_localization.tex, line 29
An explicit image parametrization. Put a=z_7(3) and x_i=z(I_q_i). Let ℓ_i be the seventh power logarithm normalized by ℓ_i(g_q_i)=1. The supporting symbols of ϕ vanish. For rational seven adic units define ℓ_7(t)=fraclog(t^6)log(3^6)pmod7 =6fract^6-17pmod7. Principal reciprocity determines the remaining coordinates: z_7(ϕ) · =0,
Exact mathematical passage
\ell_i(g_{q_i})=1
\mathrm{GL}Use in source · 2 passages
Search aliases: GL mathrm
Use in source · 07_natural_depth.tex, line 51
A local depth theorem. Here is the matrix argument controlling where this opposite class can ramify. Let Oloc be a discrete valuation ring with uniformizer π, let F∈GL_2(Oloc), Q∈Oloc^× with Q≠1, and let n≥1. Suppose N∈ M_2(Oloc) is a nonzero nilpotent matrix of the form N=μ-xy · x^2
Exact mathematical passage
F\in\mathrm{GL}_2(\Oloc)
\mathrm{GL}_2Use in source · 2 passages
Search aliases: GL_2 mathrm
Use in source · 07_natural_depth.tex, line 51
A local depth theorem. Here is the matrix argument controlling where this opposite class can ramify. Let Oloc be a discrete valuation ring with uniformizer π, let F∈GL_2(Oloc), Q∈Oloc^× with Q≠1, and let n≥1. Suppose N∈ M_2(Oloc) is a nonzero nilpotent matrix of the form N=μ-xy · x^2
Exact mathematical passage
F\in\mathrm{GL}_2(\Oloc)
\GsigUse in source · 7 passages
Search aliases: Gsig
Use in source · 19_bare_primitive.tex, line 2
A bare primitive on the justified ramification set. Fix one of the nine prime beta supports S in Section~[source reference]. Let Σ consist of S, the places above 2,3,5,7, and the two real places. The residual classes factor through Gsig. At every beta support place, χ is trivial modulo seven and the natural local class is zero. Hence its normalized cocycle is identically zero there. At the fixed places, beta is a local coboundary under the inherited contract.
Exact mathematical passage
\Gsig
H^{(1+\eps)s}Use in source · 1 passage
Search aliases: H^(1+eps)s eps
Use in source · 02_signatures.tex, line 33
gnature explicit. Let H=C^z, R=rad(ABC), s=frac1x+frac1y+frac1z. For a primitive solution, the powered terms are pairwise coprime and their radical is R. Positivity gives ABC<H^s. Assume the all epsilon abc estimate H≤ K_eps R^1+eps for every eps>0, with one constant K_eps independent of the triple. Then H<K_eps H^(1+eps)s. Whenever (1+eps)s<1, it follows that H<K_eps^1/(1-(1+eps)s). This is a conditional bound, not an unconditional result obtained from the exponent equation alone.
Exact mathematical passage
H<K_\eps H^{(1+\eps)s}.
H^1Use in source · 7 passages
Search aliases: H^1
Use in source · 07_natural_depth.tex, line 16
The inherited characteristic zero representation is the specified RM factor of the twisted Jacobian on the retained 3990mid C branch, with determinant the cyclotomic character χ. Its natural residual orientation is 0⟶ k(χ)⟶overline V⟶ k⟶0. The opposite endpoint has a nonzero class β∈ H^1(K,k(χ^-1)). A stable lattice change can alter residual inertia. Therefore properties of the natural Jacobian lattice cannot automatically be transferred to an arbitrary lattice supplied by Ribet's lemma [source …
Exact mathematical passage
\beta\in H^1(K,k(\chi^{-1}))
H^{71/105}Use in source · 1 passage
Search aliases: H^71/105
Use in source · 02_signatures.tex, line 58
Why a generic radical ceiling fails. The exponent equation gives R<H^71/105 for the signature (3,5,7). It is tempting to reverse this relation by postulating H≤ R^105/71 for every primitive sum. That universal assertion is false. The exact equality 2+109·3^10=23^5=6 436 343 has radical 2·3·109·23=15 042, and (23^5)^71>15 042^105. Its logarithmic quality is about 1.629911684. It is not a (3,5,7) counterexample. It shows that an estimate strong enough to close the desired signature must retain its…
Exact mathematical passage
R<H^{71/105}
H^sUse in source · 1 passage
Search aliases: H^s
Use in source · 02_signatures.tex, line 28
The conditional abc height bound. An abc estimate makes the role of the signature explicit. Let H=C^z, R=rad(ABC), s=frac1x+frac1y+frac1z. For a primitive solution, the powered terms are pairwise coprime and their radical is R. Positivity gives ABC<H^s. Assume the all epsilon abc estimate H≤ K_eps R^1+eps for every eps>0, with one constant K_eps independent of the triple. Then H<K_eps H^(1+eps)s. Whenever (1+eps)s<1, it follows that H<K_eps^1/(1-(1+eps)s). This is a conditional bound, not an unc…
Exact mathematical passage
ABC<H^s
H_jUse in source · 2 passages
Search aliases: H_j
Use in source · 12_fixed_seven.tex, line 37
Use the three coordinate basis X_j=[ϕ], H_j=[3], T_j=[7]. The third coordinate is valuation. The actual local scalar class in the window is κ_j=2ϵ wX_j+Λ_jH_j+7cT_j, where Λ_j=7ℓ_j(C_0)-3ℓ_j(A)-3-5d_j, ℓ_j(a)=fraclog(a^6)log(3^6)pmod7^j, d_j=fracℓ_j(5)2. At level 49, with Q_A=(A^6-1)/7 modulo 49 and q_0=(C_0^6-1)/7 modulo seven, Λ_2=47+38Q_A+14Q_A^2-7q_0pmod49. Both C_0 and the coordinate 7c[7] disappeared modulo seven. Neither may be omitted modulo forty nine.
Exact mathematical passage
X_j=[\phi],\qquad H_j=[3],\qquad T_j=[7].
h_qUse in source · 1 passage
Search aliases: h_q
Use in source · 22_affine.tex, line 36
The arithmetic target. Choose a bare global primitive p_0 from Theorem~[source reference]. This is one continuous global cochain. Its values have not been computed. Define t(p_0)∈ k^15 as follows. Its first six coordinates are the ordinary local character coordinates of p_v^fix-p_0|_v. At each support prime, put t_q(p_0)=-h_qc_q-c_qp_0(F_q)+b_qp_0(T_q), h_q=(q^2-1)/7pmod7. At support, α is zero and χ trivial, so the restriction of p_0 is an ordinary character. At fixed places only its difference…
Exact mathematical passage
\label{eq:target} t_q(p_0)=-h_qc_q-c_qp_0(F_q)+b_qp_0(T_q), \qquad h_q=(q^2-1)/7\pmod7.
h_\ellUse in source · 2 passages
Search aliases: h_ℓ ell
Use in source · 03_scaling.tex, line 45
An integral h exists in these four cases precisely under the stated divisibility conditions. Only finitely many primes occur in t or 1-t, so λ=∏_ℓℓ^h_ℓ is a well defined positive rational number. The three output supports are disjoint, proving integrality and primitivity. Every h_ℓ was forced, so two positive choices of λ would have ratio a positive rational with zero valuation at every prime. That ratio is one. figplaceholder03_valuation_packetsThe three nontrivial valuation sectors retain diff…
Exact mathematical passage
\lambda=\prod_\ell\ell^{h_\ell}
\operatorname{Hom}Use in source · 1 passage
Search aliases: Hom operatorname
Use in source · 17_local_flexibility.tex, line 4
For a finite module A, let A'=Hom(A,μ_7) and let G' be the finite group of the field cut out by that dual. The cited sufficient criterion is H^1(G',A')=0. For A=k with trivial action, A'≃ k(χ) and G'=C_6. Averaging kills its positive cohomology because six is invertible modulo seven. For A=k(χ), the source action cancels the cyclotomic target action, so the dual is trivial and G'=1. Both satisfy the criterion.
Exact mathematical passage
A'=\operatorname{Hom}(A,\mu_7)
I_2Use in source · 1 passage
Search aliases: I_2
Use in source · 05_frey.tex, line 43
[X^13]f^3 · =36A, · [X^12]f^3 · =315B^2. The diagonal lanes vanish modulo seven and the off diagonal lanes become M=0 · A^2B A · 0, M^2=A^3B I_2. These are polynomial and matrix identities. Their geometric interpretation requires the correct Cartier and Hasse--Witt conventions. On a good reduction fiber at seven they imply ordinarity when AB is nonzero modulo seven and seven rank zero when AB is zero. There is no rank one stratum in this calculation. The retained sources and the formal report di…
Exact mathematical passage
\label{eq:cartier} M=\begin{pmatrix}0&A^2B\\A&0\end{pmatrix},\qquad M^2=A^3B I_2.
I_qUse in source · 3 passages
Search aliases: I_q
Use in source · 20_localization.tex, line 13
ormalized logarithmic first digits are 5sqrt5,6, and the third coordinate is the uniformizer seven. Each of the nine inert support primes has one seventh power residue coordinate and one unramified coordinate. Real odd primary characters vanish. The total local dimension is 3+3+2·9=24 over k. The chosen g_q=ϕ+a_q generating the residue seventh power quotient supplies I_q. It need not generate the whole multiplicative group. Only its nonzero seventh power symbol and specified normalization are re…
Exact mathematical passage
I_q
I_{q_1}Use in source · 1 passage
Search aliases: I_q_1
Use in source · 20_localization.tex, line 3
The complete ordinary localization map. Let z be an ordinary k valued character of Gsig. It is a difference of bare primitives, not beta and not either primitive itself. Local class field theory gives the following coordinate order: F_2,F_3,F_5,quad7:ϕ,3,7, I_q_1,F_q_1,…,I_q_9,F_q_9. We use the local Artin convention sending a selected uniformizer to arithmetic Frobenius. At five the uniformizer is sqrt5. At support primes it is the rational prime q. Every sign in the reciprocity rows and every …
Exact mathematical passage
\label{eq:localcoords} F_2,F_3,F_5,\quad7:\phi,3,7,\quad I_{q_1},F_{q_1},\ldots,I_{q_9},F_{q_9}.
I_{q_9}Use in source · 1 passage
Search aliases: I_q_9
Use in source · 20_localization.tex, line 3
The complete ordinary localization map. Let z be an ordinary k valued character of Gsig. It is a difference of bare primitives, not beta and not either primitive itself. Local class field theory gives the following coordinate order: F_2,F_3,F_5,quad7:ϕ,3,7, I_q_1,F_q_1,…,I_q_9,F_q_9. We use the local Artin convention sending a selected uniformizer to arithmetic Frobenius. At five the uniformizer is sqrt5. At support primes it is the rational prime q. Every sign in the reciprocity rows and every …
Exact mathematical passage
\label{eq:localcoords} F_2,F_3,F_5,\quad7:\phi,3,7,\quad I_{q_1},F_{q_1},\ldots,I_{q_9},F_{q_9}.
I_{q_i}Use in source · 1 passage
Search aliases: I_q_i
Use in source · 20_localization.tex, line 29
An explicit image parametrization. Put a=z_7(3) and x_i=z(I_q_i). Let ℓ_i be the seventh power logarithm normalized by ℓ_i(g_q_i)=1. The supporting symbols of ϕ vanish. For rational seven adic units define ℓ_7(t)=fraclog(t^6)log(3^6)pmod7 =6fract^6-17pmod7. Principal reciprocity determines the remaining coordinates: z_7(ϕ) · =0,
Exact mathematical passage
x_i=z(I_{q_i})
\imUse in source · 5 passages
Search aliases: im
Use in source · 20_localization.tex, line 45
These equations define a 24 by 10 matrix J. The ten free parameters occur in their declared local coordinates, so J is injective. Direct substitution gives LJ=0. Since both im J and ker L have dimension ten, im J=ker L. Every parameter vector gives a global ordinary character. This is not merely an overapproximation by necessary equations.
Exact mathematical passage
\im J
JUse in source · 11 passages
Search aliases: J
Use in source · 09_periods.tex, line 25
[Full torsion and period powers] For the stipulated split uniformization J^an= G_m^2/Λ with rational period lattice, full J[M] is F rational if and only if μ_M⊂ F and every coordinate of a basis of Λ is an Mth power in F. The uniformization gives 0⟶μ_M^2⟶ J[M] ⟶Λ/MΛ⟶0. If all torsion points are rational, so are μ_M^2. For λ∈Λ, choose an Mth root u in the torus. Rationality of its torsion class gives g(u)/u∈Λ for every g∈ G_F. Since λ is rational, this quotient also lies in μ_M^2. The lattice is …
Exact mathematical passage
J^{\mathrm{an}}=\mathbb G_m^2/\Lambda
j^{-1}Use in source · 2 passages
Search aliases: j^-1
Use in source · 15_semilinear_image.tex, line 3
Integral semilinearity and the unlabelled image. The integral semilinear theorem supplies more than equality of residual cohomology classes. Under the retained geometric inputs, an actual map between the conjugate curves induces an integral operator U=Cσ on the Tate lattice, with ρ(jgj^-1)=Cσ(ρ(g))C^-1, Cσ(C)=-ρ(j^2), C_21∈7^nOloc. It preserves the natural endpoint, the opposite endpoint, and the stable lattice segment. The minus sign in the square comes from the actual conjugate curve map. It c…
Exact mathematical passage
\rho(jgj^{-1})=C\sigma(\rho(g))C^{-1},\qquad C\sigma(C)=-\rho(j^2),\qquad C_{21}\in7^n\Oloc.
j^2Use in source · 2 passages
Search aliases: j^2
Use in source · 15_semilinear_image.tex, line 3
Integral semilinearity and the unlabelled image. The integral semilinear theorem supplies more than equality of residual cohomology classes. Under the retained geometric inputs, an actual map between the conjugate curves induces an integral operator U=Cσ on the Tate lattice, with ρ(jgj^-1)=Cσ(ρ(g))C^-1, Cσ(C)=-ρ(j^2), C_21∈7^nOloc. It preserves the natural endpoint, the opposite endpoint, and the stable lattice segment. The minus sign in the square comes from the actual conjugate curve map. It c…
Exact mathematical passage
\rho(jgj^{-1})=C\sigma(\rho(g))C^{-1},\qquad C\sigma(C)=-\rho(j^2),\qquad C_{21}\in7^n\Oloc.
J^{\mathrm{an}}Use in source · 1 passage
Search aliases: J^an mathrm
Use in source · 09_periods.tex, line 25
[Full torsion and period powers] For the stipulated split uniformization J^an= G_m^2/Λ with rational period lattice, full J[M] is F rational if and only if μ_M⊂ F and every coordinate of a basis of Λ is an Mth power in F. The uniformization gives 0⟶μ_M^2⟶ J[M] ⟶Λ/MΛ⟶0. If all torsion points are rational, so are μ_M^2. For λ∈Λ, choose an Mth root u in the torus. Rationality of its torsion class gives g(u)/u∈Λ for every g∈ G_F. Since λ is rational, this quotient also lies in μ_M^2. The lattice is …
Exact mathematical passage
J^{\mathrm{an}}=\mathbb G_m^2/\Lambda
k^{10}Use in source · 2 passages
Search aliases: k^10
Use in source · 22_affine.tex, line 9
On the local space k^24, let E have fifteen rows. Its first six rows select all fixed local character coordinates in [source reference]. Its remaining nine rows are z⟼ c_qz(F_q)-b_qz(I_q). Define A=EJ:k^10⟶ k^15. The self conductor values and inertia conversions in Section~[source reference] are part of this operator. Dropping them would reproduce a different matrix.
Exact mathematical passage
\label{eq:A} A=EJ:k^{10}\longrightarrow k^{15}.
k^{15}Use in source · 2 passages
Search aliases: k^15
Use in source · 22_affine.tex, line 9
On the local space k^24, let E have fifteen rows. Its first six rows select all fixed local character coordinates in [source reference]. Its remaining nine rows are z⟼ c_qz(F_q)-b_qz(I_q). Define A=EJ:k^10⟶ k^15. The self conductor values and inertia conversions in Section~[source reference] are part of this operator. Dropping them would reproduce a different matrix.
Exact mathematical passage
\label{eq:A} A=EJ:k^{10}\longrightarrow k^{15}.
k^2Use in source · 1 passage
Search aliases: k^2
Use in source · 22_affine.tex, line 85
A small model of the mistake to avoid. Consider A:k→ k^2 given by A(z)=(z,0). Its kernel is zero, its rank is one, and it has a full left null functional N(u,v)=v. For target (a,b), a solution exists precisely when b=0. The map has a one dimensional image and a zero kernel, but this gives no information about the target's second coordinate. Substituting (0,0) because it is convenient manufactures solvability of a different problem.
Exact mathematical passage
A:k\to k^2
k^{24}Use in source · 1 passage
Search aliases: k^24
Use in source · 22_affine.tex, line 4
On the local space k^24, let E have fifteen rows. Its first six rows select all fixed local character coordinates in [source reference]. Its remaining nine rows are z⟼ c_qz(F_q)-b_qz(I_q). Define A=EJ:k^10⟶ k^15. The self conductor values and inertia conversions in Section~[source reference] are part of this operator. Dropping them would reproduce a different matrix.
Exact mathematical passage
k^{24}
k^9Use in source · 1 passage
Search aliases: k^9
Use in source · 19_bare_primitive.tex, line 57
For the frozen supports, each supporting local field has residue norm one modulo seven and contains μ_7. Local duality gives one k dimensional H^2 term there. The four fixed fields do not contain μ_7 and have zero H^2(K_v,k). Global dual invariants also vanish. The final portion of the Poitou--Tate sequence then identifies the restricted H^2 with k^9. The dimension nine specializes the frozen support, whereas Lemma~[source reference] held for arbitrary finite Σ.
Exact mathematical passage
k^9
K_{1/22}Use in source · 1 passage
Search aliases: K_1/22
Use in source · 02_signatures.tex, line 49
There is a stronger uniform bound across the Beal range. Apart from (3,3,3), already excluded by Fermat, one exponent is at least four. Thus s≤1/3+1/3+1/4=11/12. Take eps=1/22 and enlarge its constant to K_1/22≥1. Then (1+eps)s≤23/24, and [source reference] gives H<K_1/22^24. This conditional constant is independent of the signature. It bounds the powered terms and yields finitely many possible primitive powered triples. It does not show that the finite set is empty. Since 2^z≤ H, it also bounds…
Exact mathematical passage
K_{1/22}\ge1
K_{1/22}^{24}Use in source · 1 passage
Search aliases: K_1/22^24
Use in source · 02_signatures.tex, line 50
There is a stronger uniform bound across the Beal range. Apart from (3,3,3), already excluded by Fermat, one exponent is at least four. Thus s≤1/3+1/3+1/4=11/12. Take eps=1/22 and enlarge its constant to K_1/22≥1. Then (1+eps)s≤23/24, and [source reference] gives H<K_1/22^24. This conditional constant is independent of the signature. It bounds the powered terms and yields finitely many possible primitive powered triples. It does not show that the finite set is empty. Since 2^z≤ H, it also bounds…
Exact mathematical passage
\label{eq:uniformbealabc} H<K_{1/22}^{24}.
K_7Use in source · 1 passage
Search aliases: K_7
Use in source · 14_integral_trace.tex, line 14
To define the valuation functional used here, strictly align the actual local toric cyclotomic line at K_7 with the reference line. The aligned local representative of X has p=β=0 and upper entry q a k(χ) Kummer cocycle. Define ν_7(X) to be its [7] valuation coordinate. Remaining aligned strict gauges change q by a coboundary and a scalar multiple of residual α, both of valuation zero. Thus the coordinate is well defined under these strict equivalences. The reference b_0 has valuation zero and t…
Exact mathematical passage
K_7
K_\epsUse in source · 3 passages
Search aliases: K_eps eps
Use in source · 02_signatures.tex, line 29
The conditional abc height bound. An abc estimate makes the role of the signature explicit. Let H=C^z, R=rad(ABC), s=frac1x+frac1y+frac1z. For a primitive solution, the powered terms are pairwise coprime and their radical is R. Positivity gives ABC<H^s. Assume the all epsilon abc estimate H≤ K_eps R^1+eps for every eps>0, with one constant K_eps independent of the triple. Then H<K_eps H^(1+eps)s. Whenever (1+eps)s<1, it follows that H<K_eps^1/(1-(1+eps)s). This is a conditional bound, not an unc…
Exact mathematical passage
\label{eq:abc-assumed} H\le K_\eps R^{1+\eps}
K_\eps^{1/(1-(1+\eps)s)}Use in source · 1 passage
Search aliases: K_eps^1/(1-(1+eps)s) eps eps
Use in source · 02_signatures.tex, line 37
For a primitive solution, the powered terms are pairwise coprime and their radical is R. Positivity gives ABC<H^s. Assume the all epsilon abc estimate H≤ K_eps R^1+eps for every eps>0, with one constant K_eps independent of the triple. Then H<K_eps H^(1+eps)s. Whenever (1+eps)s<1, it follows that H<K_eps^1/(1-(1+eps)s). This is a conditional bound, not an unconditional result obtained from the exponent equation alone.
Exact mathematical passage
\label{eq:signatureheight} H<K_\eps^{1/(1-(1+\eps)s)}.
K_\eps^{105/(34-71\eps)}Use in source · 1 passage
Search aliases: K_eps^105/(34-71eps) eps eps
Use in source · 02_signatures.tex, line 43
For (3,5,7), the reciprocal sum is 71/105. Thus 0<eps<34/71 gives H<K_eps^105/(34-71eps), C<K_eps^15/(34-71eps). At eps=1/10, the latter exponent is 150/269. To turn this into a proof of absence would require the abc input, an explicit usable constant, and a complete verification below the resulting bound.
Exact mathematical passage
\label{eq:357abc} H<K_\eps^{105/(34-71\eps)},\qquad C<K_\eps^{15/(34-71\eps)}.
K_\eps^{15/(34-71\eps)}Use in source · 1 passage
Search aliases: K_eps^15/(34-71eps) eps eps
Use in source · 02_signatures.tex, line 43
For (3,5,7), the reciprocal sum is 71/105. Thus 0<eps<34/71 gives H<K_eps^105/(34-71eps), C<K_eps^15/(34-71eps). At eps=1/10, the latter exponent is 150/269. To turn this into a proof of absence would require the abc input, an explicit usable constant, and a complete verification below the resulting bound.
Exact mathematical passage
\label{eq:357abc} H<K_\eps^{105/(34-71\eps)},\qquad C<K_\eps^{15/(34-71\eps)}.
\kerUse in source · 11 passages
Search aliases: ker
Use in source · 14_integral_trace.tex, line 30
The all depth formula uses t=1+v_7(c), precision j≥ n+t+1, the probe τ_j, and a detecting g∈keroverlineρ with β(g)≠0. It is v_7(D(τ_j,g))=n+t, fracD(τ_j,g)7^n+t =fracc7^t-1fracv2ϵβ(g)pmod7. The coefficient on the right is nonzero in the detecting case. This is an exact valuation statement under the probe and actual geometric hypotheses. It does not bound n or c.
Exact mathematical passage
g\in\ker\overline\rho
LUse in source · 16 passages
Search aliases: L
Use in source · 07_natural_depth.tex, line 23
If no variable C prime had Q≡1pmod7, beta would restrict to an everywhere unramified additive class over L=K(ζ_7). Restriction is injective because [L:K]=6 is prime to seven. The inherited Masley class number result gives h(Q(ζ_35))=1 [source reference]. The field L is a subfield of index two. Extension and norm on ideal classes compose to multiplication by two, so the seven primary class group of L is zero. An everywhere unramified nonzero additive seven class is therefore impossible by class f…
Exact mathematical passage
L=K(\zeta_7)
\lfloorUse in source · 1 passage
Search aliases: lfloor
Use in source · 12_fixed_seven.tex, line 56
At higher actual lattice depth, put m=lfloor n/2rfloor+1. The inherited diagonal theorem supplies an actual global upper extension α_j for 1≤ j≤ m. In the common range 1≤ j≤min(m,7c-1), its local coordinates satisfy U_j=2ϵ wg_j, V_j=Λ_jg_j, W_j=7cg_j for one unit g_j. The exact valuation matrix preserves the valuation identity at every legitimate triangular precision 1≤ j≤ m, even beyond the proved unit period window. Consequently v_7^trunc(W_j)=min(j,1+v_7(c)). At depth one, only the residual g…
Exact mathematical passage
m=\lfloor n/2\rfloor+1
\logUse in source · 2 passages
Search aliases: log
Use in source · 12_fixed_seven.tex, line 45
Use the three coordinate basis X_j=[ϕ], H_j=[3], T_j=[7]. The third coordinate is valuation. The actual local scalar class in the window is κ_j=2ϵ wX_j+Λ_jH_j+7cT_j, where Λ_j=7ℓ_j(C_0)-3ℓ_j(A)-3-5d_j, ℓ_j(a)=fraclog(a^6)log(3^6)pmod7^j, d_j=fracℓ_j(5)2. At level 49, with Q_A=(A^6-1)/7 modulo 49 and q_0=(C_0^6-1)/7 modulo seven, Λ_2=47+38Q_A+14Q_A^2-7q_0pmod49. Both C_0 and the coordinate 7c[7] disappeared modulo seven. Neither may be omitted modulo forty nine.
Exact mathematical passage
\Lambda_j=7\ell_j(C_0)-3\ell_j(A)-3-5d_j, \quad \ell_j(a)=\frac{\log(a^6)}{\log(3^6)}\pmod{7^j}, \quad d_j=\frac{\ell_j(5)}2.
\LongrightarrowUse in source · 1 passage
Search aliases: Longrightarrow
Use in source · 03_scaling.tex, line 23
[Primitive scaling criterion] Fix a rational 0<t<1. There is a positive rational λ for which [source reference] gives primitive positive integers if and only if, for every rational prime ℓ, v_ℓ(t)>0 · Longrightarrowquad3mid v_ℓ(t), v_ℓ(1-t)>0 · Longrightarrowquad5mid v_ℓ(1-t), v_ℓ(t)<0 · Longrightarrowquad7mid -v_ℓ(t). When it exists, λ is unique. Write m=v_ℓ(t), n=v_ℓ(1-t) and h=v_ℓ(λ). Because t+(1-t)=1, the pair (m,n) belongs to one of four sectors: (m>0,n=0), (m=0,n>0), (m=n<0), (m=n=0). The…
Exact mathematical passage
v_\ell(t)>0&\quad\Longrightarrow\quad3\mid v_\ell(t),\\ v_\ell(1-t)>0&\quad\Longrightarrow\quad5\mid v_\ell(1-t),\\ v_\ell(t)<0&\quad\Longrightarrow\quad7\mid -v_\ell(t).
\mathfrak mUse in source · 1 passage
Search aliases: m mathfrak
Use in source · 09_periods.tex, line 5
At a variable place away from 2,3,5,7, write C=q^cC_0 with C_0 a unit. In the marked two node basis, the actual leading period coordinates are Π_aa · =-fracepsϕ^2675sqrt5(1+ m), Π_bb · =fracepsϕ^-2675sqrt5(1+ m), Π_ab · =Π_ba=-ϕ^-4(1+ m), where eps=C^7/A^3 and the notation means that the quotient by the displayed leading term is a principal unit. The analytic uniformization and marked smoothing comparison are inherited written inputs, with precise sources in the literature report [source referen…
Exact mathematical passage
\Pi_{aa}&=-\frac{\eps\phi^2}{675\sqrt5}(1+\mathfrak m),\\ \Pi_{bb}&=\frac{\eps\phi^{-2}}{675\sqrt5}(1+\mathfrak m),\\ \Pi_{ab}&=\Pi_{ba}=-\phi^{-4}(1+\mathfrak m),
mUse in source · 21 passages
Search aliases: m
Use in source · 03_scaling.tex, line 31
v_ℓ(t)<0 · Longrightarrowquad7mid -v_ℓ(t). When it exists, λ is unique. Write m=v_ℓ(t), n=v_ℓ(1-t) and h=v_ℓ(λ). Because t+(1-t)=1, the pair (m,n) belongs to one of four sectors: (m>0,n=0), (m=0,n>0), (m=n<0), (m=n=0). The output valuations are a_ℓ · =35h+12m+28n,
Exact mathematical passage
m=v_\ell(t)
M^2Use in source · 1 passage
Search aliases: M^2
Use in source · 05_frey.tex, line 43
[X^13]f^3 · =36A, · [X^12]f^3 · =315B^2. The diagonal lanes vanish modulo seven and the off diagonal lanes become M=0 · A^2B A · 0, M^2=A^3B I_2. These are polynomial and matrix identities. Their geometric interpretation requires the correct Cartier and Hasse--Witt conventions. On a good reduction fiber at seven they imply ordinarity when AB is nonzero modulo seven and seven rank zero when AB is zero. There is no rank one stratum in this calculation. The retained sources and the formal report di…
Exact mathematical passage
\label{eq:cartier} M=\begin{pmatrix}0&A^2B\\A&0\end{pmatrix},\qquad M^2=A^3B I_2.
M_2Use in source · 2 passages
Search aliases: M_2
Use in source · 07_natural_depth.tex, line 51
A local depth theorem. Here is the matrix argument controlling where this opposite class can ramify. Let Oloc be a discrete valuation ring with uniformizer π, let F∈GL_2(Oloc), Q∈Oloc^× with Q≠1, and let n≥1. Suppose N∈ M_2(Oloc) is a nonzero nilpotent matrix of the form N=μ-xy · x^2
Exact mathematical passage
N\in M_2(\Oloc)
\midUse in source · 20 passages
Search aliases: mid
Use in source · 01_statement.tex, line 19
with the first two bases. Primitivity becomes stronger on an actual solution than it is on an arbitrary integer triple. For a solution of [source reference], every prime dividing two bases divides the third. Consequently a primitive solution has pairwise coprime bases. If a prime p divides A and B, the equation gives pmid C^z, hence pmid C. If p divides A and C, it gives pmid B^y, hence pmid B. The third case is the same. Thus a prime shared by any pair would divide the gcd of all three. Beal is…
Exact mathematical passage
p\mid C^z
\minUse in source · 3 passages
Search aliases: min
Use in source · 07_natural_depth.tex, line 39
Define n=min_g:ρ(g)_21≠0v_7(ρ(g)_21). This positive integer is attained. It is an invariant of the fixed natural frame, not a truncation selected for a computation. Every Oloc lattice stable under ρ is preserved by the two projectors, so up to homothety it has the form Λ_j=Oloc e_1+Oloc 7^j e_2. The unit upper monodromy forces j≥0. The attained lower entry minimum forces j≤ n. Conversely every j in that interval is stable. Thus the stable lattices form a finite segment with the natural and oppos…
Exact mathematical passage
\label{eq:actualdepth} n=\min_{g:\rho(g)_{21}\ne0}v_7(\rho(g)_{21}).
\min_{g:\rho(g)_{21}\ne0}Use in source · 1 passage
Search aliases: min_g:ρ(g)_21≠0 min rho ne
Use in source · 07_natural_depth.tex, line 39
Define n=min_g:ρ(g)_21≠0v_7(ρ(g)_21). This positive integer is attained. It is an invariant of the fixed natural frame, not a truncation selected for a computation. Every Oloc lattice stable under ρ is preserved by the two projectors, so up to homothety it has the form Λ_j=Oloc e_1+Oloc 7^j e_2. The unit upper monodromy forces j≥0. The attained lower entry minimum forces j≤ n. Conversely every j in that interval is stable. Thus the stable lattices form a finite segment with the natural and oppos…
Exact mathematical passage
\label{eq:actualdepth} n=\min_{g:\rho(g)_{21}\ne0}v_7(\rho(g)_{21}).
NUse in source · 28 passages
Search aliases: N
Use in source · 06_modular_labels.tex, line 15
From point counts to a common polynomial. For a smooth genus two curve over F_q, let N_1,N_2 count its points over F_q,F_q^2, including infinity. Write s_1=q+1-N_1, s_2=fracs_1^2-(q^2+1-N_2)2. Its Frobenius polynomial is T^4-s_1T^3+s_2T^2-qs_1T+q^2. In the RM setting, the rank two traces are the roots of X^2-s_1X+(s_2-2q). The two roots belong to one common quartic. Choosing one root from one fiber and another root from an unrelated fiber can enlarge a necessary locus, but does not give a compat…
Exact mathematical passage
N_1,N_2
nUse in source · 43 passages
Search aliases: n
Use in source · 02_signatures.tex, line 11
the same for the three exponents. [Prime or four reduction] If a primitive solution of [source reference] exists, there is a primitive solution whose three exponents each belong to {4}∪{p:p is an odd prime}. Conversely, absence of primitive solutions for every ordered signature in this set implies Beal. Let n>2. If n has an odd prime divisor p, write D^n=(D^n/p)^p. If it has no odd prime divisor, it is a power of two at least four, so write D^n=(D^n/4)^4. Apply this operation to each term separa…
Exact mathematical passage
n>2
N^2Use in source · 1 passage
Search aliases: N^2
Use in source · 18_ramification.tex, line 5
item[H2.] At every variable bad place the actual inertia has I+N with N^2=0 and N in 7M_2(Oloc). Its primitive line factorization and same frame Frobenius obey the toric equations. Good inertia is trivial. item[H3.] Outside the four fixed rational primes 2,3,5,7, every possible bad place divides C. The reference is unramified there. item[H4.] At each fixed place over 2,3,5,7, the actual rank two trace is χ+1, and a local element has χ-1 a unit modulo seven. item[H5.] The four frozen beta support…
Exact mathematical passage
N^2=0
N_1Use in source · 2 passages
Search aliases: N_1
Use in source · 06_modular_labels.tex, line 15
From point counts to a common polynomial. For a smooth genus two curve over F_q, let N_1,N_2 count its points over F_q,F_q^2, including infinity. Write s_1=q+1-N_1, s_2=fracs_1^2-(q^2+1-N_2)2. Its Frobenius polynomial is T^4-s_1T^3+s_2T^2-qs_1T+q^2. In the RM setting, the rank two traces are the roots of X^2-s_1X+(s_2-2q). The two roots belong to one common quartic. Choosing one root from one fiber and another root from an unrelated fiber can enlarge a necessary locus, but does not give a compat…
Exact mathematical passage
N_1,N_2
N_2Use in source · 2 passages
Search aliases: N_2
Use in source · 06_modular_labels.tex, line 15
From point counts to a common polynomial. For a smooth genus two curve over F_q, let N_1,N_2 count its points over F_q,F_q^2, including infinity. Write s_1=q+1-N_1, s_2=fracs_1^2-(q^2+1-N_2)2. Its Frobenius polynomial is T^4-s_1T^3+s_2T^2-qs_1T+q^2. In the RM setting, the rank two traces are the roots of X^2-s_1X+(s_2-2q). The two roots belong to one common quartic. Choosing one root from one fiber and another root from an unrelated fiber can enlarge a necessary locus, but does not give a compat…
Exact mathematical passage
N_1,N_2
N_{21}Use in source · 1 passage
Search aliases: N_21
Use in source · 07_natural_depth.tex, line 56
-y^2 · xy, where μ∈Oloc∖{0}, (x,y)∈Oloc^2 is primitive, t=v(μ)≥0, and FN=QN, NF=N, F_21∈π^nOloc, v(N_21)=n. Set h=v(Q-1). Then n=t+2s, s=v(y),qquad0≤ s≤ h. The lower nilpotent entry is -μ y^2, so its valuation gives n=t+2s. The common image and kernel line of N is generated by (x,y)^T. The two identities for F show that F-I has image in this line and kills no extra quotient direction. Write F=I+x
Exact mathematical passage
FN=QN,\qquad NF=N,\qquad F_{21}\in\pi^n\Oloc,\qquad v(N_{21})=n.
\operatorname{Nm}Use in source · 2 passages
Search aliases: Nm operatorname
Use in source · 10_reciprocity.tex, line 43
If the actual polynomial is P(X)=X^4-SX^3+BX^2-QSX+Q^2, the correction u=(τ-Q-1)/7 has Tr(u)=fracS-2(Q+1)7, Nm(u)=frac(B-2Q)-(Q+1)S+(Q+1)^249 modulo seven, under the stated divisibility assumptions. The two implementations extract these values using different methods, including an integer trace discriminant and an enumeration of k by trace and norm. Neither assigns a Frobenius value at a beta conductor prime by pretending beta is unramified there. Such conductor values require the separate self …
Exact mathematical passage
\operatorname{Tr}(u)=\frac{S-2(Q+1)}7,\qquad \operatorname{Nm}(u)=\frac{(B-2Q)-(Q+1)S+(Q+1)^2}{49}
\operatorname{Nm}_{\F_{q^2}/\F_q}Use in source · 1 passage
Search aliases: Nm_F_q^2/F_q operatorname F F
Use in source · 21_ramified_frobenius.tex, line 37
The ordinary inertia generator differs from the frozen twisted residue convention. Its beta value is c_q=s_qℓ_q(Nm_F_q^2/F_qg_q) (q≡1pmod7), and c_q=s_qℓ_q(g_q) for q≡-1pmod7. The conversion factors are rrrrr 197:5 · 307:2 · 337:5 · 547:4 · 673:1
Exact mathematical passage
c_q=s_q\ell_q(\operatorname{Nm}_{\F_{q^2}/\F_q}g_q) \quad(q\equiv1\pmod7),
\nmidUse in source · 3 passages
Search aliases: nmid
Use in source · 05_frey.tex, line 51
What the direct irreducibility argument proves. The canonical recovered result is absolute irreducibility of the associated residual rank two RM representation when 14nmid C, subject to its declared realization, conductor, local weight, and finite flat inputs. Its two branches use different information.
Exact mathematical passage
14\nmid C
\notUse in source · 3 passages
Search aliases: not
Use in source · 07_natural_depth.tex, line 21
At seven, the specified twist makes the toric characters χ and one, and cyclotomic inertia is nontrivial. At a variable bad place with Qnot≡1pmod7, the same argument applies. At the places over two, three and five, a separate local cohomology calculation excludes opposite extensions. For a non seven local field of residue cardinality Q, an inertia cocycle with coefficients k(χ^-1) can be nonzero only when Q^2≡1pmod7: the tame Frobenius relation multiplies its inertia value by Q, while the coeffi…
Exact mathematical passage
Q\not\equiv1\pmod7
\Oloc^2Use in source · 1 passage
Search aliases: Oloc^2 Oloc
Use in source · 07_natural_depth.tex, line 55
-y^2 · xy, where μ∈Oloc∖{0}, (x,y)∈Oloc^2 is primitive, t=v(μ)≥0, and FN=QN, NF=N, F_21∈π^nOloc, v(N_21)=n. Set h=v(Q-1). Then n=t+2s, s=v(y),qquad0≤ s≤ h. The lower nilpotent entry is -μ y^2, so its valuation gives n=t+2s. The common image and kernel line of N is generated by (x,y)^T. The two identities for F show that F-I has image in this line and kills no extra quotient direction. Write F=I+x
Exact mathematical passage
(x,y)\in\Oloc^2
\Oloc^\timesUse in source · 1 passage
Search aliases: Oloc^× Oloc times
Use in source · 07_natural_depth.tex, line 51
A local depth theorem. Here is the matrix argument controlling where this opposite class can ramify. Let Oloc be a discrete valuation ring with uniformizer π, let F∈GL_2(Oloc), Q∈Oloc^× with Q≠1, and let n≥1. Suppose N∈ M_2(Oloc) is a nonzero nilpotent matrix of the form N=μ-xy · x^2
Exact mathematical passage
Q\in\Oloc^\times
\Oloc_FUse in source · 5 passages
Search aliases: Oloc_F Oloc
Use in source · 12_fixed_seven.tex, line 2
The fixed seven period target. At seven, principal units are not automatically seventh powers. The residue class of 8=1+7 is a simple warning. The away from seven period simplification cannot be copied without an error bound. The inherited integral marked family supplies an error in 1+7^7cOloc_F, where c=v_7(C)≥1 and F=Q_7(sqrt5). That specific error is sufficiently deep for the stated period window.
Exact mathematical passage
1+7^{7c}\Oloc_F
\overline VUse in source · 1 passage
Search aliases: overline V overline
Use in source · 07_natural_depth.tex, line 13
The inherited characteristic zero representation is the specified RM factor of the twisted Jacobian on the retained 3990mid C branch, with determinant the cyclotomic character χ. Its natural residual orientation is 0⟶ k(χ)⟶overline V⟶ k⟶0. The opposite endpoint has a nonzero class β∈ H^1(K,k(χ^-1)). A stable lattice change can alter residual inertia. Therefore properties of the natural Jacobian lattice cannot automatically be transferred to an arbitrary lattice supplied by Ribet's lemma [source …
Exact mathematical passage
0\longrightarrow k(\chi)\longrightarrow\overline V\longrightarrow k\longrightarrow0.
\overline{\sqrt5}Use in source · 2 passages
Search aliases: overlinesqrt5 overline sqrt
Use in source · 12_fixed_seven.tex, line 5
Let X=[ϕ] and H=[3] in the mod seven local Kummer group. Exact logarithmic first digits give L(ϕ)=5overlinesqrt5, L(3)=6, L(sqrt5)=3, L(u)=fracu^48-17pmod7. The two unit coordinates X,H are independent, and [sqrt5]=4H. For a rational seven adic unit A, put q_7(A)=(A^6-1)/7 modulo seven. The local marked period comparison has P=ZI+2XR, Z=(3q_7(A)+5)H, with R from [source reference].
Exact mathematical passage
L(\phi)=5\overline{\sqrt5},\qquad L(3)=6,\qquad L(\sqrt5)=3, \quad L(u)=\frac{u^{48}-1}{7}\pmod7.
\overline\rhoUse in source · 4 passages
Search aliases: overlineρ overline rho
Use in source · 13_deformation.tex, line 14
beta(g) · -p(g)∈sl_2(k). The determinant condition makes X traceless. Multiplication shows that ρ is a representation if and only if X(gh)=X(g)+Ad(overlineρ_0(g))X(h). For overlineρ_0(g)=(x · a
Exact mathematical passage
X(gh)=X(g)+\Ad(\overline\rho_0(g))X(h).
\overline\rho_0Use in source · 3 passages
Search aliases: overlineρ_0 overline rho
Use in source · 13_deformation.tex, line 14
beta(g) · -p(g)∈sl_2(k). The determinant condition makes X traceless. Multiplication shows that ρ is a representation if and only if X(gh)=X(g)+Ad(overlineρ_0(g))X(h). For overlineρ_0(g)=(x · a
Exact mathematical passage
X(gh)=X(g)+\Ad(\overline\rho_0(g))X(h).
\overline\phiUse in source · 1 passage
Search aliases: overlineϕ overline phi
Use in source · 05_frey.tex, line 55
The global unit step uses ϕ=(1+sqrt5)/2, whose residue has order sixteen in F_49^×. The local weight possibilities at seven are 0,1,7,8, up to a common reciprocity sign. Applying reciprocity to the eighth power of each constituent gives overlineϕ^8e=1 and rules out the mixed weights 1,7. The determinant norm relation δ^8=χ^2 then leaves one constituent unramified at seven, without assuming an unproved rank two determinant normalization in this argument. Its eighth power is everywhere unramified,…
Exact mathematical passage
\overline\phi^{8e}=1
\overline\phi^{8e}Use in source · 1 passage
Search aliases: overlineϕ^8e overline phi
Use in source · 05_frey.tex, line 55
The global unit step uses ϕ=(1+sqrt5)/2, whose residue has order sixteen in F_49^×. The local weight possibilities at seven are 0,1,7,8, up to a common reciprocity sign. Applying reciprocity to the eighth power of each constituent gives overlineϕ^8e=1 and rules out the mixed weights 1,7. The determinant norm relation δ^8=χ^2 then leaves one constituent unramified at seven, without assuming an unproved rank two determinant normalization in this argument. Its eighth power is everywhere unramified,…
Exact mathematical passage
\overline\phi^{8e}=1
PUse in source · 6 passages
Search aliases: P
Use in source · 08_global_coefficients.tex, line 24
[Discriminant collapse] Suppose a good Frobenius polynomial has integral coefficients and factorization P(T)=(T^2-τ T+Q)(T^2-σ(τ)T+Q) =T^4-ST^3+C_2T^2-QST+Q^2, where Oloc/Z_7 is unramified quadratic, σ is its nontrivial automorphism, all complex roots have absolute value sqrt Q, and τ≡ Q+1pmod7^rOloc. Then the integer D=S^2-4(C_2-2Q) satisfies 7^2rmid D and 0≤ D≤16Q. If 7^2r>16Q, then P(T)=(T^2-aT+Q)^2, a∈Z, |a|≤2sqrt Q, a≡ Q+1pmod7^r. Expansion gives S=τ+σ(τ), C_2-2Q=τσ(τ), and D=(τ-σ(τ))^2. Th…
Exact mathematical passage
P(T)=(T^2-\tau T+Q)(T^2-\sigma(\tau)T+Q) =T^4-ST^3+C_2T^2-QST+Q^2,
p'_0Use in source · 2 passages
Search aliases: p'_0
Use in source · 22_affine.tex, line 68
Now replace the origin by p'_0=p_0+z_0,char, where its localization is Jz_0. At the fixed places, the target differences decrease by the selected coordinates of this character. At support, [source reference] decreases by c_qz_0,char(F_q)-b_qz_0,char(T_q). Therefore t(p'_0)=t(p_0)-Az_0. The quotient class and Nt are unchanged. The coordinate basis for N itself is optional, but its vanishing criterion and the quotient class are canonical. The result concerns the stated first stage. It does not ide…
Exact mathematical passage
p'_0=p_0+z_{0,\mathrm{char}}
P^{-1}Use in source · 1 passage
Search aliases: P^-1
Use in source · 14_integral_trace.tex, line 17
The actual period valuation matrix is 7cI, so this probe has ρ(τ_j)=I+7cPE_12P^-1pmod7^j, where the same local toric line has the required alignment with the natural line. At actual depth one and a detecting g in the residual kernel, [source reference] gives D(τ,g)=49ν_7(X)β(g)pmod343. The actual fixed seven period comparison requires ν_7(X)=cfracv2ϵ in F_7. If β(g)≠0, the mixed trace determines this value.
Exact mathematical passage
\rho(\tau_j)=I+7cPE_{12}P^{-1}\pmod{7^j},
p^{\mathrm{fix}}Use in source · 2 passages
Search aliases: p^fix mathrm
Use in source · FORMAL_STATUS.md, line 118
β(g)=(χ(g)^-1-1)λ, p^fix(g)=λχ(g)^-1α(g).
Exact mathematical passage
\beta(g)=(\chi(g)^{-1}-1)\lambda, \qquad p^{\mathrm{fix}}(g)=\lambda\chi(g)^{-1}\alpha(g).
p^nUse in source · 1 passage
Search aliases: p^n
Use in source · 04_local_strata.tex, line 40
al derivatives are 6a^3u^5,qquad10b^5v^9, -14c^7w^13. In a primitive residue solution, at least two bases are units. At an odd prime, at most one of 6,10,14 is divisible by p. Thus at least one unit variable has unit derivative. Lift the other variables to nonzero integers, using p for a zero residue if necessary. Successive correction in the selected variable by multiples of p^n kills one new digit at each stage. The derivative remains a unit, so the corrections converge to a nonzero root with …
Exact mathematical passage
p^n
p_0Use in source · 30 passages
Search aliases: p_0
Use in source · 17_local_flexibility.tex, line 13
If p_0 is one global bare primitive and p_v are desired finite local primitives, their differences p_v-p_0|_v are ordinary characters. For trivial coefficients there are no nonzero coboundaries. Surjectivity supplies one global character matching every prescribed finite difference exactly. Then p=p_0+z meets those local equations, but may ramify at new primes.
Exact mathematical passage
p_0
p_vUse in source · 2 passages
Search aliases: p_v
Use in source · 17_local_flexibility.tex, line 13
If p_0 is one global bare primitive and p_v are desired finite local primitives, their differences p_v-p_0|_v are ordinary characters. For trivial coefficients there are no nonzero coboundaries. Surjectivity supplies one global character matching every prescribed finite difference exactly. Then p=p_0+z meets those local equations, but may ramify at new primes.
Exact mathematical passage
p_v
p_v^{\mathrm{fix}}Use in source · 6 passages
Search aliases: p_v^fix mathrm
Use in source · 16_secondary.tex, line 47
At a fixed place where beta is the coboundary of λ_v and p=p_v^fix, the explicit local upper primitive q_v^fix(g)=-λ_vχ(g)^-1α(g)^2 satisfies d_χ q_v^fix=R_p by direct expansion. At support, α=0 also kills the local upper right side. These facts prevent treating a formal secondary obstruction as an established nonzero obstruction in the actual local branches. Full prescribed upper values and geometric compatibility remain separate.
Exact mathematical passage
p=p_v^{\mathrm{fix}}
\pmodUse in source · 37 passages
Search aliases: pmod
Use in source · 04_local_strata.tex, line 51
If p≡1pmod4, two of the three signs agree, so one of these conditions holds. If p≡3pmod4 and all three fail, then α=β=-γ. The two remaining sign patterns can be handled by the universal identity of Proposition~[source reference].
Exact mathematical passage
p\equiv1\pmod4
\mathrm{PV}Use in source · 2 passages
Search aliases: PV mathrm
Use in source · 05_frey.tex, line 10
gives Z^2=f_A,B(X)=X^6+10BX^3+12AX+5B^2. This is the genus two family used in the recovered source. The comparison with the family of Pacetti and Villagra Torcomian is a rational change of coordinates, with their variables translated as (a,b,c,p)=(B,-C,A,7) [source reference]. Explicitly, put t=-B^5/A^3, d=-B^2/A, and substitute x_PV=d/X, y_PV=tZ/X^3 into their equation y^2=5x^6-12x^5+10tx^3+t^2. Multiplication by X^6/t^2 gives [source reference]. The map is rational for AB≠0. It does not on its…
Exact mathematical passage
x_{\mathrm{PV}}=d/X
QUse in source · 55 passages
Search aliases: Q
Use in source · 07_natural_depth.tex, line 21
At seven, the specified twist makes the toric characters χ and one, and cyclotomic inertia is nontrivial. At a variable bad place with Qnot≡1pmod7, the same argument applies. At the places over two, three and five, a separate local cohomology calculation excludes opposite extensions. For a non seven local field of residue cardinality Q, an inertia cocycle with coefficients k(χ^-1) can be nonzero only when Q^2≡1pmod7: the tame Frobenius relation multiplies its inertia value by Q, while the coeffi…
Exact mathematical passage
Q\not\equiv1\pmod7
\QUse in source · 13 passages
Search aliases: Q
Use in source · 05_frey.tex, line 53
When C is odd, the integral model has good reduction at two. Over the norm four prime of K=Q(sqrt5), the full Frobenius polynomial is either (T^2+T+4)^2 or (T^2+4)^2. A hypothetical degree one constituent, after the conductor and global unit argument, must have eighth power one. Neither quadratic has a root with eighth power one modulo seven. For T^2+4, a root has square three, hence eighth power four. For T^2+T+4, reduction of powers gives fourth power five, again eighth power four. The contrad…
Exact mathematical passage
K=\Q(\sqrt5)
\mathfrak qUse in source · 1 passage
Search aliases: q mathfrak
Use in source · 06_modular_labels.tex, line 4
Suppose a Hecke eigenvalue a_ q is replaced by its minimal polynomial. A resultant against a possible curve trace polynomial asks whether some conjugate can match some allowable trace. Repeating the test at a different prime can choose a different conjugate. If the original representation requires one residue embedding for every prime, this relaxed sequence of tests may preserve candidates that the labelled comparison would exclude. An exclusion obtained from the relaxation remains valid. Surviv…
Exact mathematical passage
a_{\mathfrak q}
qUse in source · 99 passages
Search aliases: q
Use in source · 06_modular_labels.tex, line 9
Two exact local restrictions were established under the stated modular and multiplicative prime comparisons. A rank two trace zero comparison at a prime above eleven forces 11mid A. At a prime above nineteen it forces 19mid B. In either case the full four dimensional reduction to check is (T^2+q)^2pmod7, which is stronger than zero four dimensional trace. At bad q, the multiplicative comparison gives ±(q+1), nonzero modulo seven for these two primes. At good q, the direct curve table selects the…
Exact mathematical passage
(T^2+q)^2\pmod7,
Q^{-1}Use in source · 1 passage
Search aliases: Q^-1
Use in source · 07_natural_depth.tex, line 21
e same argument applies. At the places over two, three and five, a separate local cohomology calculation excludes opposite extensions. For a non seven local field of residue cardinality Q, an inertia cocycle with coefficients k(χ^-1) can be nonzero only when Q^2≡1pmod7: the tame Frobenius relation multiplies its inertia value by Q, while the coefficient action multiplies it by Q^-1. The unramified quotient is zero when Qnot≡1. Thus Qnot≡±1 forces local H^1 to vanish. Here the relevant norms are …
Exact mathematical passage
Q^{-1}
Q^2Use in source · 3 passages
Search aliases: Q^2
Use in source · 07_natural_depth.tex, line 21
es the toric characters χ and one, and cyclotomic inertia is nontrivial. At a variable bad place with Qnot≡1pmod7, the same argument applies. At the places over two, three and five, a separate local cohomology calculation excludes opposite extensions. For a non seven local field of residue cardinality Q, an inertia cocycle with coefficients k(χ^-1) can be nonzero only when Q^2≡1pmod7: the tame Frobenius relation multiplies its inertia value by Q, while the coefficient action multiplies it by Q^-…
Exact mathematical passage
Q^2\equiv1\pmod7
q^2Use in source · 6 passages
Search aliases: q^2
Use in source · 06_modular_labels.tex, line 15
From point counts to a common polynomial. For a smooth genus two curve over F_q, let N_1,N_2 count its points over F_q,F_q^2, including infinity. Write s_1=q+1-N_1, s_2=fracs_1^2-(q^2+1-N_2)2. Its Frobenius polynomial is T^4-s_1T^3+s_2T^2-qs_1T+q^2. In the RM setting, the rank two traces are the roots of X^2-s_1X+(s_2-2q). The two roots belong to one common quartic. Choosing one root from one fiber and another root from an unrelated fiber can enlarge a necessary locus, but does not give a compat…
Exact mathematical passage
\F_q,\F_{q^2}
q^cUse in source · 1 passage
Search aliases: q^c
Use in source · 09_periods.tex, line 4
At a variable place away from 2,3,5,7, write C=q^cC_0 with C_0 a unit. In the marked two node basis, the actual leading period coordinates are Π_aa · =-fracepsϕ^2675sqrt5(1+ m),
Exact mathematical passage
C=q^cC_0
q^fUse in source · 2 passages
Search aliases: q^f
Use in source · 07_natural_depth.tex, line 25
At least one variable prime above a divisor of C must consequently have norm one modulo seven. A rational prime splits in K when it is one or four modulo five, giving residue degree one. At an inert prime the degree is two. Combining q^f≡1pmod7 with these splitting possibilities gives exactly qpmod35∈{1,8,13,22,27,29}. None of 2,3,5,7,19, the primes already in 3990, supplies this new variable prime. The conclusion asks for an eligible additional prime, not divisibility by a particular fixed prim…
Exact mathematical passage
q^f\equiv1\pmod7
q_0Use in source · 3 passages
Search aliases: q_0
Use in source · 12_fixed_seven.tex, line 50
d H_j=[3], T_j=[7]. The third coordinate is valuation. The actual local scalar class in the window is κ_j=2ϵ wX_j+Λ_jH_j+7cT_j, where Λ_j=7ℓ_j(C_0)-3ℓ_j(A)-3-5d_j, ℓ_j(a)=fraclog(a^6)log(3^6)pmod7^j, d_j=fracℓ_j(5)2. At level 49, with Q_A=(A^6-1)/7 modulo 49 and q_0=(C_0^6-1)/7 modulo seven, Λ_2=47+38Q_A+14Q_A^2-7q_0pmod49. Both C_0 and the coordinate 7c[7] disappeared modulo seven. Neither may be omitted modulo forty nine.
Exact mathematical passage
q_0=(C_0^6-1)/7
q_1Use in source · 3 passages
Search aliases: q_1
Use in source · 20_localization.tex, line 3
The complete ordinary localization map. Let z be an ordinary k valued character of Gsig. It is a difference of bare primitives, not beta and not either primitive itself. Local class field theory gives the following coordinate order: F_2,F_3,F_5,quad7:ϕ,3,7, I_q_1,F_q_1,…,I_q_9,F_q_9. We use the local Artin convention sending a selected uniformizer to arithmetic Frobenius. At five the uniformizer is sqrt5. At support primes it is the rational prime q. Every sign in the reciprocity rows and every …
Exact mathematical passage
\label{eq:localcoords} F_2,F_3,F_5,\quad7:\phi,3,7,\quad I_{q_1},F_{q_1},\ldots,I_{q_9},F_{q_9}.
\Q_7Use in source · 4 passages
Search aliases: Q_7 Q
Use in source · 12_fixed_seven.tex, line 2
The fixed seven period target. At seven, principal units are not automatically seventh powers. The residue class of 8=1+7 is a simple warning. The away from seven period simplification cannot be copied without an error bound. The inherited integral marked family supplies an error in 1+7^7cOloc_F, where c=v_7(C)≥1 and F=Q_7(sqrt5). That specific error is sufficiently deep for the stated period window.
Exact mathematical passage
F=\Q_7(\sqrt5)
q_7Use in source · 7 passages
Search aliases: q_7
Use in source · 12_fixed_seven.tex, line 9
Let X=[ϕ] and H=[3] in the mod seven local Kummer group. Exact logarithmic first digits give L(ϕ)=5overlinesqrt5, L(3)=6, L(sqrt5)=3, L(u)=fracu^48-17pmod7. The two unit coordinates X,H are independent, and [sqrt5]=4H. For a rational seven adic unit A, put q_7(A)=(A^6-1)/7 modulo seven. The local marked period comparison has P=ZI+2XR, Z=(3q_7(A)+5)H, with R from [source reference].
Exact mathematical passage
q_7(A)=(A^6-1)/7
q_9Use in source · 3 passages
Search aliases: q_9
Use in source · 20_localization.tex, line 3
The complete ordinary localization map. Let z be an ordinary k valued character of Gsig. It is a difference of bare primitives, not beta and not either primitive itself. Local class field theory gives the following coordinate order: F_2,F_3,F_5,quad7:ϕ,3,7, I_q_1,F_q_1,…,I_q_9,F_q_9. We use the local Artin convention sending a selected uniformizer to arithmetic Frobenius. At five the uniformizer is sqrt5. At support primes it is the rational prime q. Every sign in the reciprocity rows and every …
Exact mathematical passage
\label{eq:localcoords} F_2,F_3,F_5,\quad7:\phi,3,7,\quad I_{q_1},F_{q_1},\ldots,I_{q_9},F_{q_9}.
Q_AUse in source · 2 passages
Search aliases: Q_A
Use in source · 12_fixed_seven.tex, line 50
dinate basis X_j=[ϕ], H_j=[3], T_j=[7]. The third coordinate is valuation. The actual local scalar class in the window is κ_j=2ϵ wX_j+Λ_jH_j+7cT_j, where Λ_j=7ℓ_j(C_0)-3ℓ_j(A)-3-5d_j, ℓ_j(a)=fraclog(a^6)log(3^6)pmod7^j, d_j=fracℓ_j(5)2. At level 49, with Q_A=(A^6-1)/7 modulo 49 and q_0=(C_0^6-1)/7 modulo seven, Λ_2=47+38Q_A+14Q_A^2-7q_0pmod49. Both C_0 and the coordinate 7c[7] disappeared modulo seven. Neither may be omitted modulo forty nine.
Exact mathematical passage
Q_A=(A^6-1)/7
Q_A^2Use in source · 1 passage
Search aliases: Q_A^2
Use in source · 12_fixed_seven.tex, line 51
te is valuation. The actual local scalar class in the window is κ_j=2ϵ wX_j+Λ_jH_j+7cT_j, where Λ_j=7ℓ_j(C_0)-3ℓ_j(A)-3-5d_j, ℓ_j(a)=fraclog(a^6)log(3^6)pmod7^j, d_j=fracℓ_j(5)2. At level 49, with Q_A=(A^6-1)/7 modulo 49 and q_0=(C_0^6-1)/7 modulo seven, Λ_2=47+38Q_A+14Q_A^2-7q_0pmod49. Both C_0 and the coordinate 7c[7] disappeared modulo seven. Neither may be omitted modulo forty nine.
Exact mathematical passage
\label{eq:level49} \Lambda_2=47+38Q_A+14Q_A^2-7q_0\pmod{49}.
q_iUse in source · 3 passages
Search aliases: q_i
Use in source · 20_localization.tex, line 29
An explicit image parametrization. Put a=z_7(3) and x_i=z(I_q_i). Let ℓ_i be the seventh power logarithm normalized by ℓ_i(g_q_i)=1. The supporting symbols of ϕ vanish. For rational seven adic units define ℓ_7(t)=fraclog(t^6)log(3^6)pmod7 =6fract^6-17pmod7. Principal reciprocity determines the remaining coordinates: z_7(ϕ) · =0,
Exact mathematical passage
x_i=z(I_{q_i})
q_v^{\mathrm{fix}}Use in source · 2 passages
Search aliases: q_v^fix mathrm
Use in source · 16_secondary.tex, line 48
At a fixed place where beta is the coboundary of λ_v and p=p_v^fix, the explicit local upper primitive q_v^fix(g)=-λ_vχ(g)^-1α(g)^2 satisfies d_χ q_v^fix=R_p by direct expansion. At support, α=0 also kills the local upper right side. These facts prevent treating a formal secondary obstruction as an established nonzero obstruction in the actual local branches. Full prescribed upper values and geometric compatibility remain separate.
Exact mathematical passage
q_v^{\mathrm{fix}}(g)=-\lambda_v\chi(g)^{-1}\alpha(g)^2
rUse in source · 27 passages
Search aliases: r
Use in source · 03_scaling.tex, line 43
In the first sector these weighted valuations are 105h+36m,105h+35m,105h+35m. Therefore h=-m/3, and the output is (m/3,0,0). In the second sector they are 105h+84n,105h+85n,105h+84n. This forces h=-4n/5 and output (0,n/5,0). In the third sector write m=n=-r with r>0. The weighted valuations are 105h-120r,105h-120r,105h-119r. Thus h=8r/7 and the output is (0,0,r/7). In the final sector h=0.
Exact mathematical passage
m=n=-r
R^{1+\eps}Use in source · 1 passage
Search aliases: R^1+eps eps
Use in source · 02_signatures.tex, line 29
The conditional abc height bound. An abc estimate makes the role of the signature explicit. Let H=C^z, R=rad(ABC), s=frac1x+frac1y+frac1z. For a primitive solution, the powered terms are pairwise coprime and their radical is R. Positivity gives ABC<H^s. Assume the all epsilon abc estimate H≤ K_eps R^1+eps for every eps>0, with one constant K_eps independent of the triple. Then H<K_eps H^(1+eps)s. Whenever (1+eps)s<1, it follows that H<K_eps^1/(1-(1+eps)s). This is a conditional bound, not an unc…
Exact mathematical passage
\label{eq:abc-assumed} H\le K_\eps R^{1+\eps}
R^{105/71}Use in source · 1 passage
Search aliases: R^105/71
Use in source · 02_signatures.tex, line 58
Why a generic radical ceiling fails. The exponent equation gives R<H^71/105 for the signature (3,5,7). It is tempting to reverse this relation by postulating H≤ R^105/71 for every primitive sum. That universal assertion is false. The exact equality 2+109·3^10=23^5=6 436 343 has radical 2·3·109·23=15 042, and (23^5)^71>15 042^105. Its logarithmic quality is about 1.629911684. It is not a (3,5,7) counterexample. It shows that an estimate strong enough to close the desired signature must retain its…
Exact mathematical passage
H\le R^{105/71}
r^7Use in source · 1 passage
Search aliases: r^7
Use in source · 10_reciprocity.tex, line 54
Its mechanism retains the actual joint finite quotient. The three Kummer classes are independent after cyclotomic restriction. Distinct cyclotomic eigenspaces separate the natural and opposite images. Chebotarev then realizes joint vectors at good split primes, where the same quartic at the paired K places forces products to be coefficient conjugate. The resulting conditions are u^7=-u and r^7=r. In the descended subcase beta has an F_7 representative.
Exact mathematical passage
r^7=r
R_{l,q}Use in source · 1 passage
Search aliases: R_l,q
Use in source · 10_reciprocity.tex, line 15
For degree two, the indices b and -b represent the same prime. Frobenius sends a residue x to x^q, and ℓ_q(x^q)=-ℓ_q(x). Therefore (-b)ℓ_q(x^q)=bℓ_q(x). The functional is well defined on prime ideals, not just on the six embeddings. Principal reciprocity gives, for a test prime generator π_l outside the conductor, R_l,q=-∑_b∈reps_qbℓ_q(π_l(z_q^b)) =-f_q^-1∑_b=1^6bℓ_q(π_l(z_q^b)), where f_q=1 or two. Using all six terms without dividing by two in the second case would double count each prime. Con…
Exact mathematical passage
\label{eq:orbitreciprocity} R_{l,q}=-\sum_{b\in\mathrm{reps}_q}b\ell_q(\pi_l(z_q^b)) =-f_q^{-1}\sum_{b=1}^6b\ell_q(\pi_l(z_q^b)),
\radUse in source · 1 passage
Search aliases: rad
Use in source · 02_signatures.tex, line 25
The conditional abc height bound. An abc estimate makes the role of the signature explicit. Let H=C^z, R=rad(ABC), s=frac1x+frac1y+frac1z. For a primitive solution, the powered terms are pairwise coprime and their radical is R. Positivity gives ABC<H^s. Assume the all epsilon abc estimate H≤ K_eps R^1+eps for every eps>0, with one constant K_eps independent of the triple. Then H<K_eps H^(1+eps)s. Whenever (1+eps)s<1, it follows that H<K_eps^1/(1-(1+eps)s). This is a conditional bound, not an unc…
Exact mathematical passage
H=C^z,\qquad R=\rad(ABC),\qquad s=\frac1x+\frac1y+\frac1z.
\operatorname{rank}Use in source · 3 passages
Search aliases: rank operatorname
Use in source · 22_affine.tex, line 18
The homogeneous results. The complete exact matrices, row reductions, null bases, and rank minors give crrrl Pattern · rankA · dim_kker A · dim_kcoker A · Remaining difference support
Exact mathematical passage
\operatorname{rank}A
\mathrm{reps}Use in source · 2 passages
Search aliases: reps mathrm
Use in source · 10_reciprocity.tex, line 15
For degree two, the indices b and -b represent the same prime. Frobenius sends a residue x to x^q, and ℓ_q(x^q)=-ℓ_q(x). Therefore (-b)ℓ_q(x^q)=bℓ_q(x). The functional is well defined on prime ideals, not just on the six embeddings. Principal reciprocity gives, for a test prime generator π_l outside the conductor, R_l,q=-∑_b∈reps_qbℓ_q(π_l(z_q^b)) =-f_q^-1∑_b=1^6bℓ_q(π_l(z_q^b)), where f_q=1 or two. Using all six terms without dividing by two in the second case would double count each prime. Con…
Exact mathematical passage
\label{eq:orbitreciprocity} R_{l,q}=-\sum_{b\in\mathrm{reps}_q}b\ell_q(\pi_l(z_q^b)) =-f_q^{-1}\sum_{b=1}^6b\ell_q(\pi_l(z_q^b)),
\mathrm{reps}_qUse in source · 1 passage
Search aliases: reps_q mathrm
Use in source · 10_reciprocity.tex, line 15
For degree two, the indices b and -b represent the same prime. Frobenius sends a residue x to x^q, and ℓ_q(x^q)=-ℓ_q(x). Therefore (-b)ℓ_q(x^q)=bℓ_q(x). The functional is well defined on prime ideals, not just on the six embeddings. Principal reciprocity gives, for a test prime generator π_l outside the conductor, R_l,q=-∑_b∈reps_qbℓ_q(π_l(z_q^b)) =-f_q^-1∑_b=1^6bℓ_q(π_l(z_q^b)), where f_q=1 or two. Using all six terms without dividing by two in the second case would double count each prime. Con…
Exact mathematical passage
\label{eq:orbitreciprocity} R_{l,q}=-\sum_{b\in\mathrm{reps}_q}b\ell_q(\pi_l(z_q^b)) =-f_q^{-1}\sum_{b=1}^6b\ell_q(\pi_l(z_q^b)),
\mathrm{reps}_rUse in source · 1 passage
Search aliases: reps_r mathrm
Use in source · 21_ramified_frobenius.tex, line 23
n q place, write π_q=q u_q with u_q a unit. The same norm identity gives ℓ_q(u_q)=-∑_b=2^6ℓ_q(π_q(z_q^b)). Principal reciprocity on π_q therefore supplies the own conductor column D_q,q=∑_b=2^6(1-b)ℓ_q(π_q(z_q^b)). For r≠ q, the column is D_q,r=-∑_b∈reps_rbℓ_r(π_q(z_r^b)). The degree two representative convention is the one proved in [source reference]. These formulas use the same arithmetic Frobenius principal idele convention as the ordinary localization rows.
Exact mathematical passage
\label{eq:othercolumn} D_{q,r}=-\sum_{b\in\mathrm{reps}_r}b\ell_r(\pi_q(z_r^b)).
\rfloorUse in source · 1 passage
Search aliases: rfloor
Use in source · 12_fixed_seven.tex, line 56
At higher actual lattice depth, put m=lfloor n/2rfloor+1. The inherited diagonal theorem supplies an actual global upper extension α_j for 1≤ j≤ m. In the common range 1≤ j≤min(m,7c-1), its local coordinates satisfy U_j=2ϵ wg_j, V_j=Λ_jg_j, W_j=7cg_j for one unit g_j. The exact valuation matrix preserves the valuation identity at every legitimate triangular precision 1≤ j≤ m, even beyond the proved unit period window. Consequently v_7^trunc(W_j)=min(j,1+v_7(c)). At depth one, only the residual g…
Exact mathematical passage
m=\lfloor n/2\rfloor+1
SUse in source · 10 passages
Search aliases: S
Use in source · 08_global_coefficients.tex, line 24
[Discriminant collapse] Suppose a good Frobenius polynomial has integral coefficients and factorization P(T)=(T^2-τ T+Q)(T^2-σ(τ)T+Q) =T^4-ST^3+C_2T^2-QST+Q^2, where Oloc/Z_7 is unramified quadratic, σ is its nontrivial automorphism, all complex roots have absolute value sqrt Q, and τ≡ Q+1pmod7^rOloc. Then the integer D=S^2-4(C_2-2Q) satisfies 7^2rmid D and 0≤ D≤16Q. If 7^2r>16Q, then P(T)=(T^2-aT+Q)^2, a∈Z, |a|≤2sqrt Q, a≡ Q+1pmod7^r. Expansion gives S=τ+σ(τ), C_2-2Q=τσ(τ), and D=(τ-σ(τ))^2. Th…
Exact mathematical passage
P(T)=(T^2-\tau T+Q)(T^2-\sigma(\tau)T+Q) =T^4-ST^3+C_2T^2-QST+Q^2,
s^{12}Use in source · 1 passage
Search aliases: s^12
Use in source · 03_scaling.tex, line 5
A rational parametrization and its integral defect. For the signature (3,5,7), rational points are easy to manufacture. Primitive integer points are not. The following polynomial identity makes the difference exact. [Universal power identity] In any commutative ring, put s=1-t and A_0=t^12s^28, B_0=t^7s^17, C_0=t^5s^12. Then A_0^3+B_0^5=C_0^7. Weighted scaling (A,B,C)=(λ^35A_0,λ^21B_0,λ^15C_0) preserves this equality. The two terms on the left are t^36s^84 and t^35s^85. Their sum is t^35s^84(t+s…
Exact mathematical passage
A_0=t^{12}s^{28},\qquad B_0=t^7s^{17},\qquad C_0=t^5s^{12}.
s^{17}Use in source · 1 passage
Search aliases: s^17
Use in source · 03_scaling.tex, line 5
A rational parametrization and its integral defect. For the signature (3,5,7), rational points are easy to manufacture. Primitive integer points are not. The following polynomial identity makes the difference exact. [Universal power identity] In any commutative ring, put s=1-t and A_0=t^12s^28, B_0=t^7s^17, C_0=t^5s^12. Then A_0^3+B_0^5=C_0^7. Weighted scaling (A,B,C)=(λ^35A_0,λ^21B_0,λ^15C_0) preserves this equality. The two terms on the left are t^36s^84 and t^35s^85. Their sum is t^35s^84(t+s…
Exact mathematical passage
A_0=t^{12}s^{28},\qquad B_0=t^7s^{17},\qquad C_0=t^5s^{12}.
S^2Use in source · 1 passage
Search aliases: S^2
Use in source · 08_global_coefficients.tex, line 29
beginlemma[Discriminant collapse] Suppose a good Frobenius polynomial has integral coefficients and factorization P(T)=(T^2-τ T+Q)(T^2-σ(τ)T+Q) =T^4-ST^3+C_2T^2-QST+Q^2, where Oloc/Z_7 is unramified quadratic, σ is its nontrivial automorphism, all complex roots have absolute value sqrt Q, and τ≡ Q+1pmod7^rOloc. Then the integer D=S^2-4(C_2-2Q) satisfies 7^2rmid D and 0≤ D≤16Q. If 7^2r>16Q, then P(T)=(T^2-aT+Q)^2, a∈Z, |a|≤2sqrt Q, a≡ Q+1pmod7^r. Expansion gives S=τ+σ(τ), C_2-2Q=τσ(τ), and D=(τ-σ…
Exact mathematical passage
D=S^2-4(C_2-2Q)
s^{28}Use in source · 1 passage
Search aliases: s^28
Use in source · 03_scaling.tex, line 5
A rational parametrization and its integral defect. For the signature (3,5,7), rational points are easy to manufacture. Primitive integer points are not. The following polynomial identity makes the difference exact. [Universal power identity] In any commutative ring, put s=1-t and A_0=t^12s^28, B_0=t^7s^17, C_0=t^5s^12. Then A_0^3+B_0^5=C_0^7. Weighted scaling (A,B,C)=(λ^35A_0,λ^21B_0,λ^15C_0) preserves this equality. The two terms on the left are t^36s^84 and t^35s^85. Their sum is t^35s^84(t+s…
Exact mathematical passage
A_0=t^{12}s^{28},\qquad B_0=t^7s^{17},\qquad C_0=t^5s^{12}.
s^{84}Use in source · 2 passages
Search aliases: s^84
Use in source · 03_scaling.tex, line 15
Universal power identity] In any commutative ring, put s=1-t and A_0=t^12s^28, B_0=t^7s^17, C_0=t^5s^12. Then A_0^3+B_0^5=C_0^7. Weighted scaling (A,B,C)=(λ^35A_0,λ^21B_0,λ^15C_0) preserves this equality. The two terms on the left are t^36s^84 and t^35s^85. Their sum is t^35s^84(t+s)=t^35s^84, which is C_0^7. The three scaling weights multiply the powered terms by the same factor λ^105. Over a field, if t and 1-t are nonzero, all three bases are nonzero and A_0^3/C_0^7=t. Thus every regular para…
Exact mathematical passage
t^{36}s^{84}
s^{85}Use in source · 1 passage
Search aliases: s^85
Use in source · 03_scaling.tex, line 15
ntity] In any commutative ring, put s=1-t and A_0=t^12s^28, B_0=t^7s^17, C_0=t^5s^12. Then A_0^3+B_0^5=C_0^7. Weighted scaling (A,B,C)=(λ^35A_0,λ^21B_0,λ^15C_0) preserves this equality. The two terms on the left are t^36s^84 and t^35s^85. Their sum is t^35s^84(t+s)=t^35s^84, which is C_0^7. The three scaling weights multiply the powered terms by the same factor λ^105. Over a field, if t and 1-t are nonzero, all three bases are nonzero and A_0^3/C_0^7=t. Thus every regular parameter is locally po…
Exact mathematical passage
t^{35}s^{85}
s_1Use in source · 3 passages
Search aliases: s_1
Use in source · 06_modular_labels.tex, line 16
From point counts to a common polynomial. For a smooth genus two curve over F_q, let N_1,N_2 count its points over F_q,F_q^2, including infinity. Write s_1=q+1-N_1, s_2=fracs_1^2-(q^2+1-N_2)2. Its Frobenius polynomial is T^4-s_1T^3+s_2T^2-qs_1T+q^2. In the RM setting, the rank two traces are the roots of X^2-s_1X+(s_2-2q). The two roots belong to one common quartic. Choosing one root from one fiber and another root from an unrelated fiber can enlarge a necessary locus, but does not give a compat…
Exact mathematical passage
s_1=q+1-N_1,\qquad s_2=\frac{s_1^2-(q^2+1-N_2)}2.
s_1^2Use in source · 1 passage
Search aliases: s_1^2
Use in source · 06_modular_labels.tex, line 16
From point counts to a common polynomial. For a smooth genus two curve over F_q, let N_1,N_2 count its points over F_q,F_q^2, including infinity. Write s_1=q+1-N_1, s_2=fracs_1^2-(q^2+1-N_2)2. Its Frobenius polynomial is T^4-s_1T^3+s_2T^2-qs_1T+q^2. In the RM setting, the rank two traces are the roots of X^2-s_1X+(s_2-2q). The two roots belong to one common quartic. Choosing one root from one fiber and another root from an unrelated fiber can enlarge a necessary locus, but does not give a compat…
Exact mathematical passage
s_1=q+1-N_1,\qquad s_2=\frac{s_1^2-(q^2+1-N_2)}2.
s_2Use in source · 3 passages
Search aliases: s_2
Use in source · 06_modular_labels.tex, line 16
From point counts to a common polynomial. For a smooth genus two curve over F_q, let N_1,N_2 count its points over F_q,F_q^2, including infinity. Write s_1=q+1-N_1, s_2=fracs_1^2-(q^2+1-N_2)2. Its Frobenius polynomial is T^4-s_1T^3+s_2T^2-qs_1T+q^2. In the RM setting, the rank two traces are the roots of X^2-s_1X+(s_2-2q). The two roots belong to one common quartic. Choosing one root from one fiber and another root from an unrelated fiber can enlarge a necessary locus, but does not give a compat…
Exact mathematical passage
s_1=q+1-N_1,\qquad s_2=\frac{s_1^2-(q^2+1-N_2)}2.
s_qUse in source · 4 passages
Search aliases: s_q
Use in source · 10_reciprocity.tex, line 8
the restriction is additive and satisfies γ(δ_a gδ_a^-1)=a^-1γ(g). At a conductor prime q≡1pmod7, there are six degree one primes in M. At q≡-1pmod7, there are three degree two primes. If z_q is a chosen seventh root of unity and ℓ_q is the corresponding seventh power logarithm, the local functional at the embedding ζ_7↦ z_q^b is b s_qℓ_q.
Exact mathematical passage
b s_q\ell_q
s_rUse in source · 1 passage
Search aliases: s_r
Use in source · 21_ramified_frobenius.tex, line 31
Because each support prime is inert in K/Q, restriction at a rational q≡1pmod7 multiplies the compatible rational Frobenius value by two: b_q=2∑_rD_q,rs_r. For q≡-1pmod7, the compatible squared rational Frobenius has beta value zero, from the cocycle factor 1+χ(q)^-1=0. This does not say beta is unramified there.
Exact mathematical passage
\label{eq:betaF} b_q=2\sum_rD_{q,r}s_r.
\ShaUse in source · 3 passages
Search aliases: Sha
Use in source · 19_bare_primitive.tex, line 14
[The dual Kummer kernel] For every finite Σ containing the prime above seven and all real places, Sha^1_Σ(K,μ_7) =ker(H^1(Gsig,μ_7) ⟶∏_v∈ΣH^1(K_v,μ_7))=0. The discriminant of K is five. Minkowski's ideal bound is sqrt5/2<2, so every ideal class has a representative of norm one. Thus Cl(K)=1 [source reference].
Exact mathematical passage
\Sha^1_\Sigma(K,\mu_7) =\ker\left(H^1(\Gsig,\mu_7) \longrightarrow\prod_{v\in\Sigma}H^1(K_v,\mu_7)\right)=0.
\Sha^1_\SigmaUse in source · 2 passages
Search aliases: Sha^1_Σ Sha Sigma
Use in source · 19_bare_primitive.tex, line 14
[The dual Kummer kernel] For every finite Σ containing the prime above seven and all real places, Sha^1_Σ(K,μ_7) =ker(H^1(Gsig,μ_7) ⟶∏_v∈ΣH^1(K_v,μ_7))=0. The discriminant of K is five. Minkowski's ideal bound is sqrt5/2<2, so every ideal class has a representative of norm one. Thus Cl(K)=1 [source reference].
Exact mathematical passage
\Sha^1_\Sigma(K,\mu_7) =\ker\left(H^1(\Gsig,\mu_7) \longrightarrow\prod_{v\in\Sigma}H^1(K_v,\mu_7)\right)=0.
\Sha^2_\SigmaUse in source · 1 passage
Search aliases: Sha^2_Σ Sha Sigma
Use in source · 19_bare_primitive.tex, line 47
Finite set Poitou--Tate duality pairs this group perfectly with Sha^2_Σ(K,k). Every prime dividing the module order belongs to Σ, as required by the cited theorem. Odd primary real Tate terms vanish. Thus localization H^2(Gsig,k)⟶⊕_v∈ΣH^2(K_v,k) is injective [source reference].
Exact mathematical passage
\Sha^2_\Sigma(K,k)
\mathfrak{sl}Use in source · 1 passage
Search aliases: sl mathfrak
Use in source · 13_deformation.tex, line 9
Every determinant χ framed lift of this residual representation has a unique expression ρ(g)=(I+7X(g))ρ_0(g), X(g)=p(g) · q(g) β(g) · -p(g)∈sl_2(k). The determinant condition makes X traceless. Multiplication shows that ρ is a representation if and only if X(gh)=X(g)+Ad(overlineρ_0(g))X(h). For overlineρ_0(g)=(x · a
Exact mathematical passage
\label{eq:adjoint} \rho(g)=(I+7X(g))\rho_0(g),\qquad X(g)=\begin{pmatrix}p(g)&q(g)\\\beta(g)&-p(g)\end{pmatrix}\in\mathfrak{sl}_2(k).
\mathfrak{sl}_2Use in source · 1 passage
Search aliases: sl_2 mathfrak
Use in source · 13_deformation.tex, line 9
Every determinant χ framed lift of this residual representation has a unique expression ρ(g)=(I+7X(g))ρ_0(g), X(g)=p(g) · q(g) β(g) · -p(g)∈sl_2(k). The determinant condition makes X traceless. Multiplication shows that ρ is a representation if and only if X(gh)=X(g)+Ad(overlineρ_0(g))X(h). For overlineρ_0(g)=(x · a
Exact mathematical passage
\label{eq:adjoint} \rho(g)=(I+7X(g))\rho_0(g),\qquad X(g)=\begin{pmatrix}p(g)&q(g)\\\beta(g)&-p(g)\end{pmatrix}\in\mathfrak{sl}_2(k).
TUse in source · 43 passages
Search aliases: T
Use in source · 05_frey.tex, line 53
When C is odd, the integral model has good reduction at two. Over the norm four prime of K=Q(sqrt5), the full Frobenius polynomial is either (T^2+T+4)^2 or (T^2+4)^2. A hypothetical degree one constituent, after the conductor and global unit argument, must have eighth power one. Neither quadratic has a root with eighth power one modulo seven. For T^2+4, a root has square three, hence eighth power four. For T^2+T+4, reduction of powers gives fourth power five, again eighth power four. The contrad…
Exact mathematical passage
(T^2+T+4)^2
tUse in source · 59 passages
Search aliases: t
Use in source · 03_scaling.tex, line 4
A rational parametrization and its integral defect. For the signature (3,5,7), rational points are easy to manufacture. Primitive integer points are not. The following polynomial identity makes the difference exact. [Universal power identity] In any commutative ring, put s=1-t and A_0=t^12s^28, B_0=t^7s^17, C_0=t^5s^12. Then A_0^3+B_0^5=C_0^7. Weighted scaling (A,B,C)=(λ^35A_0,λ^21B_0,λ^15C_0) preserves this equality. The two terms on the left are t^36s^84 and t^35s^85. Their sum is t^35s^84(t+s…
Exact mathematical passage
s=1-t
t^{-1}Use in source · 1 passage
Search aliases: t^-1
Use in source · 06_modular_labels.tex, line 6
The Block 47 audit identified this mechanism in the pinned Pacetti--Villagra Torcomian code. It also identified a parameter dictionary issue at eleven: the unlabelled multisets agreed, but individual labels matched after t↦ t^-1. A refinement attached to the wrong labels could incorrectly combine local constraints. The new reader therefore distinguishes direct point counts on [source reference] from imported hypergeometric parameter tables.
Exact mathematical passage
t\mapsto t^{-1}
t^{12}Use in source · 1 passage
Search aliases: t^12
Use in source · 03_scaling.tex, line 5
A rational parametrization and its integral defect. For the signature (3,5,7), rational points are easy to manufacture. Primitive integer points are not. The following polynomial identity makes the difference exact. [Universal power identity] In any commutative ring, put s=1-t and A_0=t^12s^28, B_0=t^7s^17, C_0=t^5s^12. Then A_0^3+B_0^5=C_0^7. Weighted scaling (A,B,C)=(λ^35A_0,λ^21B_0,λ^15C_0) preserves this equality. The two terms on the left are t^36s^84 and t^35s^85. Their sum is t^35s^84(t+s…
Exact mathematical passage
A_0=t^{12}s^{28},\qquad B_0=t^7s^{17},\qquad C_0=t^5s^{12}.
T^2Use in source · 10 passages
Search aliases: T^2
Use in source · 05_frey.tex, line 53
When C is odd, the integral model has good reduction at two. Over the norm four prime of K=Q(sqrt5), the full Frobenius polynomial is either (T^2+T+4)^2 or (T^2+4)^2. A hypothetical degree one constituent, after the conductor and global unit argument, must have eighth power one. Neither quadratic has a root with eighth power one modulo seven. For T^2+4, a root has square three, hence eighth power four. For T^2+T+4, reduction of powers gives fourth power five, again eighth power four. The contrad…
Exact mathematical passage
(T^2+T+4)^2
t^2Use in source · 2 passages
Search aliases: t^2
Use in source · 05_frey.tex, line 10
12AX+5B^2. This is the genus two family used in the recovered source. The comparison with the family of Pacetti and Villagra Torcomian is a rational change of coordinates, with their variables translated as (a,b,c,p)=(B,-C,A,7) [source reference]. Explicitly, put t=-B^5/A^3, d=-B^2/A, and substitute x_PV=d/X, y_PV=tZ/X^3 into their equation y^2=5x^6-12x^5+10tx^3+t^2. Multiplication by X^6/t^2 gives [source reference]. The map is rational for AB≠0. It does not on its own verify a twist, integral …
Exact mathematical passage
y^2=5x^6-12x^5+10tx^3+t^2
T^3Use in source · 3 passages
Search aliases: T^3
Use in source · 06_modular_labels.tex, line 21
From point counts to a common polynomial. For a smooth genus two curve over F_q, let N_1,N_2 count its points over F_q,F_q^2, including infinity. Write s_1=q+1-N_1, s_2=fracs_1^2-(q^2+1-N_2)2. Its Frobenius polynomial is T^4-s_1T^3+s_2T^2-qs_1T+q^2. In the RM setting, the rank two traces are the roots of X^2-s_1X+(s_2-2q). The two roots belong to one common quartic. Choosing one root from one fiber and another root from an unrelated fiber can enlarge a necessary locus, but does not give a compat…
Exact mathematical passage
\label{eq:quartic} T^4-s_1T^3+s_2T^2-qs_1T+q^2.
t^{35}Use in source · 2 passages
Search aliases: t^35
Use in source · 03_scaling.tex, line 15
ntity] In any commutative ring, put s=1-t and A_0=t^12s^28, B_0=t^7s^17, C_0=t^5s^12. Then A_0^3+B_0^5=C_0^7. Weighted scaling (A,B,C)=(λ^35A_0,λ^21B_0,λ^15C_0) preserves this equality. The two terms on the left are t^36s^84 and t^35s^85. Their sum is t^35s^84(t+s)=t^35s^84, which is C_0^7. The three scaling weights multiply the powered terms by the same factor λ^105. Over a field, if t and 1-t are nonzero, all three bases are nonzero and A_0^3/C_0^7=t. Thus every regular parameter is locally po…
Exact mathematical passage
t^{35}s^{85}
t^{36}Use in source · 1 passage
Search aliases: t^36
Use in source · 03_scaling.tex, line 15
Universal power identity] In any commutative ring, put s=1-t and A_0=t^12s^28, B_0=t^7s^17, C_0=t^5s^12. Then A_0^3+B_0^5=C_0^7. Weighted scaling (A,B,C)=(λ^35A_0,λ^21B_0,λ^15C_0) preserves this equality. The two terms on the left are t^36s^84 and t^35s^85. Their sum is t^35s^84(t+s)=t^35s^84, which is C_0^7. The three scaling weights multiply the powered terms by the same factor λ^105. Over a field, if t and 1-t are nonzero, all three bases are nonzero and A_0^3/C_0^7=t. Thus every regular para…
Exact mathematical passage
t^{36}s^{84}
T^4Use in source · 3 passages
Search aliases: T^4
Use in source · 06_modular_labels.tex, line 21
From point counts to a common polynomial. For a smooth genus two curve over F_q, let N_1,N_2 count its points over F_q,F_q^2, including infinity. Write s_1=q+1-N_1, s_2=fracs_1^2-(q^2+1-N_2)2. Its Frobenius polynomial is T^4-s_1T^3+s_2T^2-qs_1T+q^2. In the RM setting, the rank two traces are the roots of X^2-s_1X+(s_2-2q). The two roots belong to one common quartic. Choosing one root from one fiber and another root from an unrelated fiber can enlarge a necessary locus, but does not give a compat…
Exact mathematical passage
\label{eq:quartic} T^4-s_1T^3+s_2T^2-qs_1T+q^2.
t^5Use in source · 1 passage
Search aliases: t^5
Use in source · 03_scaling.tex, line 5
A rational parametrization and its integral defect. For the signature (3,5,7), rational points are easy to manufacture. Primitive integer points are not. The following polynomial identity makes the difference exact. [Universal power identity] In any commutative ring, put s=1-t and A_0=t^12s^28, B_0=t^7s^17, C_0=t^5s^12. Then A_0^3+B_0^5=C_0^7. Weighted scaling (A,B,C)=(λ^35A_0,λ^21B_0,λ^15C_0) preserves this equality. The two terms on the left are t^36s^84 and t^35s^85. Their sum is t^35s^84(t+s…
Exact mathematical passage
A_0=t^{12}s^{28},\qquad B_0=t^7s^{17},\qquad C_0=t^5s^{12}.
t^6Use in source · 1 passage
Search aliases: t^6
Use in source · 20_localization.tex, line 30
An explicit image parametrization. Put a=z_7(3) and x_i=z(I_q_i). Let ℓ_i be the seventh power logarithm normalized by ℓ_i(g_q_i)=1. The supporting symbols of ϕ vanish. For rational seven adic units define ℓ_7(t)=fraclog(t^6)log(3^6)pmod7 =6fract^6-17pmod7. Principal reciprocity determines the remaining coordinates: z_7(ϕ) · =0,
Exact mathematical passage
\ell_7(t)=\frac{\log(t^6)}{\log(3^6)}\pmod7 =6\frac{t^6-1}{7}\pmod7.
t^7Use in source · 1 passage
Search aliases: t^7
Use in source · 03_scaling.tex, line 5
A rational parametrization and its integral defect. For the signature (3,5,7), rational points are easy to manufacture. Primitive integer points are not. The following polynomial identity makes the difference exact. [Universal power identity] In any commutative ring, put s=1-t and A_0=t^12s^28, B_0=t^7s^17, C_0=t^5s^12. Then A_0^3+B_0^5=C_0^7. Weighted scaling (A,B,C)=(λ^35A_0,λ^21B_0,λ^15C_0) preserves this equality. The two terms on the left are t^36s^84 and t^35s^85. Their sum is t^35s^84(t+s…
Exact mathematical passage
A_0=t^{12}s^{28},\qquad B_0=t^7s^{17},\qquad C_0=t^5s^{12}.
T^mUse in source · 1 passage
Search aliases: T^m
Use in source · 17_local_flexibility.tex, line 37
Changing F to T^mF adds mc to b and mp(T) to p(F), so the left side of [source reference] is unchanged. A consistent Frobenius lift change cannot remove the affine right side.
Exact mathematical passage
T^mF
T^QUse in source · 1 passage
Search aliases: T^Q
Use in source · 07_natural_depth.tex, line 90
If α∪β were a coboundary in the endpoint case, the two adjacent entries would extend to a representation in the three by three upper unitriangular group over k. That group has exponent seven. The tame relation FTF^-1=T^Q with Q≡1pmod7 would force the images to commute. Their central commutator is α(F)β(T)-α(T)β(F)=α(F)β(T)≠0. This contradiction proves local nonvanishing at the endpoint. The zero below the endpoint and nonzero at it are statements about the same labelled local representation.
Exact mathematical passage
FTF^{-1}=T^Q
T_aUse in source · 2 passages
Search aliases: T_a
Use in source · 15_semilinear_image.tex, line 35
The image group contains upper and lower elements T_a,G_b with tr(T_aG_b)-tr(T_a)-tr(G_b)+2=49abpmod343. All leading residue values occur as the elements vary. This flexibility does not allow a fixed local uniformizer probe to be replaced by any convenient upper element. Preimages in the Galois group may have different local membership and different arithmetic labels. Knowledge of the unlabelled image is therefore weaker than the quotient map and compatible local embeddings required by the mixed…
Exact mathematical passage
T_a,G_b
T_jUse in source · 2 passages
Search aliases: T_j
Use in source · 12_fixed_seven.tex, line 37
Use the three coordinate basis X_j=[ϕ], H_j=[3], T_j=[7]. The third coordinate is valuation. The actual local scalar class in the window is κ_j=2ϵ wX_j+Λ_jH_j+7cT_j, where Λ_j=7ℓ_j(C_0)-3ℓ_j(A)-3-5d_j, ℓ_j(a)=fraclog(a^6)log(3^6)pmod7^j, d_j=fracℓ_j(5)2. At level 49, with Q_A=(A^6-1)/7 modulo 49 and q_0=(C_0^6-1)/7 modulo seven, Λ_2=47+38Q_A+14Q_A^2-7q_0pmod49. Both C_0 and the coordinate 7c[7] disappeared modulo seven. Neither may be omitted modulo forty nine.
Exact mathematical passage
X_j=[\phi],\qquad H_j=[3],\qquad T_j=[7].
T_qUse in source · 3 passages
Search aliases: T_q
Use in source · 22_affine.tex, line 36
The arithmetic target. Choose a bare global primitive p_0 from Theorem~[source reference]. This is one continuous global cochain. Its values have not been computed. Define t(p_0)∈ k^15 as follows. Its first six coordinates are the ordinary local character coordinates of p_v^fix-p_0|_v. At each support prime, put t_q(p_0)=-h_qc_q-c_qp_0(F_q)+b_qp_0(T_q), h_q=(q^2-1)/7pmod7. At support, α is zero and χ trivial, so the restriction of p_0 is an ordinary character. At fixed places only its difference…
Exact mathematical passage
\label{eq:target} t_q(p_0)=-h_qc_q-c_qp_0(F_q)+b_qp_0(T_q), \qquad h_q=(q^2-1)/7\pmod7.
t_qUse in source · 2 passages
Search aliases: t_q
Use in source · 22_affine.tex, line 36
The arithmetic target. Choose a bare global primitive p_0 from Theorem~[source reference]. This is one continuous global cochain. Its values have not been computed. Define t(p_0)∈ k^15 as follows. Its first six coordinates are the ordinary local character coordinates of p_v^fix-p_0|_v. At each support prime, put t_q(p_0)=-h_qc_q-c_qp_0(F_q)+b_qp_0(T_q), h_q=(q^2-1)/7pmod7. At support, α is zero and χ trivial, so the restriction of p_0 is an ordinary character. At fixed places only its difference…
Exact mathematical passage
\label{eq:target} t_q(p_0)=-h_qc_q-c_qp_0(F_q)+b_qp_0(T_q), \qquad h_q=(q^2-1)/7\pmod7.
\operatorname{Tr}Use in source · 1 passage
Search aliases: Tr operatorname
Use in source · 10_reciprocity.tex, line 43
If the actual polynomial is P(X)=X^4-SX^3+BX^2-QSX+Q^2, the correction u=(τ-Q-1)/7 has Tr(u)=fracS-2(Q+1)7, Nm(u)=frac(B-2Q)-(Q+1)S+(Q+1)^249 modulo seven, under the stated divisibility assumptions. The two implementations extract these values using different methods, including an integer trace discriminant and an enumeration of k by trace and norm. Neither assigns a Frobenius value at a beta conductor prime by pretending beta is unramified there. Such conductor values require the separate self …
Exact mathematical passage
\operatorname{Tr}(u)=\frac{S-2(Q+1)}7,\qquad \operatorname{Nm}(u)=\frac{(B-2Q)-(Q+1)S+(Q+1)^2}{49}
\trUse in source · 6 passages
Search aliases: tr
Use in source · 10_reciprocity.tex, line 26
7c · d, det F=Q. Put d=1+7D and tr(F)=Q+1+7T. Then a=Q+7(T-D), and the determinant equation gives bc=T+(Q-1)D+7TD-7D^2. Modulo seven this becomes fractr(F)-Q-17=α(F)β(F) in k. This is the coefficient field rank two trace of the actual representation. It is not half the rational four dimensional trace. The hypothesis Q≡1pmod7 makes the extra diagonal term disappear. At other residue norms, [source reference] must be retained.
Exact mathematical passage
\tr(F)=Q+1+7T
\tr_{\Oloc}Use in source · 1 passage
Search aliases: tr_Oloc tr Oloc
Use in source · 14_integral_trace.tex, line 2
A mixed trace that detects the valuation. The valuation coordinate can be read from a joint word in the same representation. For T(g)=tr_Olocρ(g), define D(τ,g)=T(τ g)-T(τ)-T(g)+2. The elementary identity D(τ,g)=tr((ρ(τ)-I)(ρ(g)-I)) shows why the product label matters. Two lists of individual traces do not determine the trace of the product.
Exact mathematical passage
T(g)=\tr_{\Oloc}\rho(g)
\mathrm{trunc}Use in source · 1 passage
Search aliases: trunc mathrm
Use in source · 12_fixed_seven.tex, line 57
orem supplies an actual global upper extension α_j for 1≤ j≤ m. In the common range 1≤ j≤min(m,7c-1), its local coordinates satisfy U_j=2ϵ wg_j, V_j=Λ_jg_j, W_j=7cg_j for one unit g_j. The exact valuation matrix preserves the valuation identity at every legitimate triangular precision 1≤ j≤ m, even beyond the proved unit period window. Consequently v_7^trunc(W_j)=min(j,1+v_7(c)). At depth one, only the residual global triangular precision is supplied. Replacing the actual mod forty nine represen…
Exact mathematical passage
v_7^{\mathrm{trunc}}(W_j)=\min(j,1+v_7(c)).
UUse in source · 5 passages
Search aliases: U
Use in source · 12_fixed_seven.tex, line 56
At higher actual lattice depth, put m=lfloor n/2rfloor+1. The inherited diagonal theorem supplies an actual global upper extension α_j for 1≤ j≤ m. In the common range 1≤ j≤min(m,7c-1), its local coordinates satisfy U_j=2ϵ wg_j, V_j=Λ_jg_j, W_j=7cg_j for one unit g_j. The exact valuation matrix preserves the valuation identity at every legitimate triangular precision 1≤ j≤ m, even beyond the proved unit period window. Consequently v_7^trunc(W_j)=min(j,1+v_7(c)). At depth one, only the residual g…
Exact mathematical passage
U_j=2\epsilon wg_j
u^2Use in source · 1 passage
Search aliases: u^2
Use in source · 04_local_strata.tex, line 3
The complete local squareclass test. Write a hypothetical primitive solution as A=au^2, B=bv^2, C=cw^2, where a,b,c are positive pairwise coprime squarefree integers. The equation becomes a^3u^6+b^5v^10=c^7w^14. At a prime p, a local solution is called primitive when at most one base is divisible by p. Roots and bases are required to be nonzero in Z_p. The roots may depend on p. This last freedom is exactly why local existence can be weaker than an integer solution.
Exact mathematical passage
A=au^2,\qquad B=bv^2,\qquad C=cw^2,
u^3Use in source · 1 passage
Search aliases: u^3
Use in source · 05_frey.tex, line 30
At a variable prime dividing C, primitivity makes A,B units. In the residue field put u=-B^2/A. Then u^3=B, u^5=-A, and f_A,B(uX)=u^6(X^2+X-1)^2(X^2-2X+5). The discriminants of the two quadratics are 5 and -16, and their resultant is 45. Away from 2,3,5, the special fiber has exactly two ordinary nodes, rational normalization, and smooth infinity. This factorization is the local geometric entry point for the period and monodromy calculations.
Exact mathematical passage
u^3=B
u^{48}Use in source · 1 passage
Search aliases: u^48
Use in source · 12_fixed_seven.tex, line 5
Let X=[ϕ] and H=[3] in the mod seven local Kummer group. Exact logarithmic first digits give L(ϕ)=5overlinesqrt5, L(3)=6, L(sqrt5)=3, L(u)=fracu^48-17pmod7. The two unit coordinates X,H are independent, and [sqrt5]=4H. For a rational seven adic unit A, put q_7(A)=(A^6-1)/7 modulo seven. The local marked period comparison has P=ZI+2XR, Z=(3q_7(A)+5)H, with R from [source reference].
Exact mathematical passage
L(\phi)=5\overline{\sqrt5},\qquad L(3)=6,\qquad L(\sqrt5)=3, \quad L(u)=\frac{u^{48}-1}{7}\pmod7.
u^5Use in source · 2 passages
Search aliases: u^5
Use in source · 04_local_strata.tex, line 37
Odd prime lifting. For the polynomial on the left of [source reference] minus its right side, the partial derivatives are 6a^3u^5,qquad10b^5v^9, -14c^7w^13. In a primitive residue solution, at least two bases are units. At an odd prime, at most one of 6,10,14 is divisible by p. Thus at least one unit variable has unit derivative. Lift the other variables to nonzero integers, using p for a zero residue if necessary. Successive correction in the selected variable by multiples of p^n kills one new …
Exact mathematical passage
6a^3u^5,\qquad10b^5v^9,\qquad-14c^7w^{13}.
u^6Use in source · 4 passages
Search aliases: u^6
Use in source · 04_local_strata.tex, line 7
The complete local squareclass test. Write a hypothetical primitive solution as A=au^2, B=bv^2, C=cw^2, where a,b,c are positive pairwise coprime squarefree integers. The equation becomes a^3u^6+b^5v^10=c^7w^14. At a prime p, a local solution is called primitive when at most one base is divisible by p. Roots and bases are required to be nonzero in Z_p. The roots may depend on p. This last freedom is exactly why local existence can be weaker than an integer solution.
Exact mathematical passage
\label{eq:strata} a^3u^6+b^5v^{10}=c^7w^{14}.
u^7Use in source · 1 passage
Search aliases: u^7
Use in source · 10_reciprocity.tex, line 54
Its mechanism retains the actual joint finite quotient. The three Kummer classes are independent after cyclotomic restriction. Distinct cyclotomic eigenspaces separate the natural and opposite images. Chebotarev then realizes joint vectors at good split primes, where the same quartic at the paired K places forces products to be coefficient conjugate. The resulting conditions are u^7=-u and r^7=r. In the descended subcase beta has an F_7 representative.
Exact mathematical passage
u^7=-u
U_3Use in source · 1 passage
Search aliases: U_3
Use in source · 16_secondary.tex, line 34
An exact control separating the two stages. Take the finite group U_3(F_7) with coordinates (a,b,c) and multiplication (a,b,c)(A,B,C)=(a+A,b+B,c+C+aB). Use trivial cyclotomic action, α=a, β=b, and p=c. Then dp=-α∪β, so the first obstruction vanishes. The elements x=(1,0,0) and z=(0,0,1) commute, but R_p(x,z)=2, R_p(z,x)=0. For every ordinary cochain q, commuting arguments give dq(x,z)=dq(z,x). Hence no q solves the second equation. This is a finite group control, not an absolute Galois group or …
Exact mathematical passage
U_3(\F_7)
U_aUse in source · 1 passage
Search aliases: U_a
Use in source · 14_integral_trace.tex, line 40
Let U_a=(1 · 7a 0 · 1) and V=(1 · 0
Exact mathematical passage
U_a=\left(\begin{smallmatrix}1&7a\\0&1\end{smallmatrix}\right)
U_jUse in source · 1 passage
Search aliases: U_j
Use in source · 12_fixed_seven.tex, line 56
At higher actual lattice depth, put m=lfloor n/2rfloor+1. The inherited diagonal theorem supplies an actual global upper extension α_j for 1≤ j≤ m. In the common range 1≤ j≤min(m,7c-1), its local coordinates satisfy U_j=2ϵ wg_j, V_j=Λ_jg_j, W_j=7cg_j for one unit g_j. The exact valuation matrix preserves the valuation identity at every legitimate triangular precision 1≤ j≤ m, even beyond the proved unit period window. Consequently v_7^trunc(W_j)=min(j,1+v_7(c)). At depth one, only the residual g…
Exact mathematical passage
U_j=2\epsilon wg_j
u_qUse in source · 3 passages
Search aliases: u_q
Use in source · 21_ramified_frobenius.tex, line 14
At the chosen q place, write π_q=q u_q with u_q a unit. The same norm identity gives ℓ_q(u_q)=-∑_b=2^6ℓ_q(π_q(z_q^b)). Principal reciprocity on π_q therefore supplies the own conductor column D_q,q=∑_b=2^6(1-b)ℓ_q(π_q(z_q^b)). For r≠ q, the column is D_q,r=-∑_b∈reps_rbℓ_r(π_q(z_r^b)). The degree two representative convention is the one proved in [source reference]. These formulas use the same arithmetic Frobenius principal idele convention as the ordinary localization rows.
Exact mathematical passage
\pi_q=q u_q
VUse in source · 2 passages
Search aliases: V
Use in source · 12_fixed_seven.tex, line 56
At higher actual lattice depth, put m=lfloor n/2rfloor+1. The inherited diagonal theorem supplies an actual global upper extension α_j for 1≤ j≤ m. In the common range 1≤ j≤min(m,7c-1), its local coordinates satisfy U_j=2ϵ wg_j, V_j=Λ_jg_j, W_j=7cg_j for one unit g_j. The exact valuation matrix preserves the valuation identity at every legitimate triangular precision 1≤ j≤ m, even beyond the proved unit period window. Consequently v_7^trunc(W_j)=min(j,1+v_7(c)). At depth one, only the residual g…
Exact mathematical passage
V_j=\Lambda_jg_j
v^{10}Use in source · 5 passages
Search aliases: v^10
Use in source · 04_local_strata.tex, line 7
The complete local squareclass test. Write a hypothetical primitive solution as A=au^2, B=bv^2, C=cw^2, where a,b,c are positive pairwise coprime squarefree integers. The equation becomes a^3u^6+b^5v^10=c^7w^14. At a prime p, a local solution is called primitive when at most one base is divisible by p. Roots and bases are required to be nonzero in Z_p. The roots may depend on p. This last freedom is exactly why local existence can be weaker than an integer solution.
Exact mathematical passage
\label{eq:strata} a^3u^6+b^5v^{10}=c^7w^{14}.
v^2Use in source · 2 passages
Search aliases: v^2
Use in source · 04_local_strata.tex, line 3
The complete local squareclass test. Write a hypothetical primitive solution as A=au^2, B=bv^2, C=cw^2, where a,b,c are positive pairwise coprime squarefree integers. The equation becomes a^3u^6+b^5v^10=c^7w^14. At a prime p, a local solution is called primitive when at most one base is divisible by p. Roots and bases are required to be nonzero in Z_p. The roots may depend on p. This last freedom is exactly why local existence can be weaker than an integer solution.
Exact mathematical passage
A=au^2,\qquad B=bv^2,\qquad C=cw^2,
v^9Use in source · 1 passage
Search aliases: v^9
Use in source · 04_local_strata.tex, line 37
Odd prime lifting. For the polynomial on the left of [source reference] minus its right side, the partial derivatives are 6a^3u^5,qquad10b^5v^9, -14c^7w^13. In a primitive residue solution, at least two bases are units. At an odd prime, at most one of 6,10,14 is divisible by p. Thus at least one unit variable has unit derivative. Lift the other variables to nonzero integers, using p for a zero residue if necessary. Successive correction in the selected variable by multiples of p^n kills one new …
Exact mathematical passage
6a^3u^5,\qquad10b^5v^9,\qquad-14c^7w^{13}.
v_7Use in source · 8 passages
Search aliases: v_7
Use in source · 07_natural_depth.tex, line 39
Define n=min_g:ρ(g)_21≠0v_7(ρ(g)_21). This positive integer is attained. It is an invariant of the fixed natural frame, not a truncation selected for a computation. Every Oloc lattice stable under ρ is preserved by the two projectors, so up to homothety it has the form Λ_j=Oloc e_1+Oloc 7^j e_2. The unit upper monodromy forces j≥0. The attained lower entry minimum forces j≤ n. Conversely every j in that interval is stable. Thus the stable lattices form a finite segment with the natural and oppos…
Exact mathematical passage
\label{eq:actualdepth} n=\min_{g:\rho(g)_{21}\ne0}v_7(\rho(g)_{21}).
v_7^{\mathrm{trunc}}Use in source · 1 passage
Search aliases: v_7^trunc mathrm
Use in source · 12_fixed_seven.tex, line 57
orem supplies an actual global upper extension α_j for 1≤ j≤ m. In the common range 1≤ j≤min(m,7c-1), its local coordinates satisfy U_j=2ϵ wg_j, V_j=Λ_jg_j, W_j=7cg_j for one unit g_j. The exact valuation matrix preserves the valuation identity at every legitimate triangular precision 1≤ j≤ m, even beyond the proved unit period window. Consequently v_7^trunc(W_j)=min(j,1+v_7(c)). At depth one, only the residual global triangular precision is supplied. Replacing the actual mod forty nine represen…
Exact mathematical passage
v_7^{\mathrm{trunc}}(W_j)=\min(j,1+v_7(c)).
V_jUse in source · 1 passage
Search aliases: V_j
Use in source · 12_fixed_seven.tex, line 56
At higher actual lattice depth, put m=lfloor n/2rfloor+1. The inherited diagonal theorem supplies an actual global upper extension α_j for 1≤ j≤ m. In the common range 1≤ j≤min(m,7c-1), its local coordinates satisfy U_j=2ϵ wg_j, V_j=Λ_jg_j, W_j=7cg_j for one unit g_j. The exact valuation matrix preserves the valuation identity at every legitimate triangular precision 1≤ j≤ m, even beyond the proved unit period window. Consequently v_7^trunc(W_j)=min(j,1+v_7(c)). At depth one, only the residual g…
Exact mathematical passage
V_j=\Lambda_jg_j
v_qUse in source · 4 passages
Search aliases: v_q
Use in source · 07_natural_depth.tex, line 77
For the natural curve, the variable node thickness is 7v_q(C). The natural monodromy content is consequently t=1+v_7(v_q(C)). This inherited geometric bridge uses the ordinary node smoothing parameter and the identification of the graph pairing with the period valuation [source reference]. Applied to an actually ramified beta prime, [source reference] becomes n=1+v_7(v_q(C))+2s,qquad0≤ s≤ v_7(q^f-1). Here f is the residue degree in K. If 7nmid v_q(C) at every eligible prime, the actual depth is …
Exact mathematical passage
7v_q(C)
v_\ellUse in source · 4 passages
Search aliases: v_ℓ ell
Use in source · 03_scaling.tex, line 23
[Primitive scaling criterion] Fix a rational 0<t<1. There is a positive rational λ for which [source reference] gives primitive positive integers if and only if, for every rational prime ℓ, v_ℓ(t)>0 · Longrightarrowquad3mid v_ℓ(t), v_ℓ(1-t)>0 · Longrightarrowquad5mid v_ℓ(1-t), v_ℓ(t)<0 · Longrightarrowquad7mid -v_ℓ(t). When it exists, λ is unique. Write m=v_ℓ(t), n=v_ℓ(1-t) and h=v_ℓ(λ). Because t+(1-t)=1, the pair (m,n) belongs to one of four sectors: (m>0,n=0), (m=0,n>0), (m=n<0), (m=n=0). The…
Exact mathematical passage
v_\ell(t)>0&\quad\Longrightarrow\quad3\mid v_\ell(t),\\ v_\ell(1-t)>0&\quad\Longrightarrow\quad5\mid v_\ell(1-t),\\ v_\ell(t)<0&\quad\Longrightarrow\quad7\mid -v_\ell(t).
WUse in source · 2 passages
Search aliases: W
Use in source · 12_fixed_seven.tex, line 56
At higher actual lattice depth, put m=lfloor n/2rfloor+1. The inherited diagonal theorem supplies an actual global upper extension α_j for 1≤ j≤ m. In the common range 1≤ j≤min(m,7c-1), its local coordinates satisfy U_j=2ϵ wg_j, V_j=Λ_jg_j, W_j=7cg_j for one unit g_j. The exact valuation matrix preserves the valuation identity at every legitimate triangular precision 1≤ j≤ m, even beyond the proved unit period window. Consequently v_7^trunc(W_j)=min(j,1+v_7(c)). At depth one, only the residual g…
Exact mathematical passage
W_j=7cg_j
wUse in source · 27 passages
Search aliases: w
Use in source · 04_local_strata.tex, line 3
The complete local squareclass test. Write a hypothetical primitive solution as A=au^2, B=bv^2, C=cw^2, where a,b,c are positive pairwise coprime squarefree integers. The equation becomes a^3u^6+b^5v^10=c^7w^14. At a prime p, a local solution is called primitive when at most one base is divisible by p. Roots and bases are required to be nonzero in Z_p. The roots may depend on p. This last freedom is exactly why local existence can be weaker than an integer solution.
Exact mathematical passage
A=au^2,\qquad B=bv^2,\qquad C=cw^2,
w^{-14}Use in source · 1 passage
Search aliases: w^-14
Use in source · 04_local_strata.tex, line 42
If pmid a, primitivity forces the other two bases to be units. Reduction gives b^5v^10=c^7w^14. The quadratic character of this equality proves the necessary symbol condition. Conversely, v^10w^-14 ranges over the squares of F_p^×, because gcd(10,14,p-1)=2. The symbol condition lets us choose the two unit roots, then lift. The other two cases use gcd(6,14,p-1)=2 and gcd(6,10,p-1)=2. Moving a term across the equality gives the minus sign when pmid c.
Exact mathematical passage
v^{10}w^{-14}
w^{13}Use in source · 1 passage
Search aliases: w^13
Use in source · 04_local_strata.tex, line 37
Odd prime lifting. For the polynomial on the left of [source reference] minus its right side, the partial derivatives are 6a^3u^5,qquad10b^5v^9, -14c^7w^13. In a primitive residue solution, at least two bases are units. At an odd prime, at most one of 6,10,14 is divisible by p. Thus at least one unit variable has unit derivative. Lift the other variables to nonzero integers, using p for a zero residue if necessary. Successive correction in the selected variable by multiples of p^n kills one new …
Exact mathematical passage
6a^3u^5,\qquad10b^5v^9,\qquad-14c^7w^{13}.
w^{14}Use in source · 4 passages
Search aliases: w^14
Use in source · 04_local_strata.tex, line 7
The complete local squareclass test. Write a hypothetical primitive solution as A=au^2, B=bv^2, C=cw^2, where a,b,c are positive pairwise coprime squarefree integers. The equation becomes a^3u^6+b^5v^10=c^7w^14. At a prime p, a local solution is called primitive when at most one base is divisible by p. Roots and bases are required to be nonzero in Z_p. The roots may depend on p. This last freedom is exactly why local existence can be weaker than an integer solution.
Exact mathematical passage
\label{eq:strata} a^3u^6+b^5v^{10}=c^7w^{14}.
w^2Use in source · 2 passages
Search aliases: w^2
Use in source · 04_local_strata.tex, line 3
The complete local squareclass test. Write a hypothetical primitive solution as A=au^2, B=bv^2, C=cw^2, where a,b,c are positive pairwise coprime squarefree integers. The equation becomes a^3u^6+b^5v^10=c^7w^14. At a prime p, a local solution is called primitive when at most one base is divisible by p. Roots and bases are required to be nonzero in Z_p. The roots may depend on p. This last freedom is exactly why local existence can be weaker than an integer solution.
Exact mathematical passage
A=au^2,\qquad B=bv^2,\qquad C=cw^2,
W_jUse in source · 2 passages
Search aliases: W_j
Use in source · 12_fixed_seven.tex, line 56
At higher actual lattice depth, put m=lfloor n/2rfloor+1. The inherited diagonal theorem supplies an actual global upper extension α_j for 1≤ j≤ m. In the common range 1≤ j≤min(m,7c-1), its local coordinates satisfy U_j=2ϵ wg_j, V_j=Λ_jg_j, W_j=7cg_j for one unit g_j. The exact valuation matrix preserves the valuation identity at every legitimate triangular precision 1≤ j≤ m, even beyond the proved unit period window. Consequently v_7^trunc(W_j)=min(j,1+v_7(c)). At depth one, only the residual g…
Exact mathematical passage
W_j=7cg_j
\widetilde zUse in source · 1 passage
Search aliases: widetilde z widetilde
Use in source · 21_ramified_frobenius.tex, line 28
The direct independent check lifts the chosen root of Φ_7 modulo q^2 by Hensel, then computes π_q(widetilde z_q)/q modulo q. It verifies the self unit rather than assigning its missing value from the norm relation. The norm calculation and Hensel calculation share the input generator but use different arithmetic routes.
Exact mathematical passage
\pi_q(\widetilde z_q)/q
\widetilde z_qUse in source · 1 passage
Search aliases: widetilde z_q widetilde
Use in source · 21_ramified_frobenius.tex, line 28
The direct independent check lifts the chosen root of Φ_7 modulo q^2 by Hensel, then computes π_q(widetilde z_q)/q modulo q. It verifies the self unit rather than assigning its missing value from the norm relation. The norm calculation and Hensel calculation share the input generator but use different arithmetic routes.
Exact mathematical passage
\pi_q(\widetilde z_q)/q
x^{-1}Use in source · 1 passage
Search aliases: x^-1
Use in source · 13_deformation.tex, line 18
0 · 1), direct conjugation sends (p,q,b) to (p+ax^-1b, xq-2ap-a^2x^-1b, x^-1b). Thus the three coupled equations are β(gh) · =β(g)+χ(g)^-1β(h),
Exact mathematical passage
(p+ax^{-1}b,\;xq-2ap-a^2x^{-1}b,\;x^{-1}b).
X^{12}Use in source · 1 passage
Search aliases: X^12
Use in source · 05_frey.tex, line 38
A characteristic seven cancellation. In characteristic seven, the Cartier coefficient calculation comes from f^3. Its relevant coefficients are [X^6]f^3 · =1575B^4, · [X^5]f^3 · =4320A^2B, [X^13]f^3 · =36A, · [X^12]f^3 · =315B^2. The diagonal lanes vanish modulo seven and the off diagonal lanes become M=0 · A^2B
Exact mathematical passage
[X^6]f^3&=1575B^4,& [X^5]f^3&=4320A^2B,\\ [X^{13}]f^3&=36A,& [X^{12}]f^3&=315B^2.
X^{13}Use in source · 1 passage
Search aliases: X^13
Use in source · 05_frey.tex, line 38
A characteristic seven cancellation. In characteristic seven, the Cartier coefficient calculation comes from f^3. Its relevant coefficients are [X^6]f^3 · =1575B^4, · [X^5]f^3 · =4320A^2B, [X^13]f^3 · =36A, · [X^12]f^3 · =315B^2. The diagonal lanes vanish modulo seven and the off diagonal lanes become M=0 · A^2B
Exact mathematical passage
[X^6]f^3&=1575B^4,& [X^5]f^3&=4320A^2B,\\ [X^{13}]f^3&=36A,& [X^{12}]f^3&=315B^2.
X^2Use in source · 4 passages
Search aliases: X^2
Use in source · 05_frey.tex, line 31
At a variable prime dividing C, primitivity makes A,B units. In the residue field put u=-B^2/A. Then u^3=B, u^5=-A, and f_A,B(uX)=u^6(X^2+X-1)^2(X^2-2X+5). The discriminants of the two quadratics are 5 and -16, and their resultant is 45. Away from 2,3,5, the special fiber has exactly two ordinary nodes, rational normalization, and smooth infinity. This factorization is the local geometric entry point for the period and monodromy calculations.
Exact mathematical passage
\label{eq:nodalfactor} f_{A,B}(uX)=u^6(X^2+X-1)^2(X^2-2X+5).
x^2Use in source · 2 passages
Search aliases: x^2
Use in source · 04_local_strata.tex, line 54
There are (p+1)/4 elements t for which both t and 1-t are nonsquares. Here is a direct count. Extend the quadratic character χ_p by zero. Counting (x-y)(x+y)=d for d≠0 gives ∑_xχ_p(x^2-d)=-1. Completing the square then gives ∑_tχ_p(t(1-t))=-χ_p(-1)=1. After omitting 0,1, both single character sums are -1. Expanding the indicator (1-χ_p(t))(1-χ_p(1-t))/4 over the remaining points gives (p+1)/4>0.
Exact mathematical passage
\sum_x\chi_p(x^2-d)=-1.
X^3Use in source · 5 passages
Search aliases: X^3
Use in source · 05_frey.tex, line 3
A curve that retains the exponent seven. The signature route uses the integral curve y^2+(X^3+B)y=2BX^3+3AX+B^2. Completing the square with Z=2y+X^3+B gives Z^2=f_A,B(X)=X^6+10BX^3+12AX+5B^2. This is the genus two family used in the recovered source. The comparison with the family of Pacetti and Villagra Torcomian is a rational change of coordinates, with their variables translated as (a,b,c,p)=(B,-C,A,7) [source reference]. Explicitly, put t=-B^5/A^3, d=-B^2/A, and substitute x_PV=d/X, y_PV=tZ/…
Exact mathematical passage
\label{eq:integralcurve} y^2+(X^3+B)y=2BX^3+3AX+B^2.
x^3Use in source · 3 passages
Search aliases: x^3
Use in source · 05_frey.tex, line 10
12AX+5B^2. This is the genus two family used in the recovered source. The comparison with the family of Pacetti and Villagra Torcomian is a rational change of coordinates, with their variables translated as (a,b,c,p)=(B,-C,A,7) [source reference]. Explicitly, put t=-B^5/A^3, d=-B^2/A, and substitute x_PV=d/X, y_PV=tZ/X^3 into their equation y^2=5x^6-12x^5+10tx^3+t^2. Multiplication by X^6/t^2 gives [source reference]. The map is rational for AB≠0. It does not on its own verify a twist, integral …
Exact mathematical passage
y^2=5x^6-12x^5+10tx^3+t^2
X^4Use in source · 1 passage
Search aliases: X^4
Use in source · 10_reciprocity.tex, line 39
If the actual polynomial is P(X)=X^4-SX^3+BX^2-QSX+Q^2, the correction u=(τ-Q-1)/7 has Tr(u)=fracS-2(Q+1)7, Nm(u)=frac(B-2Q)-(Q+1)S+(Q+1)^249 modulo seven, under the stated divisibility assumptions. The two implementations extract these values using different methods, including an integer trace discriminant and an enumeration of k by trace and norm. Neither assigns a Frobenius value at a beta conductor prime by pretending beta is unramified there. Such conductor values require the separate self …
Exact mathematical passage
P(X)=X^4-SX^3+BX^2-QSX+Q^2,
X^5Use in source · 1 passage
Search aliases: X^5
Use in source · 05_frey.tex, line 38
A characteristic seven cancellation. In characteristic seven, the Cartier coefficient calculation comes from f^3. Its relevant coefficients are [X^6]f^3 · =1575B^4, · [X^5]f^3 · =4320A^2B, [X^13]f^3 · =36A, · [X^12]f^3 · =315B^2. The diagonal lanes vanish modulo seven and the off diagonal lanes become M=0 · A^2B
Exact mathematical passage
[X^6]f^3&=1575B^4,& [X^5]f^3&=4320A^2B,\\ [X^{13}]f^3&=36A,& [X^{12}]f^3&=315B^2.
x^5Use in source · 2 passages
Search aliases: x^5
Use in source · 05_frey.tex, line 10
12AX+5B^2. This is the genus two family used in the recovered source. The comparison with the family of Pacetti and Villagra Torcomian is a rational change of coordinates, with their variables translated as (a,b,c,p)=(B,-C,A,7) [source reference]. Explicitly, put t=-B^5/A^3, d=-B^2/A, and substitute x_PV=d/X, y_PV=tZ/X^3 into their equation y^2=5x^6-12x^5+10tx^3+t^2. Multiplication by X^6/t^2 gives [source reference]. The map is rational for AB≠0. It does not on its own verify a twist, integral …
Exact mathematical passage
y^2=5x^6-12x^5+10tx^3+t^2
X^6Use in source · 3 passages
Search aliases: X^6
Use in source · 05_frey.tex, line 7
A curve that retains the exponent seven. The signature route uses the integral curve y^2+(X^3+B)y=2BX^3+3AX+B^2. Completing the square with Z=2y+X^3+B gives Z^2=f_A,B(X)=X^6+10BX^3+12AX+5B^2. This is the genus two family used in the recovered source. The comparison with the family of Pacetti and Villagra Torcomian is a rational change of coordinates, with their variables translated as (a,b,c,p)=(B,-C,A,7) [source reference]. Explicitly, put t=-B^5/A^3, d=-B^2/A, and substitute x_PV=d/X, y_PV=tZ/…
Exact mathematical passage
\label{eq:sextic} Z^2=f_{A,B}(X)=X^6+10BX^3+12AX+5B^2.
x^6Use in source · 1 passage
Search aliases: x^6
Use in source · 05_frey.tex, line 10
12AX+5B^2. This is the genus two family used in the recovered source. The comparison with the family of Pacetti and Villagra Torcomian is a rational change of coordinates, with their variables translated as (a,b,c,p)=(B,-C,A,7) [source reference]. Explicitly, put t=-B^5/A^3, d=-B^2/A, and substitute x_PV=d/X, y_PV=tZ/X^3 into their equation y^2=5x^6-12x^5+10tx^3+t^2. Multiplication by X^6/t^2 gives [source reference]. The map is rational for AB≠0. It does not on its own verify a twist, integral …
Exact mathematical passage
y^2=5x^6-12x^5+10tx^3+t^2
x^pUse in source · 1 passage
Search aliases: x^p
Use in source · 24_sources.tex, line 21
0. DOI: https://doi.org/10.4171/009-1/2110.4171/009-1/21. bibitembilu2005 Y. F. Bilu, Catalan without logarithmic forms (after Bugeaud, Hanrot and Mihuailescu), J. Th eor. Nombres Bordeaux 17 (2005), 69--85. DOI: https://doi.org/10.5802/jtnb.47810.5802/jtnb.478. bibitemdarmon-granville1995 H. Darmon and A. Granville, On the equations z^m=F(x,y) and Ax^p+By^q=Cz^r, Bull. London Math. Soc. 27 (1995), 513--543. DOI: https://doi.org/10.1112/blms/27.6.51310.1112/blms/27.6.513. bibitemdahmen-siksek202…
Exact mathematical passage
Ax^p+By^q=Cz^r
x^qUse in source · 3 passages
Search aliases: x^q
Use in source · 10_reciprocity.tex, line 10
For degree two, the indices b and -b represent the same prime. Frobenius sends a residue x to x^q, and ℓ_q(x^q)=-ℓ_q(x). Therefore (-b)ℓ_q(x^q)=bℓ_q(x). The functional is well defined on prime ideals, not just on the six embeddings. Principal reciprocity gives, for a test prime generator π_l outside the conductor, R_l,q=-∑_b∈reps_qbℓ_q(π_l(z_q^b)) =-f_q^-1∑_b=1^6bℓ_q(π_l(z_q^b)), where f_q=1 or two. Using all six terms without dividing by two in the second case would double count each prime. Con…
Exact mathematical passage
x^q
x_iUse in source · 2 passages
Search aliases: x_i
Use in source · 20_localization.tex, line 29
An explicit image parametrization. Put a=z_7(3) and x_i=z(I_q_i). Let ℓ_i be the seventh power logarithm normalized by ℓ_i(g_q_i)=1. The supporting symbols of ϕ vanish. For rational seven adic units define ℓ_7(t)=fraclog(t^6)log(3^6)pmod7 =6fract^6-17pmod7. Principal reciprocity determines the remaining coordinates: z_7(ϕ) · =0,
Exact mathematical passage
x_i=z(I_{q_i})
X_jUse in source · 2 passages
Search aliases: X_j
Use in source · 12_fixed_seven.tex, line 37
Use the three coordinate basis X_j=[ϕ], H_j=[3], T_j=[7]. The third coordinate is valuation. The actual local scalar class in the window is κ_j=2ϵ wX_j+Λ_jH_j+7cT_j, where Λ_j=7ℓ_j(C_0)-3ℓ_j(A)-3-5d_j, ℓ_j(a)=fraclog(a^6)log(3^6)pmod7^j, d_j=fracℓ_j(5)2. At level 49, with Q_A=(A^6-1)/7 modulo 49 and q_0=(C_0^6-1)/7 modulo seven, Λ_2=47+38Q_A+14Q_A^2-7q_0pmod49. Both C_0 and the coordinate 7c[7] disappeared modulo seven. Neither may be omitted modulo forty nine.
Exact mathematical passage
X_j=[\phi],\qquad H_j=[3],\qquad T_j=[7].
x_{\mathrm{PV}}Use in source · 1 passage
Search aliases: x_PV mathrm
Use in source · 05_frey.tex, line 10
gives Z^2=f_A,B(X)=X^6+10BX^3+12AX+5B^2. This is the genus two family used in the recovered source. The comparison with the family of Pacetti and Villagra Torcomian is a rational change of coordinates, with their variables translated as (a,b,c,p)=(B,-C,A,7) [source reference]. Explicitly, put t=-B^5/A^3, d=-B^2/A, and substitute x_PV=d/X, y_PV=tZ/X^3 into their equation y^2=5x^6-12x^5+10tx^3+t^2. Multiplication by X^6/t^2 gives [source reference]. The map is rational for AB≠0. It does not on its…
Exact mathematical passage
x_{\mathrm{PV}}=d/X
x_{q_1}Use in source · 1 passage
Search aliases: x_q_1
Use in source · 22_affine.tex, line 26
bottomrule figplaceholder19_affine_dimensionsThe kernel measures ambiguity after an affine solution exists. The cokernel measures possible first stage incompatibility. Neither dimension evaluates the actual target. In the global parameter order (a,x_q_1,…,x_q_9), the kernel is exactly the span of the unit vectors at the listed minus congruence primes. After all toric rows are imposed, the plus congruence inertia differences vanish. This is a theorem about differences. It does not imply that the …
Exact mathematical passage
(a,x_{q_1},\ldots,x_{q_9})
x_{q_9}Use in source · 1 passage
Search aliases: x_q_9
Use in source · 22_affine.tex, line 26
bottomrule figplaceholder19_affine_dimensionsThe kernel measures ambiguity after an affine solution exists. The cokernel measures possible first stage incompatibility. Neither dimension evaluates the actual target. In the global parameter order (a,x_q_1,…,x_q_9), the kernel is exactly the span of the unit vectors at the listed minus congruence primes. After all toric rows are imposed, the plus congruence inertia differences vanish. This is a theorem about differences. It does not imply that the …
Exact mathematical passage
(a,x_{q_1},\ldots,x_{q_9})
YUse in source · 2 passages
Search aliases: Y
Use in source · 13_deformation.tex, line 48
Strict conjugation by I+7Y changes X by Y-Ad(overlineρ_0)Y. Independent local changes can simplify representatives but do not create a common global integral frame. The affine target in the last part of this reader is formulated after one frame and one reference have been fixed.
Exact mathematical passage
I+7Y
y^2Use in source · 4 passages
Search aliases: y^2
Use in source · 05_frey.tex, line 3
A curve that retains the exponent seven. The signature route uses the integral curve y^2+(X^3+B)y=2BX^3+3AX+B^2. Completing the square with Z=2y+X^3+B gives Z^2=f_A,B(X)=X^6+10BX^3+12AX+5B^2. This is the genus two family used in the recovered source. The comparison with the family of Pacetti and Villagra Torcomian is a rational change of coordinates, with their variables translated as (a,b,c,p)=(B,-C,A,7) [source reference]. Explicitly, put t=-B^5/A^3, d=-B^2/A, and substitute x_PV=d/X, y_PV=tZ/…
Exact mathematical passage
\label{eq:integralcurve} y^2+(X^3+B)y=2BX^3+3AX+B^2.
y^3Use in source · 1 passage
Search aliases: y^3
Use in source · 24_sources.tex, line 22
efhttps://doi.org/10.5802/jtnb.47810.5802/jtnb.478. bibitemdarmon-granville1995 H. Darmon and A. Granville, On the equations z^m=F(x,y) and Ax^p+By^q=Cz^r, Bull. London Math. Soc. 27 (1995), 513--543. DOI: https://doi.org/10.1112/blms/27.6.51310.1112/blms/27.6.513. bibitemdahmen-siksek2024 S. R. Dahmen and S. Siksek, On the generalized Fermat equation x^5+y^3=z^7, author working paper, July 2024. https://few.vu.nl/~sdn249/GFE357.pdf. bibitempacetti-villagra2025 A. Pacetti and L. Villagra Torcomi…
Exact mathematical passage
x^5+y^3=z^7
y^5Use in source · 2 passages
Search aliases: y^5
Use in source · 24_sources.tex, line 8
A current source check also found Chocian's preprint of 22 September 2026, which reports a computer assisted proof for the primitive equation x^3+y^5=z^7 [source reference]. Its existence and reported scope are recorded here. This release does not independently audit that claimed proof or import it as a theorem closing the recovered project route. Earlier project descriptions of the route as incomplete should therefore be read as descriptions of this project's evidence, not as an exhaustive asse…
Exact mathematical passage
x^3+y^5=z^7
y^qUse in source · 1 passage
Search aliases: y^q
Use in source · 24_sources.tex, line 21
0. DOI: https://doi.org/10.4171/009-1/2110.4171/009-1/21. bibitembilu2005 Y. F. Bilu, Catalan without logarithmic forms (after Bugeaud, Hanrot and Mihuailescu), J. Th eor. Nombres Bordeaux 17 (2005), 69--85. DOI: https://doi.org/10.5802/jtnb.47810.5802/jtnb.478. bibitemdarmon-granville1995 H. Darmon and A. Granville, On the equations z^m=F(x,y) and Ax^p+By^q=Cz^r, Bull. London Math. Soc. 27 (1995), 513--543. DOI: https://doi.org/10.1112/blms/27.6.51310.1112/blms/27.6.513. bibitemdahmen-siksek202…
Exact mathematical passage
Ax^p+By^q=Cz^r
y_{\mathrm{PV}}Use in source · 1 passage
Search aliases: y_PV mathrm
Use in source · 05_frey.tex, line 10
Z^2=f_A,B(X)=X^6+10BX^3+12AX+5B^2. This is the genus two family used in the recovered source. The comparison with the family of Pacetti and Villagra Torcomian is a rational change of coordinates, with their variables translated as (a,b,c,p)=(B,-C,A,7) [source reference]. Explicitly, put t=-B^5/A^3, d=-B^2/A, and substitute x_PV=d/X, y_PV=tZ/X^3 into their equation y^2=5x^6-12x^5+10tx^3+t^2. Multiplication by X^6/t^2 gives [source reference]. The map is rational for AB≠0. It does not on its own v…
Exact mathematical passage
y_{\mathrm{PV}}=tZ/X^3
\ZUse in source · 9 passages
Search aliases: Z
Use in source · 04_local_strata.tex, line 10
:strata Write a hypothetical primitive solution as A=au^2, B=bv^2, C=cw^2, where a,b,c are positive pairwise coprime squarefree integers. The equation becomes a^3u^6+b^5v^10=c^7w^14. At a prime p, a local solution is called primitive when at most one base is divisible by p. Roots and bases are required to be nonzero in Z_p. The roots may depend on p. This last freedom is exactly why local existence can be weaker than an integer solution.
Exact mathematical passage
\Z_p
Z^1Use in source · 3 passages
Search aliases: Z^1
Use in source · 16_secondary.tex, line 23
Every other first primitive is p+z with z∈ Z^1(K,k) an ordinary character. Formula [source reference] gives μ_p+z-μ_p=α∪ z-z∪α. Graded commutativity in odd characteristic makes its cohomology class 2[α]∪[z]. Therefore the choice independent object at this stage is the affine coset [μ_p]+2[α]∪ H^1(K,k). There exists some first primitive for which the upper equation is soluble precisely when zero belongs to this coset. There is not one canonical class independent of the primitive without taking th…
Exact mathematical passage
z\in Z^1(K,k)
Z^2Use in source · 1 passage
Search aliases: Z^2
Use in source · 05_frey.tex, line 7
A curve that retains the exponent seven. The signature route uses the integral curve y^2+(X^3+B)y=2BX^3+3AX+B^2. Completing the square with Z=2y+X^3+B gives Z^2=f_A,B(X)=X^6+10BX^3+12AX+5B^2. This is the genus two family used in the recovered source. The comparison with the family of Pacetti and Villagra Torcomian is a rational change of coordinates, with their variables translated as (a,b,c,p)=(B,-C,A,7) [source reference]. Explicitly, put t=-B^5/A^3, d=-B^2/A, and substitute x_PV=d/X, y_PV=tZ/…
Exact mathematical passage
\label{eq:sextic} Z^2=f_{A,B}(X)=X^6+10BX^3+12AX+5B^2.
z^7Use in source · 3 passages
Search aliases: z^7
Use in source · 24_sources.tex, line 8
A current source check also found Chocian's preprint of 22 September 2026, which reports a computer assisted proof for the primitive equation x^3+y^5=z^7 [source reference]. Its existence and reported scope are recorded here. This release does not independently audit that claimed proof or import it as a theorem closing the recovered project route. Earlier project descriptions of the route as incomplete should therefore be read as descriptions of this project's evidence, not as an exhaustive asse…
Exact mathematical passage
x^3+y^5=z^7
z^mUse in source · 1 passage
Search aliases: z^m
Use in source · 24_sources.tex, line 21
ty, 2005, 325--340. DOI: https://doi.org/10.4171/009-1/2110.4171/009-1/21. bibitembilu2005 Y. F. Bilu, Catalan without logarithmic forms (after Bugeaud, Hanrot and Mihuailescu), J. Th eor. Nombres Bordeaux 17 (2005), 69--85. DOI: https://doi.org/10.5802/jtnb.47810.5802/jtnb.478. bibitemdarmon-granville1995 H. Darmon and A. Granville, On the equations z^m=F(x,y) and Ax^p+By^q=Cz^r, Bull. London Math. Soc. 27 (1995), 513--543. DOI: https://doi.org/10.1112/blms/27.6.51310.1112/blms/27.6.513. bibite…
Exact mathematical passage
z^m=F(x,y)
z^rUse in source · 1 passage
Search aliases: z^r
Use in source · 24_sources.tex, line 21
0. DOI: https://doi.org/10.4171/009-1/2110.4171/009-1/21. bibitembilu2005 Y. F. Bilu, Catalan without logarithmic forms (after Bugeaud, Hanrot and Mihuailescu), J. Th eor. Nombres Bordeaux 17 (2005), 69--85. DOI: https://doi.org/10.5802/jtnb.47810.5802/jtnb.478. bibitemdarmon-granville1995 H. Darmon and A. Granville, On the equations z^m=F(x,y) and Ax^p+By^q=Cz^r, Bull. London Math. Soc. 27 (1995), 513--543. DOI: https://doi.org/10.1112/blms/27.6.51310.1112/blms/27.6.513. bibitemdahmen-siksek202…
Exact mathematical passage
Ax^p+By^q=Cz^r
z_*Use in source · 3 passages
Search aliases: z_*
Use in source · 22_affine.tex, line 77
[Affine fibers] If Stage P has one solution with parameter z_*, its complete solution set for these linear conditions is z_*+ker A. It has 49^3,49^3,49^2,49^3 choices in patterns A through D, respectively. If the target is not in the image, the set is empty. Az=t is equivalent to A(z-z_*)=0. The exact kernel dimensions give the counts over k. An additional common pointwise semilinear j, frame, and reference condition makes the relevant minus conductor coefficients lie in wF_7. Under that extra h…
Exact mathematical passage
z_*
z_0Use in source · 2 passages
Search aliases: z_0
Use in source · 22_affine.tex, line 68
Now replace the origin by p'_0=p_0+z_0,char, where its localization is Jz_0. At the fixed places, the target differences decrease by the selected coordinates of this character. At support, [source reference] decreases by c_qz_0,char(F_q)-b_qz_0,char(T_q). Therefore t(p'_0)=t(p_0)-Az_0. The quotient class and Nt are unchanged. The coordinate basis for N itself is optional, but its vanishing criterion and the quotient class are canonical. The result concerns the stated first stage. It does not ide…
Exact mathematical passage
Jz_0
z_{0,\mathrm{char}}Use in source · 2 passages
Search aliases: z_0,char mathrm
Use in source · 22_affine.tex, line 68
Now replace the origin by p'_0=p_0+z_0,char, where its localization is Jz_0. At the fixed places, the target differences decrease by the selected coordinates of this character. At support, [source reference] decreases by c_qz_0,char(F_q)-b_qz_0,char(T_q). Therefore t(p'_0)=t(p_0)-Az_0. The quotient class and Nt are unchanged. The coordinate basis for N itself is optional, but its vanishing criterion and the quotient class are canonical. The result concerns the stated first stage. It does not ide…
Exact mathematical passage
p'_0=p_0+z_{0,\mathrm{char}}
\Z_2Use in source · 1 passage
Search aliases: Z_2 Z
Use in source · 04_local_strata.tex, line 64
For sufficiency, fix the selected even root to be one when its coefficient is even, and two when the coefficient is odd. Fix one odd root to be one. After moving the terms and dividing by the remaining coefficient power, the required remaining root power is a unit congruent to one modulo eight. Such a unit is a square in Z_2, and every odd power map on Z_2^× is bijective. It consequently has the required sixth, tenth, or fourteenth root.
Exact mathematical passage
\Z_2
\Z_2^\timesUse in source · 1 passage
Search aliases: Z_2^× Z times
Use in source · 04_local_strata.tex, line 64
For sufficiency, fix the selected even root to be one when its coefficient is even, and two when the coefficient is odd. Fix one odd root to be one. After moving the terms and dividing by the remaining coefficient power, the required remaining root power is a unit congruent to one modulo eight. Such a unit is a square in Z_2, and every odd power map on Z_2^× is bijective. It consequently has the required sixth, tenth, or fourteenth root.
Exact mathematical passage
\Z_2^\times
\Z_7Use in source · 3 passages
Search aliases: Z_7 Z
Use in source · 07_natural_depth.tex, line 7
The natural lattice and the depth of an opposite extension. The reducible branch uses two different kinds of radical. The base field is K=Q(sqrt5), ϕ=(1+sqrt5)/2. The coefficient ring and field are Oloc=Z_7[w]/(w^2-5), k=F_7[w]/(w^2-5)=F_49. The base unit ϕ defines Kummer classes. The coefficient w acts on a rank two representation. Although both involve a square root of five, they have different roles and are never identified by an implicit residue substitution.
Exact mathematical passage
\Oloc=\Z_7[w]/(w^2-5),\qquad k=\F_7[w]/(w^2-5)=\F_{49}.
z_7Use in source · 2 passages
Search aliases: z_7
Use in source · 20_localization.tex, line 29
An explicit image parametrization. Put a=z_7(3) and x_i=z(I_q_i). Let ℓ_i be the seventh power logarithm normalized by ℓ_i(g_q_i)=1. The supporting symbols of ϕ vanish. For rational seven adic units define ℓ_7(t)=fraclog(t^6)log(3^6)pmod7 =6fract^6-17pmod7. Principal reciprocity determines the remaining coordinates: z_7(ϕ) · =0,
Exact mathematical passage
a=z_7(3)
z_{\mathrm{char}}Use in source · 2 passages
Search aliases: z_char mathrm
Use in source · 22_affine.tex, line 56
A candidate p=p_0+z_char meets the fixed local condition precisely when the selected local character coordinates equal those of p_v^fix-p_0|_v. At support, substitution in [source reference] gives c_qz_char(F_q)-b_qz_char(T_q)=t_q(p_0). Together, these fifteen equations are exactly Az=t(p_0). Thus Stage P is soluble if and only if t(p_0)∈im A.
Exact mathematical passage
p=p_0+z_{\mathrm{char}}
\Z_pUse in source · 2 passages
Search aliases: Z_p Z
Use in source · 04_local_strata.tex, line 10
:strata Write a hypothetical primitive solution as A=au^2, B=bv^2, C=cw^2, where a,b,c are positive pairwise coprime squarefree integers. The equation becomes a^3u^6+b^5v^10=c^7w^14. At a prime p, a local solution is called primitive when at most one base is divisible by p. Roots and bases are required to be nonzero in Z_p. The roots may depend on p. This last freedom is exactly why local existence can be weaker than an integer solution.
Exact mathematical passage
\Z_p
z_qUse in source · 1 passage
Search aliases: z_q
Use in source · 10_reciprocity.tex, line 8
In the descended all inert regime, let M=Q(ζ_7) and let γ∈ H^1(Q,F_7(χ^-1)) restrict to the relevant beta class. Over M, the restriction is additive and satisfies γ(δ_a gδ_a^-1)=a^-1γ(g). At a conductor prime q≡1pmod7, there are six degree one primes in M. At q≡-1pmod7, there are three degree two primes. If z_q is a chosen seventh root of unity and ℓ_q is the corresponding seventh power logarithm, the local functional at the embedding ζ_7↦ z_q^b is b s_qℓ_q.
Exact mathematical passage
z_q
z_q^bUse in source · 4 passages
Search aliases: z_q^b
Use in source · 10_reciprocity.tex, line 8
ant beta class. Over M, the restriction is additive and satisfies γ(δ_a gδ_a^-1)=a^-1γ(g). At a conductor prime q≡1pmod7, there are six degree one primes in M. At q≡-1pmod7, there are three degree two primes. If z_q is a chosen seventh root of unity and ℓ_q is the corresponding seventh power logarithm, the local functional at the embedding ζ_7↦ z_q^b is b s_qℓ_q.
Exact mathematical passage
\zeta_7\mapsto z_q^b
z_r^bUse in source · 1 passage
Search aliases: z_r^b
Use in source · 21_ramified_frobenius.tex, line 23
n q place, write π_q=q u_q with u_q a unit. The same norm identity gives ℓ_q(u_q)=-∑_b=2^6ℓ_q(π_q(z_q^b)). Principal reciprocity on π_q therefore supplies the own conductor column D_q,q=∑_b=2^6(1-b)ℓ_q(π_q(z_q^b)). For r≠ q, the column is D_q,r=-∑_b∈reps_rbℓ_r(π_q(z_r^b)). The degree two representative convention is the one proved in [source reference]. These formulas use the same arithmetic Frobenius principal idele convention as the ordinary localization rows.
Exact mathematical passage
\label{eq:othercolumn} D_{q,r}=-\sum_{b\in\mathrm{reps}_r}b\ell_r(\pi_q(z_r^b)).
|Use in source · 6 passages
Search aliases: |
Use in source · 08_global_coefficients.tex, line 33
al has integral coefficients and factorization P(T)=(T^2-τ T+Q)(T^2-σ(τ)T+Q) =T^4-ST^3+C_2T^2-QST+Q^2, where Oloc/Z_7 is unramified quadratic, σ is its nontrivial automorphism, all complex roots have absolute value sqrt Q, and τ≡ Q+1pmod7^rOloc. Then the integer D=S^2-4(C_2-2Q) satisfies 7^2rmid D and 0≤ D≤16Q. If 7^2r>16Q, then P(T)=(T^2-aT+Q)^2, a∈Z, |a|≤2sqrt Q, a≡ Q+1pmod7^r. Expansion gives S=τ+σ(τ), C_2-2Q=τσ(τ), and D=(τ-σ(τ))^2. The congruence is preserved by σ, so this difference is div…
Exact mathematical passage
P(T)=(T^2-aT+Q)^2, \quad a\in\Z,\quad |a|\le2\sqrt Q, \quad a\equiv Q+1\pmod{7^r}.
|_vUse in source · 4 passages
Search aliases: |_v
Use in source · 17_local_flexibility.tex, line 13
If p_0 is one global bare primitive and p_v are desired finite local primitives, their differences p_v-p_0|_v are ordinary characters. For trivial coefficients there are no nonzero coboundaries. Surjectivity supplies one global character matching every prescribed finite difference exactly. Then p=p_0+z meets those local equations, but may ramify at new primes.
Exact mathematical passage
p_v-p_0|_v
\pmUse in source · 9 passages
Search aliases: ± pm
Use in source · 06_modular_labels.tex, line 12
rictions were established under the stated modular and multiplicative prime comparisons. A rank two trace zero comparison at a prime above eleven forces 11mid A. At a prime above nineteen it forces 19mid B. In either case the full four dimensional reduction to check is (T^2+q)^2pmod7, which is stronger than zero four dimensional trace. At bad q, the multiplicative comparison gives ±(q+1), nonzero modulo seven for these two primes. At good q, the direct curve table selects the stated base zero st…
Exact mathematical passage
\pm(q+1)
\cdotUse in source · 5 passages
Search aliases: · cdot
Use in source · 02_signatures.tex, line 59
Why a generic radical ceiling fails. The exponent equation gives R<H^71/105 for the signature (3,5,7). It is tempting to reverse this relation by postulating H≤ R^105/71 for every primitive sum. That universal assertion is false. The exact equality 2+109·3^10=23^5=6 436 343 has radical 2·3·109·23=15 042, and (23^5)^71>15 042^105. Its logarithmic quality is about 1.629911684. It is not a (3,5,7) counterexample. It shows that an estimate strong enough to close the desired signature must retain its…
Exact mathematical passage
2+109\cdot3^{10}=23^5=6\,436\,343
\alpha_jUse in source · 1 passage
Search aliases: α_j alpha
Use in source · 12_fixed_seven.tex, line 56
At higher actual lattice depth, put m=lfloor n/2rfloor+1. The inherited diagonal theorem supplies an actual global upper extension α_j for 1≤ j≤ m. In the common range 1≤ j≤min(m,7c-1), its local coordinates satisfy U_j=2ϵ wg_j, V_j=Λ_jg_j, W_j=7cg_j for one unit g_j. The exact valuation matrix preserves the valuation identity at every legitimate triangular precision 1≤ j≤ m, even beyond the proved unit period window. Consequently v_7^trunc(W_j)=min(j,1+v_7(c)). At depth one, only the residual g…
Exact mathematical passage
\alpha_j
\gammaUse in source · 5 passages
Search aliases: γ gamma
Use in source · 04_local_strata.tex, line 45
Odd primes away from the coefficients. Let α,β,γ be the quadratic characters of a,b,c, and let e=(-1/p). A primitive residue solution with one zero base exists if β=γ, α=γ, α=eβ, respectively. The same cyclic group argument proves this, and the lifting construction replaces the zero root by a nonzero multiple of p.
Exact mathematical passage
\alpha,\beta,\gamma
\deltaUse in source · 2 passages
Search aliases: δ delta
Use in source · 05_frey.tex, line 55
The global unit step uses ϕ=(1+sqrt5)/2, whose residue has order sixteen in F_49^×. The local weight possibilities at seven are 0,1,7,8, up to a common reciprocity sign. Applying reciprocity to the eighth power of each constituent gives overlineϕ^8e=1 and rules out the mixed weights 1,7. The determinant norm relation δ^8=χ^2 then leaves one constituent unramified at seven, without assuming an unproved rank two determinant normalization in this argument. Its eighth power is everywhere unramified,…
Exact mathematical passage
\delta^8=\chi^2
\delta^8Use in source · 1 passage
Search aliases: δ^8 delta
Use in source · 05_frey.tex, line 55
The global unit step uses ϕ=(1+sqrt5)/2, whose residue has order sixteen in F_49^×. The local weight possibilities at seven are 0,1,7,8, up to a common reciprocity sign. Applying reciprocity to the eighth power of each constituent gives overlineϕ^8e=1 and rules out the mixed weights 1,7. The determinant norm relation δ^8=χ^2 then leaves one constituent unramified at seven, without assuming an unproved rank two determinant normalization in this argument. Its eighth power is everywhere unramified,…
Exact mathematical passage
\delta^8=\chi^2
\delta_aUse in source · 1 passage
Search aliases: δ_a delta
Use in source · 10_reciprocity.tex, line 5
In the descended all inert regime, let M=Q(ζ_7) and let γ∈ H^1(Q,F_7(χ^-1)) restrict to the relevant beta class. Over M, the restriction is additive and satisfies γ(δ_a gδ_a^-1)=a^-1γ(g). At a conductor prime q≡1pmod7, there are six degree one primes in M. At q≡-1pmod7, there are three degree two primes. If z_q is a chosen seventh root of unity and ℓ_q is the corresponding seventh power logarithm, the local functional at the embedding ζ_7↦ z_q^b is b s_qℓ_q.
Exact mathematical passage
\gamma(\delta_a g\delta_a^{-1})=a^{-1}\gamma(g).
\delta_a^{-1}Use in source · 1 passage
Search aliases: δ_a^-1 delta
Use in source · 10_reciprocity.tex, line 5
In the descended all inert regime, let M=Q(ζ_7) and let γ∈ H^1(Q,F_7(χ^-1)) restrict to the relevant beta class. Over M, the restriction is additive and satisfies γ(δ_a gδ_a^-1)=a^-1γ(g). At a conductor prime q≡1pmod7, there are six degree one primes in M. At q≡-1pmod7, there are three degree two primes. If z_q is a chosen seventh root of unity and ℓ_q is the corresponding seventh power logarithm, the local functional at the embedding ζ_7↦ z_q^b is b s_qℓ_q.
Exact mathematical passage
\gamma(\delta_a g\delta_a^{-1})=a^{-1}\gamma(g).
\epsilonUse in source · 9 passages
Search aliases: ϵ epsilon
Use in source · 12_fixed_seven.tex, line 15
The actual coefficient action must intertwine the toric submodule and quotient period matrices. Their valuation matrix is exactly 7cI, so the two action matrices agree after cancellation over Z_7. Their reduction commutes with R and has characteristic polynomial T^2-5. The retained centralizer argument then gives one orientation ϵ∈{1,-1} and local scalar class κ_7=2ϵ wX+(3q_7(A)+5)H. Merely knowing R^2=5I would not identify this as the actual coefficient action. The period intertwiner is the ext…
Exact mathematical passage
\epsilon\in\{1,-1\}
\zetaUse in source · 9 passages
Search aliases: ζ zeta
Use in source · 07_natural_depth.tex, line 23
If no variable C prime had Q≡1pmod7, beta would restrict to an everywhere unramified additive class over L=K(ζ_7). Restriction is injective because [L:K]=6 is prime to seven. The inherited Masley class number result gives h(Q(ζ_35))=1 [source reference]. The field L is a subfield of index two. Extension and norm on ideal classes compose to multiplication by two, so the seven primary class group of L is zero. An everywhere unramified nonzero additive seven class is therefore impossible by class f…
Exact mathematical passage
L=K(\zeta_7)
\zeta_{35}Use in source · 1 passage
Search aliases: ζ_35 zeta
Use in source · 07_natural_depth.tex, line 23
If no variable C prime had Q≡1pmod7, beta would restrict to an everywhere unramified additive class over L=K(ζ_7). Restriction is injective because [L:K]=6 is prime to seven. The inherited Masley class number result gives h(Q(ζ_35))=1 [source reference]. The field L is a subfield of index two. Extension and norm on ideal classes compose to multiplication by two, so the seven primary class group of L is zero. An everywhere unramified nonzero additive seven class is therefore impossible by class f…
Exact mathematical passage
h(\Q(\zeta_{35}))=1
\zeta_7Use in source · 8 passages
Search aliases: ζ_7 zeta
Use in source · 07_natural_depth.tex, line 23
If no variable C prime had Q≡1pmod7, beta would restrict to an everywhere unramified additive class over L=K(ζ_7). Restriction is injective because [L:K]=6 is prime to seven. The inherited Masley class number result gives h(Q(ζ_35))=1 [source reference]. The field L is a subfield of index two. Extension and norm on ideal classes compose to multiplication by two, so the seven primary class group of L is zero. An everywhere unramified nonzero additive seven class is therefore impossible by class f…
Exact mathematical passage
L=K(\zeta_7)
\zeta_7^2Use in source · 2 passages
Search aliases: ζ_7^2 zeta
Use in source · 21_ramified_frobenius.tex, line 5
Let M=Q(ζ_7). For a rational conductor q≡1pmod7, choose a generator π_q of norm q for the selected degree one prime. The source gives exact coefficient vectors in 1,ζ_7,…,ζ_7^5 for q=197,337,547,673,743,883,1933. For example, π_197=ζ_7+2ζ_7^2+2ζ_7^4, and π_337=-1+ζ_7-ζ_7^2+3ζ_7^3-ζ_7^4+ζ_7^5. The norm equality identifies one prime ideal. It does not leave an unspecified product of several primes.
Exact mathematical passage
\pi_{197}=\zeta_7+2\zeta_7^2+2\zeta_7^4,
\zeta_7^3Use in source · 1 passage
Search aliases: ζ_7^3 zeta
Use in source · 21_ramified_frobenius.tex, line 9
Let M=Q(ζ_7). For a rational conductor q≡1pmod7, choose a generator π_q of norm q for the selected degree one prime. The source gives exact coefficient vectors in 1,ζ_7,…,ζ_7^5 for q=197,337,547,673,743,883,1933. For example, π_197=ζ_7+2ζ_7^2+2ζ_7^4, and π_337=-1+ζ_7-ζ_7^2+3ζ_7^3-ζ_7^4+ζ_7^5. The norm equality identifies one prime ideal. It does not leave an unspecified product of several primes.
Exact mathematical passage
\pi_{337}=-1+\zeta_7-\zeta_7^2+3\zeta_7^3-\zeta_7^4+\zeta_7^5.
\zeta_7^4Use in source · 2 passages
Search aliases: ζ_7^4 zeta
Use in source · 21_ramified_frobenius.tex, line 5
Let M=Q(ζ_7). For a rational conductor q≡1pmod7, choose a generator π_q of norm q for the selected degree one prime. The source gives exact coefficient vectors in 1,ζ_7,…,ζ_7^5 for q=197,337,547,673,743,883,1933. For example, π_197=ζ_7+2ζ_7^2+2ζ_7^4, and π_337=-1+ζ_7-ζ_7^2+3ζ_7^3-ζ_7^4+ζ_7^5. The norm equality identifies one prime ideal. It does not leave an unspecified product of several primes.
Exact mathematical passage
\pi_{197}=\zeta_7+2\zeta_7^2+2\zeta_7^4,
\zeta_7^5Use in source · 2 passages
Search aliases: ζ_7^5 zeta
Use in source · 21_ramified_frobenius.tex, line 4
Let M=Q(ζ_7). For a rational conductor q≡1pmod7, choose a generator π_q of norm q for the selected degree one prime. The source gives exact coefficient vectors in 1,ζ_7,…,ζ_7^5 for q=197,337,547,673,743,883,1933. For example, π_197=ζ_7+2ζ_7^2+2ζ_7^4, and π_337=-1+ζ_7-ζ_7^2+3ζ_7^3-ζ_7^4+ζ_7^5. The norm equality identifies one prime ideal. It does not leave an unspecified product of several primes.
Exact mathematical passage
1,\zeta_7,\ldots,\zeta_7^5
\etaUse in source · 3 passages
Search aliases: η eta
Use in source · 08_global_coefficients.tex, line 12
7^nc(g) · d(g). Modulo 7^n, the diagonal entries are characters. Determinant χ lets us write them as χη_n and η_n^-1. The natural monodromy restrictions and fixed local traces constrain the same η_n at every place. A locally chosen ray character with the right residue does not prove it occurs as this actual diagonal character.
Exact mathematical passage
\chi\eta_n
\eta_nUse in source · 2 passages
Search aliases: η_n eta
Use in source · 08_global_coefficients.tex, line 12
7^nc(g) · d(g). Modulo 7^n, the diagonal entries are characters. Determinant χ lets us write them as χη_n and η_n^-1. The natural monodromy restrictions and fixed local traces constrain the same η_n at every place. A locally chosen ray character with the right residue does not prove it occurs as this actual diagonal character.
Exact mathematical passage
\chi\eta_n
\eta_n^{-1}Use in source · 1 passage
Search aliases: η_n^-1 eta
Use in source · 08_global_coefficients.tex, line 12
7^nc(g) · d(g). Modulo 7^n, the diagonal entries are characters. Determinant χ lets us write them as χη_n and η_n^-1. The natural monodromy restrictions and fixed local traces constrain the same η_n at every place. A locally chosen ray character with the right residue does not prove it occurs as this actual diagonal character.
Exact mathematical passage
\eta_n^{-1}
\thetaUse in source · 4 passages
Search aliases: θ theta
Use in source · 17_local_flexibility.tex, line 23
The local toric hyperplane. At a beta support prime, use a pro seven tame generator T and arithmetic Frobenius F. Put c=β(T)≠0, b=β(F), h=(Q-1)/7pmod7. The local first digit normal form is M=cθ · -θ^2 1 · -θ, A=θ b · θ h-θ^2b b · h-θ b. Relative to the chosen reference, p(T)=θ c, p(F)=θ b-h. Eliminating θ gives the invariant affine equation cp(F)-bp(T)=-hc. Conversely any pair satisfying this equation determines θ=p(T)/c and reconstructs the displayed first order packet. The packet converse does…
Exact mathematical passage
M=c\begin{pmatrix}\theta&-\theta^2\\1&-\theta\end{pmatrix},\qquad A=\begin{pmatrix}\theta b&\theta h-\theta^2b\\b&h-\theta b\end{pmatrix}.
\theta^2Use in source · 1 passage
Search aliases: θ^2 theta
Use in source · 17_local_flexibility.tex, line 23
The local toric hyperplane. At a beta support prime, use a pro seven tame generator T and arithmetic Frobenius F. Put c=β(T)≠0, b=β(F), h=(Q-1)/7pmod7. The local first digit normal form is M=cθ · -θ^2 1 · -θ, A=θ b · θ h-θ^2b b · h-θ b. Relative to the chosen reference, p(T)=θ c, p(F)=θ b-h. Eliminating θ gives the invariant affine equation cp(F)-bp(T)=-hc. Conversely any pair satisfying this equation determines θ=p(T)/c and reconstructs the displayed first order packet. The packet converse does…
Exact mathematical passage
M=c\begin{pmatrix}\theta&-\theta^2\\1&-\theta\end{pmatrix},\qquad A=\begin{pmatrix}\theta b&\theta h-\theta^2b\\b&h-\theta b\end{pmatrix}.
\kappaUse in source · 4 passages
Search aliases: κ kappa
Use in source · 09_periods.tex, line 38
l. At residue characteristic different from seven, a unit is an Mth power precisely when its residue is, by simple root lifting. A general element also needs valuation divisible by M. If Q is the residue cardinality, μ_M⊂ F precisely when Mmid Q-1. Applying Proposition~[source reference] to [source reference] gives the complete local test 7^jmid Q-1,qquad7^jmid7c, ϕ∈(κ_v^×)^7^j, fracC_0^73^3A^3(sqrt5)^5∈(κ_v^×)^7^j. The valuation condition on c is not lost when replacing C by its unit part C_0.
Exact mathematical passage
\label{eq:full-local-torsion} \begin{gathered} 7^j\mid Q-1,\qquad7^j\mid7c,\\ \phi\in(\kappa_v^\times)^{7^j},\qquad \frac{C_0^7}{3^3A^3(\sqrt5)^5}\in(\kappa_v^\times)^{7^j}. \end{gathered}
\kappa_7Use in source · 1 passage
Search aliases: κ_7 kappa
Use in source · 12_fixed_seven.tex, line 16
he actual coefficient action must intertwine the toric submodule and quotient period matrices. Their valuation matrix is exactly 7cI, so the two action matrices agree after cancellation over Z_7. Their reduction commutes with R and has characteristic polynomial T^2-5. The retained centralizer argument then gives one orientation ϵ∈{1,-1} and local scalar class κ_7=2ϵ wX+(3q_7(A)+5)H. Merely knowing R^2=5I would not identify this as the actual coefficient action. The period intertwiner is the extr…
Exact mathematical passage
\label{eq:kappa7} \kappa_7=2\epsilon wX+(3q_7(A)+5)H.
\kappa_jUse in source · 1 passage
Search aliases: κ_j kappa
Use in source · 12_fixed_seven.tex, line 41
Use the three coordinate basis X_j=[ϕ], H_j=[3], T_j=[7]. The third coordinate is valuation. The actual local scalar class in the window is κ_j=2ϵ wX_j+Λ_jH_j+7cT_j, where Λ_j=7ℓ_j(C_0)-3ℓ_j(A)-3-5d_j, ℓ_j(a)=fraclog(a^6)log(3^6)pmod7^j, d_j=fracℓ_j(5)2. At level 49, with Q_A=(A^6-1)/7 modulo 49 and q_0=(C_0^6-1)/7 modulo seven, Λ_2=47+38Q_A+14Q_A^2-7q_0pmod49. Both C_0 and the coordinate 7c[7] disappeared modulo seven. Neither may be omitted modulo forty nine.
Exact mathematical passage
\label{eq:higherkappa} \kappa_j=2\epsilon wX_j+\Lambda_jH_j+7cT_j,
\kappa_v^\timesUse in source · 2 passages
Search aliases: κ_v^× kappa times
Use in source · 09_periods.tex, line 38
l. At residue characteristic different from seven, a unit is an Mth power precisely when its residue is, by simple root lifting. A general element also needs valuation divisible by M. If Q is the residue cardinality, μ_M⊂ F precisely when Mmid Q-1. Applying Proposition~[source reference] to [source reference] gives the complete local test 7^jmid Q-1,qquad7^jmid7c, ϕ∈(κ_v^×)^7^j, fracC_0^73^3A^3(sqrt5)^5∈(κ_v^×)^7^j. The valuation condition on c is not lost when replacing C by its unit part C_0.
Exact mathematical passage
\label{eq:full-local-torsion} \begin{gathered} 7^j\mid Q-1,\qquad7^j\mid7c,\\ \phi\in(\kappa_v^\times)^{7^j},\qquad \frac{C_0^7}{3^3A^3(\sqrt5)^5}\in(\kappa_v^\times)^{7^j}. \end{gathered}
\LambdaUse in source · 10 passages
Search aliases: Λ Lambda
Use in source · 07_natural_depth.tex, line 43
Define n=min_g:ρ(g)_21≠0v_7(ρ(g)_21). This positive integer is attained. It is an invariant of the fixed natural frame, not a truncation selected for a computation. Every Oloc lattice stable under ρ is preserved by the two projectors, so up to homothety it has the form Λ_j=Oloc e_1+Oloc 7^j e_2. The unit upper monodromy forces j≥0. The attained lower entry minimum forces j≤ n. Conversely every j in that interval is stable. Thus the stable lattices form a finite segment with the natural and oppos…
Exact mathematical passage
\Lambda_j=\Oloc e_1+\Oloc 7^j e_2.
\lambdaUse in source · 21 passages
Search aliases: λ lambda
Use in source · FORMAL_STATUS.md, line 118
β(g)=(χ(g)^-1-1)λ, p^fix(g)=λχ(g)^-1α(g).
Exact mathematical passage
\beta(g)=(\chi(g)^{-1}-1)\lambda, \qquad p^{\mathrm{fix}}(g)=\lambda\chi(g)^{-1}\alpha(g).
\lambda^{105}Use in source · 1 passage
Search aliases: λ^105 lambda
Use in source · 03_scaling.tex, line 15
=C_0^7. Weighted scaling (A,B,C)=(λ^35A_0,λ^21B_0,λ^15C_0) preserves this equality. The two terms on the left are t^36s^84 and t^35s^85. Their sum is t^35s^84(t+s)=t^35s^84, which is C_0^7. The three scaling weights multiply the powered terms by the same factor λ^105. Over a field, if t and 1-t are nonzero, all three bases are nonzero and A_0^3/C_0^7=t. Thus every regular parameter is locally populated. This works modulo arbitrary prime powers whenever t and 1-t are units. A finite field test th…
Exact mathematical passage
\lambda^{105}
\lambda^{15}Use in source · 1 passage
Search aliases: λ^15 lambda
Use in source · 03_scaling.tex, line 9
For the signature (3,5,7), rational points are easy to manufacture. Primitive integer points are not. The following polynomial identity makes the difference exact. [Universal power identity] In any commutative ring, put s=1-t and A_0=t^12s^28, B_0=t^7s^17, C_0=t^5s^12. Then A_0^3+B_0^5=C_0^7. Weighted scaling (A,B,C)=(λ^35A_0,λ^21B_0,λ^15C_0) preserves this equality. The two terms on the left are t^36s^84 and t^35s^85. Their sum is t^35s^84(t+s)=t^35s^84, which is C_0^7. The three scaling weight…
Exact mathematical passage
\label{eq:weightedscale} (A,B,C)=(\lambda^{35}A_0,\lambda^{21}B_0,\lambda^{15}C_0)
\lambda^{21}Use in source · 1 passage
Search aliases: λ^21 lambda
Use in source · 03_scaling.tex, line 9
For the signature (3,5,7), rational points are easy to manufacture. Primitive integer points are not. The following polynomial identity makes the difference exact. [Universal power identity] In any commutative ring, put s=1-t and A_0=t^12s^28, B_0=t^7s^17, C_0=t^5s^12. Then A_0^3+B_0^5=C_0^7. Weighted scaling (A,B,C)=(λ^35A_0,λ^21B_0,λ^15C_0) preserves this equality. The two terms on the left are t^36s^84 and t^35s^85. Their sum is t^35s^84(t+s)=t^35s^84, which is C_0^7. The three scaling weight…
Exact mathematical passage
\label{eq:weightedscale} (A,B,C)=(\lambda^{35}A_0,\lambda^{21}B_0,\lambda^{15}C_0)
\lambda^3Use in source · 1 passage
Search aliases: λ^3 lambda
Use in source · 05_frey.tex, line 21
[Polynomial discriminant] The sextic in [source reference] satisfies disc(f_A,B)=2^123^65^5(A^3+B^5)^2. For a hypothetical (3,5,7) solution this becomes 2^123^65^5C^14. Give X,A,B weights 1,5,3. Then f_λ^5A,λ^3B(λ X)=λ^6f_A,B(X). The six roots scale by λ, so their fifteen squared differences give discriminant weight thirty. The only nonnegative integer solutions of 5i+3j=30 are (i,j)=(6,0),(3,5),(0,10). The discriminant therefore has the form uA^6+vA^3B^5+wB^10.
Exact mathematical passage
f_{\lambda^5A,\lambda^3B}(\lambda X)=\lambda^6f_{A,B}(X).
\lambda^{35}Use in source · 1 passage
Search aliases: λ^35 lambda
Use in source · 03_scaling.tex, line 9
For the signature (3,5,7), rational points are easy to manufacture. Primitive integer points are not. The following polynomial identity makes the difference exact. [Universal power identity] In any commutative ring, put s=1-t and A_0=t^12s^28, B_0=t^7s^17, C_0=t^5s^12. Then A_0^3+B_0^5=C_0^7. Weighted scaling (A,B,C)=(λ^35A_0,λ^21B_0,λ^15C_0) preserves this equality. The two terms on the left are t^36s^84 and t^35s^85. Their sum is t^35s^84(t+s)=t^35s^84, which is C_0^7. The three scaling weight…
Exact mathematical passage
\label{eq:weightedscale} (A,B,C)=(\lambda^{35}A_0,\lambda^{21}B_0,\lambda^{15}C_0)
\lambda^5Use in source · 1 passage
Search aliases: λ^5 lambda
Use in source · 05_frey.tex, line 21
[Polynomial discriminant] The sextic in [source reference] satisfies disc(f_A,B)=2^123^65^5(A^3+B^5)^2. For a hypothetical (3,5,7) solution this becomes 2^123^65^5C^14. Give X,A,B weights 1,5,3. Then f_λ^5A,λ^3B(λ X)=λ^6f_A,B(X). The six roots scale by λ, so their fifteen squared differences give discriminant weight thirty. The only nonnegative integer solutions of 5i+3j=30 are (i,j)=(6,0),(3,5),(0,10). The discriminant therefore has the form uA^6+vA^3B^5+wB^10.
Exact mathematical passage
f_{\lambda^5A,\lambda^3B}(\lambda X)=\lambda^6f_{A,B}(X).
\lambda^6Use in source · 1 passage
Search aliases: λ^6 lambda
Use in source · 05_frey.tex, line 21
[Polynomial discriminant] The sextic in [source reference] satisfies disc(f_A,B)=2^123^65^5(A^3+B^5)^2. For a hypothetical (3,5,7) solution this becomes 2^123^65^5C^14. Give X,A,B weights 1,5,3. Then f_λ^5A,λ^3B(λ X)=λ^6f_A,B(X). The six roots scale by λ, so their fifteen squared differences give discriminant weight thirty. The only nonnegative integer solutions of 5i+3j=30 are (i,j)=(6,0),(3,5),(0,10). The discriminant therefore has the form uA^6+vA^3B^5+wB^10.
Exact mathematical passage
f_{\lambda^5A,\lambda^3B}(\lambda X)=\lambda^6f_{A,B}(X).
\Lambda_2Use in source · 1 passage
Search aliases: Λ_2 Lambda
Use in source · 12_fixed_seven.tex, line 51
te is valuation. The actual local scalar class in the window is κ_j=2ϵ wX_j+Λ_jH_j+7cT_j, where Λ_j=7ℓ_j(C_0)-3ℓ_j(A)-3-5d_j, ℓ_j(a)=fraclog(a^6)log(3^6)pmod7^j, d_j=fracℓ_j(5)2. At level 49, with Q_A=(A^6-1)/7 modulo 49 and q_0=(C_0^6-1)/7 modulo seven, Λ_2=47+38Q_A+14Q_A^2-7q_0pmod49. Both C_0 and the coordinate 7c[7] disappeared modulo seven. Neither may be omitted modulo forty nine.
Exact mathematical passage
\label{eq:level49} \Lambda_2=47+38Q_A+14Q_A^2-7q_0\pmod{49}.
\Lambda_jUse in source · 4 passages
Search aliases: Λ_j Lambda
Use in source · 07_natural_depth.tex, line 43
Define n=min_g:ρ(g)_21≠0v_7(ρ(g)_21). This positive integer is attained. It is an invariant of the fixed natural frame, not a truncation selected for a computation. Every Oloc lattice stable under ρ is preserved by the two projectors, so up to homothety it has the form Λ_j=Oloc e_1+Oloc 7^j e_2. The unit upper monodromy forces j≥0. The attained lower entry minimum forces j≤ n. Conversely every j in that interval is stable. Thus the stable lattices form a finite segment with the natural and oppos…
Exact mathematical passage
\Lambda_j=\Oloc e_1+\Oloc 7^j e_2.
\lambda_vUse in source · 5 passages
Search aliases: λ_v lambda
Use in source · 16_secondary.tex, line 47
At a fixed place where beta is the coboundary of λ_v and p=p_v^fix, the explicit local upper primitive q_v^fix(g)=-λ_vχ(g)^-1α(g)^2 satisfies d_χ q_v^fix=R_p by direct expansion. At support, α=0 also kills the local upper right side. These facts prevent treating a formal secondary obstruction as an established nonzero obstruction in the actual local branches. Full prescribed upper values and geometric compatibility remain separate.
Exact mathematical passage
\lambda_v
\mu_7Use in source · 6 passages
Search aliases: μ_7 mu
Use in source · 16_secondary.tex, line 45
The fixed primitive local to global theorem. For K=Q(sqrt5), localization on H^2(K,k(χ)) is injective. To see the coefficient translation, restrict to L=K(ζ_7), whose degree over K is six. Restriction followed by corestriction multiplies by six, which is invertible on seven primary cohomology. Over L, the module is two trivial F_7 copies and μ_7 is present. Kummer theory identifies each copy of H^2(L,F_7) with Br(L)[7]. Brauer--Hasse--Noether gives local to global injectivity for the Brauer grou…
Exact mathematical passage
\mu_7
\mu_MUse in source · 2 passages
Search aliases: μ_M mu
Use in source · 09_periods.tex, line 25
[Full torsion and period powers] For the stipulated split uniformization J^an= G_m^2/Λ with rational period lattice, full J[M] is F rational if and only if μ_M⊂ F and every coordinate of a basis of Λ is an Mth power in F. The uniformization gives 0⟶μ_M^2⟶ J[M] ⟶Λ/MΛ⟶0. If all torsion points are rational, so are μ_M^2. For λ∈Λ, choose an Mth root u in the torus. Rationality of its torsion class gives g(u)/u∈Λ for every g∈ G_F. Since λ is rational, this quotient also lies in μ_M^2. The lattice is …
Exact mathematical passage
\mu_M\subset F
\mu_M^2Use in source · 4 passages
Search aliases: μ_M^2 mu
Use in source · 09_periods.tex, line 29
[Full torsion and period powers] For the stipulated split uniformization J^an= G_m^2/Λ with rational period lattice, full J[M] is F rational if and only if μ_M⊂ F and every coordinate of a basis of Λ is an Mth power in F. The uniformization gives 0⟶μ_M^2⟶ J[M] ⟶Λ/MΛ⟶0. If all torsion points are rational, so are μ_M^2. For λ∈Λ, choose an Mth root u in the torus. Rationality of its torsion class gives g(u)/u∈Λ for every g∈ G_F. Since λ is rational, this quotient also lies in μ_M^2. The lattice is …
Exact mathematical passage
0\longrightarrow\mu_M^2\longrightarrow J[M] \longrightarrow\Lambda/M\Lambda\longrightarrow0.
\mu_{p+z}Use in source · 1 passage
Search aliases: μ_p+z mu
Use in source · 16_secondary.tex, line 24
Every other first primitive is p+z with z∈ Z^1(K,k) an ordinary character. Formula [source reference] gives μ_p+z-μ_p=α∪ z-z∪α. Graded commutativity in odd characteristic makes its cohomology class 2[α]∪[z]. Therefore the choice independent object at this stage is the affine coset [μ_p]+2[α]∪ H^1(K,k). There exists some first primitive for which the upper equation is soluble precisely when zero belongs to this coset. There is not one canonical class independent of the primitive without taking th…
Exact mathematical passage
\mu_{p+z}-\mu_p=\alpha\cup z-z\cup\alpha.
\nuUse in source · 3 passages
Search aliases: ν nu
Use in source · 14_integral_trace.tex, line 14
To define the valuation functional used here, strictly align the actual local toric cyclotomic line at K_7 with the reference line. The aligned local representative of X has p=β=0 and upper entry q a k(χ) Kummer cocycle. Define ν_7(X) to be its [7] valuation coordinate. Remaining aligned strict gauges change q by a coboundary and a scalar multiple of residual α, both of valuation zero. Thus the coordinate is well defined under these strict equivalences. The reference b_0 has valuation zero and t…
Exact mathematical passage
\nu_7(X)
\nu_7Use in source · 3 passages
Search aliases: ν_7 nu
Use in source · 14_integral_trace.tex, line 14
To define the valuation functional used here, strictly align the actual local toric cyclotomic line at K_7 with the reference line. The aligned local representative of X has p=β=0 and upper entry q a k(χ) Kummer cocycle. Define ν_7(X) to be its [7] valuation coordinate. Remaining aligned strict gauges change q by a coboundary and a scalar multiple of residual α, both of valuation zero. Thus the coordinate is well defined under these strict equivalences. The reference b_0 has valuation zero and t…
Exact mathematical passage
\nu_7(X)
\piUse in source · 16 passages
Search aliases: π pi
Use in source · 07_natural_depth.tex, line 51
A local depth theorem. Here is the matrix argument controlling where this opposite class can ramify. Let Oloc be a discrete valuation ring with uniformizer π, let F∈GL_2(Oloc), Q∈Oloc^× with Q≠1, and let n≥1. Suppose N∈ M_2(Oloc) is a nonzero nilpotent matrix of the form N=μ-xy · x^2
Exact mathematical passage
\pi
\pi^nUse in source · 1 passage
Search aliases: π^n pi
Use in source · 07_natural_depth.tex, line 56
-y^2 · xy, where μ∈Oloc∖{0}, (x,y)∈Oloc^2 is primitive, t=v(μ)≥0, and FN=QN, NF=N, F_21∈π^nOloc, v(N_21)=n. Set h=v(Q-1). Then n=t+2s, s=v(y),qquad0≤ s≤ h. The lower nilpotent entry is -μ y^2, so its valuation gives n=t+2s. The common image and kernel line of N is generated by (x,y)^T. The two identities for F show that F-I has image in this line and kills no extra quotient direction. Write F=I+x
Exact mathematical passage
FN=QN,\qquad NF=N,\qquad F_{21}\in\pi^n\Oloc,\qquad v(N_{21})=n.
\pi^{n-s}Use in source · 1 passage
Search aliases: π^n-s pi
Use in source · 07_natural_depth.tex, line 68
y(a,b), ax+by=Q-1. The lower entry is ya, so a∈π^n-sOloc=π^t+sOloc. Both ax and by are divisible by π^s. Their sum Q-1 is therefore divisible by π^s, giving s≤ h. The square on y explains the factor two. A bound n≤ t+h would omit the geometry of the rank one nilpotent and is false. The examples N=7^t-7^s · 1
Exact mathematical passage
a\in\pi^{n-s}\Oloc=\pi^{t+s}\Oloc
\pi^sUse in source · 2 passages
Search aliases: π^s pi
Use in source · 07_natural_depth.tex, line 68
y(a,b), ax+by=Q-1. The lower entry is ya, so a∈π^n-sOloc=π^t+sOloc. Both ax and by are divisible by π^s. Their sum Q-1 is therefore divisible by π^s, giving s≤ h. The square on y explains the factor two. A bound n≤ t+h would omit the geometry of the rank one nilpotent and is false. The examples N=7^t-7^s · 1
Exact mathematical passage
\pi^s
\pi^{t+s}Use in source · 1 passage
Search aliases: π^t+s pi
Use in source · 07_natural_depth.tex, line 68
y(a,b), ax+by=Q-1. The lower entry is ya, so a∈π^n-sOloc=π^t+sOloc. Both ax and by are divisible by π^s. Their sum Q-1 is therefore divisible by π^s, giving s≤ h. The square on y explains the factor two. A bound n≤ t+h would omit the geometry of the rank one nilpotent and is false. The examples N=7^t-7^s · 1
Exact mathematical passage
a\in\pi^{n-s}\Oloc=\pi^{t+s}\Oloc
\pi_{197}Use in source · 1 passage
Search aliases: π_197 pi
Use in source · 21_ramified_frobenius.tex, line 5
Let M=Q(ζ_7). For a rational conductor q≡1pmod7, choose a generator π_q of norm q for the selected degree one prime. The source gives exact coefficient vectors in 1,ζ_7,…,ζ_7^5 for q=197,337,547,673,743,883,1933. For example, π_197=ζ_7+2ζ_7^2+2ζ_7^4, and π_337=-1+ζ_7-ζ_7^2+3ζ_7^3-ζ_7^4+ζ_7^5. The norm equality identifies one prime ideal. It does not leave an unspecified product of several primes.
Exact mathematical passage
\pi_{197}=\zeta_7+2\zeta_7^2+2\zeta_7^4,
\pi_{337}Use in source · 1 passage
Search aliases: π_337 pi
Use in source · 21_ramified_frobenius.tex, line 9
Let M=Q(ζ_7). For a rational conductor q≡1pmod7, choose a generator π_q of norm q for the selected degree one prime. The source gives exact coefficient vectors in 1,ζ_7,…,ζ_7^5 for q=197,337,547,673,743,883,1933. For example, π_197=ζ_7+2ζ_7^2+2ζ_7^4, and π_337=-1+ζ_7-ζ_7^2+3ζ_7^3-ζ_7^4+ζ_7^5. The norm equality identifies one prime ideal. It does not leave an unspecified product of several primes.
Exact mathematical passage
\pi_{337}=-1+\zeta_7-\zeta_7^2+3\zeta_7^3-\zeta_7^4+\zeta_7^5.
\Pi_{aa}Use in source · 1 passage
Search aliases: Π_aa Pi
Use in source · 09_periods.tex, line 5
At a variable place away from 2,3,5,7, write C=q^cC_0 with C_0 a unit. In the marked two node basis, the actual leading period coordinates are Π_aa · =-fracepsϕ^2675sqrt5(1+ m), Π_bb · =fracepsϕ^-2675sqrt5(1+ m), Π_ab · =Π_ba=-ϕ^-4(1+ m), where eps=C^7/A^3 and the notation means that the quotient by the displayed leading term is a principal unit. The analytic uniformization and marked smoothing comparison are inherited written inputs, with precise sources in the literature report [source referen…
Exact mathematical passage
\Pi_{aa}&=-\frac{\eps\phi^2}{675\sqrt5}(1+\mathfrak m),\\ \Pi_{bb}&=\frac{\eps\phi^{-2}}{675\sqrt5}(1+\mathfrak m),\\ \Pi_{ab}&=\Pi_{ba}=-\phi^{-4}(1+\mathfrak m),
\Pi_{ab}Use in source · 1 passage
Search aliases: Π_ab Pi
Use in source · 09_periods.tex, line 5
At a variable place away from 2,3,5,7, write C=q^cC_0 with C_0 a unit. In the marked two node basis, the actual leading period coordinates are Π_aa · =-fracepsϕ^2675sqrt5(1+ m), Π_bb · =fracepsϕ^-2675sqrt5(1+ m), Π_ab · =Π_ba=-ϕ^-4(1+ m), where eps=C^7/A^3 and the notation means that the quotient by the displayed leading term is a principal unit. The analytic uniformization and marked smoothing comparison are inherited written inputs, with precise sources in the literature report [source referen…
Exact mathematical passage
\Pi_{aa}&=-\frac{\eps\phi^2}{675\sqrt5}(1+\mathfrak m),\\ \Pi_{bb}&=\frac{\eps\phi^{-2}}{675\sqrt5}(1+\mathfrak m),\\ \Pi_{ab}&=\Pi_{ba}=-\phi^{-4}(1+\mathfrak m),
\Pi_{ba}Use in source · 1 passage
Search aliases: Π_ba Pi
Use in source · 09_periods.tex, line 5
At a variable place away from 2,3,5,7, write C=q^cC_0 with C_0 a unit. In the marked two node basis, the actual leading period coordinates are Π_aa · =-fracepsϕ^2675sqrt5(1+ m), Π_bb · =fracepsϕ^-2675sqrt5(1+ m), Π_ab · =Π_ba=-ϕ^-4(1+ m), where eps=C^7/A^3 and the notation means that the quotient by the displayed leading term is a principal unit. The analytic uniformization and marked smoothing comparison are inherited written inputs, with precise sources in the literature report [source referen…
Exact mathematical passage
\Pi_{aa}&=-\frac{\eps\phi^2}{675\sqrt5}(1+\mathfrak m),\\ \Pi_{bb}&=\frac{\eps\phi^{-2}}{675\sqrt5}(1+\mathfrak m),\\ \Pi_{ab}&=\Pi_{ba}=-\phi^{-4}(1+\mathfrak m),
\Pi_{bb}Use in source · 1 passage
Search aliases: Π_bb Pi
Use in source · 09_periods.tex, line 5
At a variable place away from 2,3,5,7, write C=q^cC_0 with C_0 a unit. In the marked two node basis, the actual leading period coordinates are Π_aa · =-fracepsϕ^2675sqrt5(1+ m), Π_bb · =fracepsϕ^-2675sqrt5(1+ m), Π_ab · =Π_ba=-ϕ^-4(1+ m), where eps=C^7/A^3 and the notation means that the quotient by the displayed leading term is a principal unit. The analytic uniformization and marked smoothing comparison are inherited written inputs, with precise sources in the literature report [source referen…
Exact mathematical passage
\Pi_{aa}&=-\frac{\eps\phi^2}{675\sqrt5}(1+\mathfrak m),\\ \Pi_{bb}&=\frac{\eps\phi^{-2}}{675\sqrt5}(1+\mathfrak m),\\ \Pi_{ab}&=\Pi_{ba}=-\phi^{-4}(1+\mathfrak m),
\pi_lUse in source · 2 passages
Search aliases: π_l pi
Use in source · 10_reciprocity.tex, line 14
For degree two, the indices b and -b represent the same prime. Frobenius sends a residue x to x^q, and ℓ_q(x^q)=-ℓ_q(x). Therefore (-b)ℓ_q(x^q)=bℓ_q(x). The functional is well defined on prime ideals, not just on the six embeddings. Principal reciprocity gives, for a test prime generator π_l outside the conductor, R_l,q=-∑_b∈reps_qbℓ_q(π_l(z_q^b)) =-f_q^-1∑_b=1^6bℓ_q(π_l(z_q^b)), where f_q=1 or two. Using all six terms without dividing by two in the second case would double count each prime. Con…
Exact mathematical passage
\pi_l
\pi_qUse in source · 7 passages
Search aliases: π_q pi
Use in source · 21_ramified_frobenius.tex, line 4
Let M=Q(ζ_7). For a rational conductor q≡1pmod7, choose a generator π_q of norm q for the selected degree one prime. The source gives exact coefficient vectors in 1,ζ_7,…,ζ_7^5 for q=197,337,547,673,743,883,1933. For example, π_197=ζ_7+2ζ_7^2+2ζ_7^4, and π_337=-1+ζ_7-ζ_7^2+3ζ_7^3-ζ_7^4+ζ_7^5. The norm equality identifies one prime ideal. It does not leave an unspecified product of several primes.
Exact mathematical passage
\pi_q
\rho_0Use in source · 2 passages
Search aliases: ρ_0 rho
Use in source · 13_deformation.tex, line 3
The coupled adjoint equations. Choose one reference Kummer representation over Oloc/49, ρ_0(g)=χ(g) · b_0(g) 0 · 1, whose reduction has the same residual α as the actual natural representation. The reference is built by lifting the coefficients of the same global classes [ϕ],[3],[sqrt5]. It has no seven valuation coordinate at the fixed place. It is an auxiliary representation, not the actual one.
Exact mathematical passage
\label{eq:rho0} \rho_0(g)=\begin{pmatrix}\chi(g)&b_0(g)\\0&1\end{pmatrix},
\SigmaUse in source · 15 passages
Search aliases: Σ Sigma
Use in source · 19_bare_primitive.tex, line 2
A bare primitive on the justified ramification set. Fix one of the nine prime beta supports S in Section~[source reference]. Let Σ consist of S, the places above 2,3,5,7, and the two real places. The residual classes factor through Gsig. At every beta support place, χ is trivial modulo seven and the natural local class is zero. Hence its normalized cocycle is identically zero there. At the fixed places, beta is a local coboundary under the inherited contract.
Exact mathematical passage
\Sigma
\sigmaUse in source · 9 passages
Search aliases: σ sigma
Use in source · 08_global_coefficients.tex, line 24
[Discriminant collapse] Suppose a good Frobenius polynomial has integral coefficients and factorization P(T)=(T^2-τ T+Q)(T^2-σ(τ)T+Q) =T^4-ST^3+C_2T^2-QST+Q^2, where Oloc/Z_7 is unramified quadratic, σ is its nontrivial automorphism, all complex roots have absolute value sqrt Q, and τ≡ Q+1pmod7^rOloc. Then the integer D=S^2-4(C_2-2Q) satisfies 7^2rmid D and 0≤ D≤16Q. If 7^2r>16Q, then P(T)=(T^2-aT+Q)^2, a∈Z, |a|≤2sqrt Q, a≡ Q+1pmod7^r. Expansion gives S=τ+σ(τ), C_2-2Q=τσ(τ), and D=(τ-σ(τ))^2. Th…
Exact mathematical passage
P(T)=(T^2-\tau T+Q)(T^2-\sigma(\tau)T+Q) =T^4-ST^3+C_2T^2-QST+Q^2,
\tauUse in source · 16 passages
Search aliases: τ tau
Use in source · 08_global_coefficients.tex, line 17
At good auxiliary primes, the actual rank two trace obeys a precision condition of the form τ≡ Q+1pmod7^rOloc. The exponent r is supplied by the depth and local diagonal estimates. It is not inferred merely from a rank four congruence. This stronger comparison yields a particularly useful universal lemma.
Exact mathematical passage
\tau\equiv Q+1\pmod{7^r\Oloc}.
\tau_jUse in source · 4 passages
Search aliases: τ_j tau
Use in source · 14_integral_trace.tex, line 12
The local Kummer basis [ϕ],[3],[7] supplies a unit annihilating probe τ_j after adjoining the required 7^j roots of unity. It kills the unit coordinates and evaluates the uniformizer coordinate to one. The inherited construction uses the cyclotomic action to show restriction remains injective. At higher cyclotomic levels this is not a blanket prime to seven degree argument, because the extension also has a seven primary part.
Exact mathematical passage
\tau_j
\PhiUse in source · 1 passage
Search aliases: Φ Phi
Use in source · 21_ramified_frobenius.tex, line 28
The direct independent check lifts the chosen root of Φ_7 modulo q^2 by Hensel, then computes π_q(widetilde z_q)/q modulo q. It verifies the self unit rather than assigning its missing value from the norm relation. The norm calculation and Hensel calculation share the input generator but use different arithmetic routes.
Exact mathematical passage
\Phi_7
\phi^{-2}Use in source · 1 passage
Search aliases: ϕ^-2 phi
Use in source · 09_periods.tex, line 5
At a variable place away from 2,3,5,7, write C=q^cC_0 with C_0 a unit. In the marked two node basis, the actual leading period coordinates are Π_aa · =-fracepsϕ^2675sqrt5(1+ m), Π_bb · =fracepsϕ^-2675sqrt5(1+ m), Π_ab · =Π_ba=-ϕ^-4(1+ m), where eps=C^7/A^3 and the notation means that the quotient by the displayed leading term is a principal unit. The analytic uniformization and marked smoothing comparison are inherited written inputs, with precise sources in the literature report [source referen…
Exact mathematical passage
\Pi_{aa}&=-\frac{\eps\phi^2}{675\sqrt5}(1+\mathfrak m),\\ \Pi_{bb}&=\frac{\eps\phi^{-2}}{675\sqrt5}(1+\mathfrak m),\\ \Pi_{ab}&=\Pi_{ba}=-\phi^{-4}(1+\mathfrak m),
\phi^{-4}Use in source · 1 passage
Search aliases: ϕ^-4 phi
Use in source · 09_periods.tex, line 5
At a variable place away from 2,3,5,7, write C=q^cC_0 with C_0 a unit. In the marked two node basis, the actual leading period coordinates are Π_aa · =-fracepsϕ^2675sqrt5(1+ m), Π_bb · =fracepsϕ^-2675sqrt5(1+ m), Π_ab · =Π_ba=-ϕ^-4(1+ m), where eps=C^7/A^3 and the notation means that the quotient by the displayed leading term is a principal unit. The analytic uniformization and marked smoothing comparison are inherited written inputs, with precise sources in the literature report [source referen…
Exact mathematical passage
\Pi_{aa}&=-\frac{\eps\phi^2}{675\sqrt5}(1+\mathfrak m),\\ \Pi_{bb}&=\frac{\eps\phi^{-2}}{675\sqrt5}(1+\mathfrak m),\\ \Pi_{ab}&=\Pi_{ba}=-\phi^{-4}(1+\mathfrak m),
\phi^2Use in source · 1 passage
Search aliases: ϕ^2 phi
Use in source · 09_periods.tex, line 5
At a variable place away from 2,3,5,7, write C=q^cC_0 with C_0 a unit. In the marked two node basis, the actual leading period coordinates are Π_aa · =-fracepsϕ^2675sqrt5(1+ m), Π_bb · =fracepsϕ^-2675sqrt5(1+ m), Π_ab · =Π_ba=-ϕ^-4(1+ m), where eps=C^7/A^3 and the notation means that the quotient by the displayed leading term is a principal unit. The analytic uniformization and marked smoothing comparison are inherited written inputs, with precise sources in the literature report [source referen…
Exact mathematical passage
\Pi_{aa}&=-\frac{\eps\phi^2}{675\sqrt5}(1+\mathfrak m),\\ \Pi_{bb}&=\frac{\eps\phi^{-2}}{675\sqrt5}(1+\mathfrak m),\\ \Pi_{ab}&=\Pi_{ba}=-\phi^{-4}(1+\mathfrak m),
\phi^4Use in source · 1 passage
Search aliases: ϕ^4 phi
Use in source · 05_frey.tex, line 57
When C is even and seven does not divide C, the two adic torus calculation replaces good reduction at two. The global character has Frobenius value -1 at two. Reciprocity on ϕ^4 and 2/7 transports that value to seven. The finite flat comparison at good seven, together with [source reference], gives the value A^3B. Thus A^3B=-1 modulo seven. Since both bases are nonzero, B=-A^3 and A^3+B^5=0 modulo seven, contradicting 7nmid C.
Exact mathematical passage
\phi^4
\phi^{48}Use in source · 1 passage
Search aliases: ϕ^48 phi
Use in source · 19_bare_primitive.tex, line 28
At F=Q_7(sqrt5), exact arithmetic modulo 49 gives fracϕ^48-17=5overlinesqrt5≠0 in F_49. If a unit is a seventh power, its forty eighth power is one modulo forty nine. Indeed, any unit has forty eighth power in 1+7Oloc_F, and the seventh power of this group is contained in 1+49Oloc_F. Formula [source reference] shows that ϕ is not a seventh power in F.
Exact mathematical passage
\label{eq:phi-nonseventh} \frac{\phi^{48}-1}{7}=5\overline{\sqrt5}\ne0\quad\text{in }\F_{49}.
\phi^eUse in source · 3 passages
Search aliases: ϕ^e phi
Use in source · 19_bare_primitive.tex, line 23
Its units are {±ϕ^e:e∈Z}. For an elementary verification, scale a unit by a power of ϕ and by a sign so that its first real embedding lies in [1,ϕ). If its value a were strictly between the endpoints, a norm minus one unit would have integral trace a-a^-1 strictly between zero and one. A norm plus one unit would have trace a+a^-1 strictly between two and sqrt5<3. Both are impossible. The normalized unit is therefore one, proving the assertion.
Exact mathematical passage
\{\pm\phi^e:e\in\Z\}
\Phi_7Use in source · 1 passage
Search aliases: Φ_7 Phi
Use in source · 21_ramified_frobenius.tex, line 28
The direct independent check lifts the chosen root of Φ_7 modulo q^2 by Hensel, then computes π_q(widetilde z_q)/q modulo q. It verifies the self unit rather than assigning its missing value from the norm relation. The norm calculation and Hensel calculation share the input generator but use different arithmetic routes.
Exact mathematical passage
\Phi_7
\chi^{-1}Use in source · 5 passages
Search aliases: χ^-1 chi
Use in source · 07_natural_depth.tex, line 16
The inherited characteristic zero representation is the specified RM factor of the twisted Jacobian on the retained 3990mid C branch, with determinant the cyclotomic character χ. Its natural residual orientation is 0⟶ k(χ)⟶overline V⟶ k⟶0. The opposite endpoint has a nonzero class β∈ H^1(K,k(χ^-1)). A stable lattice change can alter residual inertia. Therefore properties of the natural Jacobian lattice cannot automatically be transferred to an arbitrary lattice supplied by Ribet's lemma [source …
Exact mathematical passage
\beta\in H^1(K,k(\chi^{-1}))
\chi^2Use in source · 1 passage
Search aliases: χ^2 chi
Use in source · 05_frey.tex, line 55
The global unit step uses ϕ=(1+sqrt5)/2, whose residue has order sixteen in F_49^×. The local weight possibilities at seven are 0,1,7,8, up to a common reciprocity sign. Applying reciprocity to the eighth power of each constituent gives overlineϕ^8e=1 and rules out the mixed weights 1,7. The determinant norm relation δ^8=χ^2 then leaves one constituent unramified at seven, without assuming an unproved rank two determinant normalization in this argument. Its eighth power is everywhere unramified,…
Exact mathematical passage
\delta^8=\chi^2
\chi_pUse in source · 4 passages
Search aliases: χ_p chi
Use in source · 04_local_strata.tex, line 53
There are (p+1)/4 elements t for which both t and 1-t are nonsquares. Here is a direct count. Extend the quadratic character χ_p by zero. Counting (x-y)(x+y)=d for d≠0 gives ∑_xχ_p(x^2-d)=-1. Completing the square then gives ∑_tχ_p(t(1-t))=-χ_p(-1)=1. After omitting 0,1, both single character sums are -1. Expanding the indicator (1-χ_p(t))(1-χ_p(1-t))/4 over the remaining points gives (p+1)/4>0.
Exact mathematical passage
\chi_p
\ldotsUse in source · 4 passages
Search aliases: … ldots
Use in source · 20_localization.tex, line 3
The complete ordinary localization map. Let z be an ordinary k valued character of Gsig. It is a difference of bare primitives, not beta and not either primitive itself. Local class field theory gives the following coordinate order: F_2,F_3,F_5,quad7:ϕ,3,7, I_q_1,F_q_1,…,I_q_9,F_q_9. We use the local Artin convention sending a selected uniformizer to arithmetic Frobenius. At five the uniformizer is sqrt5. At support primes it is the rational prime q. Every sign in the reciprocity rows and every …
Exact mathematical passage
\label{eq:localcoords} F_2,F_3,F_5,\quad7:\phi,3,7,\quad I_{q_1},F_{q_1},\ldots,I_{q_9},F_{q_9}.
\ellUse in source · 24 passages
Search aliases: ℓ ell
Use in source · 03_scaling.tex, line 22
[Primitive scaling criterion] Fix a rational 0<t<1. There is a positive rational λ for which [source reference] gives primitive positive integers if and only if, for every rational prime ℓ, v_ℓ(t)>0 · Longrightarrowquad3mid v_ℓ(t),
Exact mathematical passage
\ell
\ell^{h_\ell}Use in source · 1 passage
Search aliases: ℓ^h_ℓ ell ell
Use in source · 03_scaling.tex, line 45
An integral h exists in these four cases precisely under the stated divisibility conditions. Only finitely many primes occur in t or 1-t, so λ=∏_ℓℓ^h_ℓ is a well defined positive rational number. The three output supports are disjoint, proving integrality and primitivity. Every h_ℓ was forced, so two positive choices of λ would have ratio a positive rational with zero valuation at every prime. That ratio is one. figplaceholder03_valuation_packetsThe three nontrivial valuation sectors retain diff…
Exact mathematical passage
\lambda=\prod_\ell\ell^{h_\ell}
\ell_7Use in source · 2 passages
Search aliases: ℓ_7 ell
Use in source · 20_localization.tex, line 30
An explicit image parametrization. Put a=z_7(3) and x_i=z(I_q_i). Let ℓ_i be the seventh power logarithm normalized by ℓ_i(g_q_i)=1. The supporting symbols of ϕ vanish. For rational seven adic units define ℓ_7(t)=fraclog(t^6)log(3^6)pmod7 =6fract^6-17pmod7. Principal reciprocity determines the remaining coordinates: z_7(ϕ) · =0,
Exact mathematical passage
\ell_7(t)=\frac{\log(t^6)}{\log(3^6)}\pmod7 =6\frac{t^6-1}{7}\pmod7.
\ell_iUse in source · 3 passages
Search aliases: ℓ_i ell
Use in source · 20_localization.tex, line 29
An explicit image parametrization. Put a=z_7(3) and x_i=z(I_q_i). Let ℓ_i be the seventh power logarithm normalized by ℓ_i(g_q_i)=1. The supporting symbols of ϕ vanish. For rational seven adic units define ℓ_7(t)=fraclog(t^6)log(3^6)pmod7 =6fract^6-17pmod7. Principal reciprocity determines the remaining coordinates: z_7(ϕ) · =0,
Exact mathematical passage
\ell_i
\ell_jUse in source · 1 passage
Search aliases: ℓ_j ell
Use in source · 12_fixed_seven.tex, line 45
Use the three coordinate basis X_j=[ϕ], H_j=[3], T_j=[7]. The third coordinate is valuation. The actual local scalar class in the window is κ_j=2ϵ wX_j+Λ_jH_j+7cT_j, where Λ_j=7ℓ_j(C_0)-3ℓ_j(A)-3-5d_j, ℓ_j(a)=fraclog(a^6)log(3^6)pmod7^j, d_j=fracℓ_j(5)2. At level 49, with Q_A=(A^6-1)/7 modulo 49 and q_0=(C_0^6-1)/7 modulo seven, Λ_2=47+38Q_A+14Q_A^2-7q_0pmod49. Both C_0 and the coordinate 7c[7] disappeared modulo seven. Neither may be omitted modulo forty nine.
Exact mathematical passage
\Lambda_j=7\ell_j(C_0)-3\ell_j(A)-3-5d_j, \quad \ell_j(a)=\frac{\log(a^6)}{\log(3^6)}\pmod{7^j}, \quad d_j=\frac{\ell_j(5)}2.
\ell_qUse in source · 9 passages
Search aliases: ℓ_q ell
Use in source · 10_reciprocity.tex, line 8
regime, let M=Q(ζ_7) and let γ∈ H^1(Q,F_7(χ^-1)) restrict to the relevant beta class. Over M, the restriction is additive and satisfies γ(δ_a gδ_a^-1)=a^-1γ(g). At a conductor prime q≡1pmod7, there are six degree one primes in M. At q≡-1pmod7, there are three degree two primes. If z_q is a chosen seventh root of unity and ℓ_q is the corresponding seventh power logarithm, the local functional at the embedding ζ_7↦ z_q^b is b s_qℓ_q.
Exact mathematical passage
\ell_q
\ell_rUse in source · 1 passage
Search aliases: ℓ_r ell
Use in source · 21_ramified_frobenius.tex, line 23
n q place, write π_q=q u_q with u_q a unit. The same norm identity gives ℓ_q(u_q)=-∑_b=2^6ℓ_q(π_q(z_q^b)). Principal reciprocity on π_q therefore supplies the own conductor column D_q,q=∑_b=2^6(1-b)ℓ_q(π_q(z_q^b)). For r≠ q, the column is D_q,r=-∑_b∈reps_rbℓ_r(π_q(z_r^b)). The degree two representative convention is the one proved in [source reference]. These formulas use the same arithmetic Frobenius principal idele convention as the ordinary localization rows.
Exact mathematical passage
\label{eq:othercolumn} D_{q,r}=-\sum_{b\in\mathrm{reps}_r}b\ell_r(\pi_q(z_r^b)).
\toUse in source · 2 passages
Search aliases: → to
Use in source · 19_bare_primitive.tex, line 38
[Conditional restricted bare primitive] Assume the stated residual and local properties for α∈ Z^1(Gsig,k(χ)) and β∈ Z^1(Gsig,k(χ^-1)). Then there is a normalized continuous cochain p_0:Gsig→ k satisfying dp_0=-α∪β. No auxiliary ramification outside Σ is needed. The Cartier dual of trivial F_7 is μ_7. The nondegenerate F_7 trace pairing on k identifies the dual of trivial k with k(χ). Lemma~[source reference] in two F_7 coordinates therefore gives Sha^1_Σ(K,k(χ))=0.
Exact mathematical passage
p_0:\Gsig\to k
\mapstoUse in source · 2 passages
Search aliases: ↦ mapsto
Use in source · 06_modular_labels.tex, line 6
The Block 47 audit identified this mechanism in the pinned Pacetti--Villagra Torcomian code. It also identified a parameter dictionary issue at eleven: the unlabelled multisets agreed, but individual labels matched after t↦ t^-1. A refinement attached to the wrong labels could incorrectly combine local constraints. The new reader therefore distinguishes direct point counts on [source reference] from imported hypergeometric parameter tables.
Exact mathematical passage
t\mapsto t^{-1}
\prodUse in source · 3 passages
Search aliases: ∏ prod
Use in source · 03_scaling.tex, line 45
An integral h exists in these four cases precisely under the stated divisibility conditions. Only finitely many primes occur in t or 1-t, so λ=∏_ℓℓ^h_ℓ is a well defined positive rational number. The three output supports are disjoint, proving integrality and primitivity. Every h_ℓ was forced, so two positive choices of λ would have ratio a positive rational with zero valuation at every prime. That ratio is one. figplaceholder03_valuation_packetsThe three nontrivial valuation sectors retain diff…
Exact mathematical passage
\lambda=\prod_\ell\ell^{h_\ell}
\prod_{v\in S}Use in source · 1 passage
Search aliases: ∏_v∈ S prod in
Use in source · 17_local_flexibility.tex, line 7
Consequently the unrestricted maps H^1(K,k)⟶∏_v∈ SH^1(K_v,k), H^1(K,k(χ))⟶∏_v∈ SH^1(K_v,k(χ)) are surjective for every finite set S under the cited localization theorem. The precise source formulation and its fixed additional finite set refinement are recorded in the literature audit. This statement does not keep the character unramified at every other place.
Exact mathematical passage
H^1(K,k)\longrightarrow\prod_{v\in S}H^1(K_v,k),\qquad H^1(K,k(\chi))\longrightarrow\prod_{v\in S}H^1(K_v,k(\chi))
\prod_{v\in\Sigma}Use in source · 1 passage
Search aliases: ∏_v∈Σ prod in Sigma
Use in source · 19_bare_primitive.tex, line 14
[The dual Kummer kernel] For every finite Σ containing the prime above seven and all real places, Sha^1_Σ(K,μ_7) =ker(H^1(Gsig,μ_7) ⟶∏_v∈ΣH^1(K_v,μ_7))=0. The discriminant of K is five. Minkowski's ideal bound is sqrt5/2<2, so every ideal class has a representative of norm one. Thus Cl(K)=1 [source reference].
Exact mathematical passage
\Sha^1_\Sigma(K,\mu_7) =\ker\left(H^1(\Gsig,\mu_7) \longrightarrow\prod_{v\in\Sigma}H^1(K_v,\mu_7)\right)=0.
\prod_\ellUse in source · 1 passage
Search aliases: ∏_ℓ prod ell
Use in source · 03_scaling.tex, line 45
An integral h exists in these four cases precisely under the stated divisibility conditions. Only finitely many primes occur in t or 1-t, so λ=∏_ℓℓ^h_ℓ is a well defined positive rational number. The three output supports are disjoint, proving integrality and primitivity. Every h_ℓ was forced, so two positive choices of λ would have ratio a positive rational with zero valuation at every prime. That ratio is one. figplaceholder03_valuation_packetsThe three nontrivial valuation sectors retain diff…
Exact mathematical passage
\lambda=\prod_\ell\ell^{h_\ell}
\sumUse in source · 8 passages
Search aliases: ∑ sum
Use in source · 04_local_strata.tex, line 54
There are (p+1)/4 elements t for which both t and 1-t are nonsquares. Here is a direct count. Extend the quadratic character χ_p by zero. Counting (x-y)(x+y)=d for d≠0 gives ∑_xχ_p(x^2-d)=-1. Completing the square then gives ∑_tχ_p(t(1-t))=-χ_p(-1)=1. After omitting 0,1, both single character sums are -1. Expanding the indicator (1-χ_p(t))(1-χ_p(1-t))/4 over the remaining points gives (p+1)/4>0.
Exact mathematical passage
\sum_x\chi_p(x^2-d)=-1.
\sum_{b=1}^6Use in source · 1 passage
Search aliases: ∑_b=1^6 sum
Use in source · 10_reciprocity.tex, line 15
For degree two, the indices b and -b represent the same prime. Frobenius sends a residue x to x^q, and ℓ_q(x^q)=-ℓ_q(x). Therefore (-b)ℓ_q(x^q)=bℓ_q(x). The functional is well defined on prime ideals, not just on the six embeddings. Principal reciprocity gives, for a test prime generator π_l outside the conductor, R_l,q=-∑_b∈reps_qbℓ_q(π_l(z_q^b)) =-f_q^-1∑_b=1^6bℓ_q(π_l(z_q^b)), where f_q=1 or two. Using all six terms without dividing by two in the second case would double count each prime. Con…
Exact mathematical passage
\label{eq:orbitreciprocity} R_{l,q}=-\sum_{b\in\mathrm{reps}_q}b\ell_q(\pi_l(z_q^b)) =-f_q^{-1}\sum_{b=1}^6b\ell_q(\pi_l(z_q^b)),
\sum_{b=2}^6Use in source · 2 passages
Search aliases: ∑_b=2^6 sum
Use in source · 21_ramified_frobenius.tex, line 15
At the chosen q place, write π_q=q u_q with u_q a unit. The same norm identity gives ℓ_q(u_q)=-∑_b=2^6ℓ_q(π_q(z_q^b)). Principal reciprocity on π_q therefore supplies the own conductor column D_q,q=∑_b=2^6(1-b)ℓ_q(π_q(z_q^b)). For r≠ q, the column is D_q,r=-∑_b∈reps_rbℓ_r(π_q(z_r^b)). The degree two representative convention is the one proved in [source reference]. These formulas use the same arithmetic Frobenius principal idele convention as the ordinary localization rows.
Exact mathematical passage
\label{eq:selfunit} \ell_q(u_q)=-\sum_{b=2}^6\ell_q(\pi_q(z_q^b)).
\sum_{b\in\mathrm{reps}_q}Use in source · 1 passage
Search aliases: ∑_b∈reps_q sum in mathrm
Use in source · 10_reciprocity.tex, line 15
For degree two, the indices b and -b represent the same prime. Frobenius sends a residue x to x^q, and ℓ_q(x^q)=-ℓ_q(x). Therefore (-b)ℓ_q(x^q)=bℓ_q(x). The functional is well defined on prime ideals, not just on the six embeddings. Principal reciprocity gives, for a test prime generator π_l outside the conductor, R_l,q=-∑_b∈reps_qbℓ_q(π_l(z_q^b)) =-f_q^-1∑_b=1^6bℓ_q(π_l(z_q^b)), where f_q=1 or two. Using all six terms without dividing by two in the second case would double count each prime. Con…
Exact mathematical passage
\label{eq:orbitreciprocity} R_{l,q}=-\sum_{b\in\mathrm{reps}_q}b\ell_q(\pi_l(z_q^b)) =-f_q^{-1}\sum_{b=1}^6b\ell_q(\pi_l(z_q^b)),
\sum_{b\in\mathrm{reps}_r}Use in source · 1 passage
Search aliases: ∑_b∈reps_r sum in mathrm
Use in source · 21_ramified_frobenius.tex, line 23
n q place, write π_q=q u_q with u_q a unit. The same norm identity gives ℓ_q(u_q)=-∑_b=2^6ℓ_q(π_q(z_q^b)). Principal reciprocity on π_q therefore supplies the own conductor column D_q,q=∑_b=2^6(1-b)ℓ_q(π_q(z_q^b)). For r≠ q, the column is D_q,r=-∑_b∈reps_rbℓ_r(π_q(z_r^b)). The degree two representative convention is the one proved in [source reference]. These formulas use the same arithmetic Frobenius principal idele convention as the ordinary localization rows.
Exact mathematical passage
\label{eq:othercolumn} D_{q,r}=-\sum_{b\in\mathrm{reps}_r}b\ell_r(\pi_q(z_r^b)).
\sum_iUse in source · 1 passage
Search aliases: ∑_i sum
Use in source · 20_localization.tex, line 35
An explicit image parametrization. Put a=z_7(3) and x_i=z(I_q_i). Let ℓ_i be the seventh power logarithm normalized by ℓ_i(g_q_i)=1. The supporting symbols of ϕ vanish. For rational seven adic units define ℓ_7(t)=fraclog(t^6)log(3^6)pmod7 =6fract^6-17pmod7. Principal reciprocity determines the remaining coordinates: z_7(ϕ) · =0, z(F_2) · =-5a-∑_iℓ_i(2)x_i, z(F_3) · =-a-∑_iℓ_i(3)x_i, z(F_5) · =-4a-∑_iℓ_i(sqrt5)x_i, z_7(7) · =-∑_iℓ_i(7)x_i, z(F_q) · =-ℓ_7(q)a-∑_i:q_i≠ qℓ_i(q)x_i. At its own conduc…
Exact mathematical passage
z_7(\phi)&=0,\label{eq:Jphi}\\ z(F_2)&=-5a-\sum_i\ell_i(2)x_i,\label{eq:J2}\\ z(F_3)&=-a-\sum_i\ell_i(3)x_i,\label{eq:J3}\\ z(F_5)&=-4a-\sum_i\ell_i(\sqrt5)x_i,\label{eq:J5}\\ z_7(7)&=-\sum_i\ell_i(7)x_i,\label{eq:J7}\\ z(F_q)&=-\ell_7(q)a-\sum_{i:q_i\ne q}\ell_i(q)x_i.\label{eq:Jq}
\sum_{i:q_i\ne q}Use in source · 1 passage
Search aliases: ∑_i:q_i≠ q sum ne
Use in source · 20_localization.tex, line 35
An explicit image parametrization. Put a=z_7(3) and x_i=z(I_q_i). Let ℓ_i be the seventh power logarithm normalized by ℓ_i(g_q_i)=1. The supporting symbols of ϕ vanish. For rational seven adic units define ℓ_7(t)=fraclog(t^6)log(3^6)pmod7 =6fract^6-17pmod7. Principal reciprocity determines the remaining coordinates: z_7(ϕ) · =0, z(F_2) · =-5a-∑_iℓ_i(2)x_i, z(F_3) · =-a-∑_iℓ_i(3)x_i, z(F_5) · =-4a-∑_iℓ_i(sqrt5)x_i, z_7(7) · =-∑_iℓ_i(7)x_i, z(F_q) · =-ℓ_7(q)a-∑_i:q_i≠ qℓ_i(q)x_i. At its own conduc…
Exact mathematical passage
z_7(\phi)&=0,\label{eq:Jphi}\\ z(F_2)&=-5a-\sum_i\ell_i(2)x_i,\label{eq:J2}\\ z(F_3)&=-a-\sum_i\ell_i(3)x_i,\label{eq:J3}\\ z(F_5)&=-4a-\sum_i\ell_i(\sqrt5)x_i,\label{eq:J5}\\ z_7(7)&=-\sum_i\ell_i(7)x_i,\label{eq:J7}\\ z(F_q)&=-\ell_7(q)a-\sum_{i:q_i\ne q}\ell_i(q)x_i.\label{eq:Jq}
\sum_rUse in source · 1 passage
Search aliases: ∑_r sum
Use in source · 21_ramified_frobenius.tex, line 31
Because each support prime is inert in K/Q, restriction at a rational q≡1pmod7 multiplies the compatible rational Frobenius value by two: b_q=2∑_rD_q,rs_r. For q≡-1pmod7, the compatible squared rational Frobenius has beta value zero, from the cocycle factor 1+χ(q)^-1=0. This does not say beta is unramified there.
Exact mathematical passage
\label{eq:betaF} b_q=2\sum_rD_{q,r}s_r.
\sum_tUse in source · 1 passage
Search aliases: ∑_t sum
Use in source · 04_local_strata.tex, line 57
There are (p+1)/4 elements t for which both t and 1-t are nonsquares. Here is a direct count. Extend the quadratic character χ_p by zero. Counting (x-y)(x+y)=d for d≠0 gives ∑_xχ_p(x^2-d)=-1. Completing the square then gives ∑_tχ_p(t(1-t))=-χ_p(-1)=1. After omitting 0,1, both single character sums are -1. Expanding the indicator (1-χ_p(t))(1-χ_p(1-t))/4 over the remaining points gives (p+1)/4>0.
Exact mathematical passage
\sum_t\chi_p(t(1-t))=-\chi_p(-1)=1
\sum_xUse in source · 1 passage
Search aliases: ∑_x sum
Use in source · 04_local_strata.tex, line 54
There are (p+1)/4 elements t for which both t and 1-t are nonsquares. Here is a direct count. Extend the quadratic character χ_p by zero. Counting (x-y)(x+y)=d for d≠0 gives ∑_xχ_p(x^2-d)=-1. Completing the square then gives ∑_tχ_p(t(1-t))=-χ_p(-1)=1. After omitting 0,1, both single character sums are -1. Expanding the indicator (1-χ_p(t))(1-χ_p(1-t))/4 over the remaining points gives (p+1)/4>0.
Exact mathematical passage
\sum_x\chi_p(x^2-d)=-1.
\setminusUse in source · 1 passage
Search aliases: ∖ setminus
Use in source · 07_natural_depth.tex, line 55
-y^2 · xy, where μ∈Oloc∖{0}, (x,y)∈Oloc^2 is primitive, t=v(μ)≥0, and FN=QN, NF=N, F_21∈π^nOloc, v(N_21)=n. Set h=v(Q-1). Then n=t+2s, s=v(y),qquad0≤ s≤ h. The lower nilpotent entry is -μ y^2, so its valuation gives n=t+2s. The common image and kernel line of N is generated by (x,y)^T. The two identities for F show that F-I has image in this line and kills no extra quotient direction. Write F=I+x
Exact mathematical passage
\mu\in\Oloc\setminus\{0\}
\simeqUse in source · 1 passage
Search aliases: ≃ simeq
Use in source · 17_local_flexibility.tex, line 4
For a finite module A, let A'=Hom(A,μ_7) and let G' be the finite group of the field cut out by that dual. The cited sufficient criterion is H^1(G',A')=0. For A=k with trivial action, A'≃ k(χ) and G'=C_6. Averaging kills its positive cohomology because six is invertible modulo seven. For A=k(χ), the source action cancels the cyclotomic target action, so the dual is trivial and G'=1. Both satisfy the criterion.
Exact mathematical passage
A'\simeq k(\chi)
\neUse in source · 15 passages
Search aliases: ≠ ne
Use in source · 04_local_strata.tex, line 53
There are (p+1)/4 elements t for which both t and 1-t are nonsquares. Here is a direct count. Extend the quadratic character χ_p by zero. Counting (x-y)(x+y)=d for d≠0 gives ∑_xχ_p(x^2-d)=-1. Completing the square then gives ∑_tχ_p(t(1-t))=-χ_p(-1)=1. After omitting 0,1, both single character sums are -1. Expanding the indicator (1-χ_p(t))(1-χ_p(1-t))/4 over the remaining points gives (p+1)/4>0.
Exact mathematical passage
d\ne0
\equivUse in source · 33 passages
Search aliases: ≡ equiv
Use in source · 04_local_strata.tex, line 28
midrule A · b,c · b≡ c
Exact mathematical passage
b\equiv c
\leUse in source · 17 passages
Search aliases: ≤ le
Use in source · 02_signatures.tex, line 29
The conditional abc height bound. An abc estimate makes the role of the signature explicit. Let H=C^z, R=rad(ABC), s=frac1x+frac1y+frac1z. For a primitive solution, the powered terms are pairwise coprime and their radical is R. Positivity gives ABC<H^s. Assume the all epsilon abc estimate H≤ K_eps R^1+eps for every eps>0, with one constant K_eps independent of the triple. Then H<K_eps H^(1+eps)s. Whenever (1+eps)s<1, it follows that H<K_eps^1/(1-(1+eps)s). This is a conditional bound, not an unc…
Exact mathematical passage
\label{eq:abc-assumed} H\le K_\eps R^{1+\eps}
\geUse in source · 13 passages
Search aliases: ≥ ge
Use in source · 01_statement.tex, line 30
The positivity assumptions also carry information. They imply A^x<C^z, B^y<C^z, C≥2. The last inequality follows because A^x+B^y≥2. A base on the left may equal 1, so a finite classification of powered values need not be a finite classification of exponent labels. The equality 1^x=1 holds for every x. This small point is relevant when comparing a theorem for one fixed signature with a statement uniform in all exponents.
Exact mathematical passage
\label{eq:heightbasic} A^x<C^z,\qquad B^y<C^z,\qquad C\ge2.
\subsetUse in source · 2 passages
Search aliases: ⊂ subset
Use in source · 09_periods.tex, line 25
[Full torsion and period powers] For the stipulated split uniformization J^an= G_m^2/Λ with rational period lattice, full J[M] is F rational if and only if μ_M⊂ F and every coordinate of a basis of Λ is an Mth power in F. The uniformization gives 0⟶μ_M^2⟶ J[M] ⟶Λ/MΛ⟶0. If all torsion points are rational, so are μ_M^2. For λ∈Λ, choose an Mth root u in the torus. Rationality of its torsion class gives g(u)/u∈Λ for every g∈ G_F. Since λ is rational, this quotient also lies in μ_M^2. The lattice is …
Exact mathematical passage
\mu_M\subset F
\bigoplusUse in source · 1 passage
Search aliases: ⊕ bigoplus
Use in source · 19_bare_primitive.tex, line 48
Finite set Poitou--Tate duality pairs this group perfectly with Sha^2_Σ(K,k). Every prime dividing the module order belongs to Σ, as required by the cited theorem. Odd primary real Tate terms vanish. Thus localization H^2(Gsig,k)⟶⊕_v∈ΣH^2(K_v,k) is injective [source reference].
Exact mathematical passage
\label{eq:restrictedH2} H^2(\Gsig,k)\longrightarrow\bigoplus_{v\in\Sigma}H^2(K_v,k)
\oplusUse in source · 1 passage
Search aliases: ⊕ oplus
Use in source · 15_semilinear_image.tex, line 33
The abelianization is k⊕ B. The independent global Kummer class [7] cannot factor through that abelianization with its required cyclotomic weight, giving the stated global noncontainment of the uniformizer Kummer extension in the torsion field over K_∞. This field statement does not imply local noncontainment or zero local period valuation.
Exact mathematical passage
k\oplus B
\bigoplus_{v\in\Sigma}Use in source · 1 passage
Search aliases: ⊕_v∈Σ bigoplus in Sigma
Use in source · 19_bare_primitive.tex, line 48
Finite set Poitou--Tate duality pairs this group perfectly with Sha^2_Σ(K,k). Every prime dividing the module order belongs to Σ, as required by the cited theorem. Odd primary real Tate terms vanish. Thus localization H^2(Gsig,k)⟶⊕_v∈ΣH^2(K_v,k) is injective [source reference].
Exact mathematical passage
\label{eq:restrictedH2} H^2(\Gsig,k)\longrightarrow\bigoplus_{v\in\Sigma}H^2(K_v,k)
\longrightarrowUse in source · 6 passages
Search aliases: ⟶ longrightarrow
Use in source · 07_natural_depth.tex, line 13
The inherited characteristic zero representation is the specified RM factor of the twisted Jacobian on the retained 3990mid C branch, with determinant the cyclotomic character χ. Its natural residual orientation is 0⟶ k(χ)⟶overline V⟶ k⟶0. The opposite endpoint has a nonzero class β∈ H^1(K,k(χ^-1)). A stable lattice change can alter residual inertia. Therefore properties of the natural Jacobian lattice cannot automatically be transferred to an arbitrary lattice supplied by Ribet's lemma [source …
Exact mathematical passage
0\longrightarrow k(\chi)\longrightarrow\overline V\longrightarrow k\longrightarrow0.
\longmapstoUse in source · 1 passage
Search aliases: ⟼ longmapsto
Use in source · 22_affine.tex, line 5
On the local space k^24, let E have fifteen rows. Its first six rows select all fixed local character coordinates in [source reference]. Its remaining nine rows are z⟼ c_qz(F_q)-b_qz(I_q). Define A=EJ:k^10⟶ k^15. The self conductor values and inertia conversions in Section~[source reference] are part of this operator. Dropping them would reproduce a different matrix.
Exact mathematical passage
z\longmapsto c_qz(F_q)-b_qz(I_q).
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- research/beal-reader/README.md · 0 mathematical passages
- research/beal-reader/evidence/FORMAL_STATUS.md · 3 mathematical passages
- research/beal-reader/paper/chapters/00_reading.tex · 1 mathematical passage
- research/beal-reader/paper/chapters/01_statement.tex · 30 mathematical passages
- research/beal-reader/paper/chapters/02_signatures.tex · 50 mathematical passages
- research/beal-reader/paper/chapters/03_scaling.tex · 60 mathematical passages
- research/beal-reader/paper/chapters/04_local_strata.tex · 88 mathematical passages
- research/beal-reader/paper/chapters/05_frey.tex · 69 mathematical passages
- research/beal-reader/paper/chapters/06_modular_labels.tex · 15 mathematical passages
- research/beal-reader/paper/chapters/07_natural_depth.tex · 98 mathematical passages
- research/beal-reader/paper/chapters/08_global_coefficients.tex · 54 mathematical passages
- research/beal-reader/paper/chapters/09_periods.tex · 59 mathematical passages
- research/beal-reader/paper/chapters/10_reciprocity.tex · 44 mathematical passages
- research/beal-reader/paper/chapters/11_bounded_patterns.tex · 28 mathematical passages
- research/beal-reader/paper/chapters/12_fixed_seven.tex · 65 mathematical passages
- research/beal-reader/paper/chapters/13_deformation.tex · 34 mathematical passages
- research/beal-reader/paper/chapters/14_integral_trace.tex · 43 mathematical passages
- research/beal-reader/paper/chapters/15_semilinear_image.tex · 28 mathematical passages
- research/beal-reader/paper/chapters/16_secondary.tex · 57 mathematical passages
- research/beal-reader/paper/chapters/17_local_flexibility.tex · 31 mathematical passages
- research/beal-reader/paper/chapters/18_ramification.tex · 38 mathematical passages
- research/beal-reader/paper/chapters/19_bare_primitive.tex · 84 mathematical passages
- research/beal-reader/paper/chapters/20_localization.tex · 42 mathematical passages
- research/beal-reader/paper/chapters/21_ramified_frobenius.tex · 44 mathematical passages
- research/beal-reader/paper/chapters/22_affine.tex · 66 mathematical passages
- research/beal-reader/paper/chapters/23_evidence.tex · 3 mathematical passages
- research/beal-reader/paper/chapters/24_sources.tex · 8 mathematical passages
- research/beal-reader/paper/main.tex · 0 mathematical passages
- research/beal-reader/shared/preamble.tex · 0 mathematical passages
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