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Hilbert 12: cubic divisor development
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627 symbol spellings in 903 mathematical passages across 16 selected source files. Subscripts and superscripts have their own entries alongside their base symbols.
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+Introduction passage · 164 passages
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Introduction passage · rigidity.tex, line 71
has been constructed as a_Δ(v)=α e^2π im^Tve^h(v), m∈ Z^3, h mean zero, and its based ratio is actually identified with U_Γ/U_rm alg. Write h^#(v)=overlineh(overline v). It suffices to establish [source reference] for h-h^#. The theorem gives h-h^#=g-g∘ M. The gauge e^-g/2 replaces the periodic logarithm by (h+h^#)/2, real on the real torus. Since Mp_3=p_3 modulo the lattice, this periodic gauge has equal values at the two endpoints, leaving fraca_Δ(p_3)a_Δ(0)=e^2π im_2/3e^real number. After the…
Exact mathematical passage
(h+h^\#)/2
=Introduction passage · 408 passages
Search aliases: =
Introduction passage · polarization.tex, line 30
The actual arithmetic candidate U=U_rm alg is defined by the retained rational formulas in nolinkurlprior/ARITHMETIC_INPUT.md. This block starts from those formulas. No gamma value is used to find or fit coefficients.
Exact mathematical passage
U=U_{\rm alg}
aIntroduction passage · 132 passages
Search aliases: a
Introduction passage · polarization.tex, line 30
The actual arithmetic candidate U=U_rm alg is defined by the retained rational formulas in nolinkurlprior/ARITHMETIC_INPUT.md. This block starts from those formulas. No gamma value is used to find or fit coefficients.
Exact mathematical passage
U=U_{\rm alg}
a_\DeltaIntroduction passage · 2 passages
Search aliases: a_Δ Delta
Introduction passage · rigidity.tex, line 67
Phase-only compatibility and the missing identification. Suppose a genuine nonzero relative multiplier has been constructed as a_Δ(v)=α e^2π im^Tve^h(v), m∈ Z^3, h mean zero, and its based ratio is actually identified with U_Γ/U_rm alg. Write h^#(v)=overlineh(overline v). It suffices to establish [source reference] for h-h^#. The theorem gives h-h^#=g-g∘ M. The gauge e^-g/2 replaces the periodic logarithm by (h+h^#)/2, real on the real torus. Since Mp_3=p_3 modulo the lattice, this periodic gaug…
Exact mathematical passage
a_\Delta(v)=\alpha e^{2\pi im^Tv}e^{h(v)},\qquad m\in\mathbb Z^3,\quad h\text{ mean zero},
eIntroduction passage · 46 passages
Search aliases: e
Introduction passage · rigidity.tex, line 67
Phase-only compatibility and the missing identification. Suppose a genuine nonzero relative multiplier has been constructed as a_Δ(v)=α e^2π im^Tve^h(v), m∈ Z^3, h mean zero, and its based ratio is actually identified with U_Γ/U_rm alg. Write h^#(v)=overlineh(overline v). It suffices to establish [source reference] for h-h^#. The theorem gives h-h^#=g-g∘ M. The gauge e^-g/2 replaces the periodic logarithm by (h+h^#)/2, real on the real torus. Since Mp_3=p_3 modulo the lattice, this periodic gaug…
Exact mathematical passage
a_\Delta(v)=\alpha e^{2\pi im^Tv}e^{h(v)},\qquad m\in\mathbb Z^3,\quad h\text{ mean zero},
e^{-g/2}Introduction passage · 1 passage
Search aliases: e^-g/2
Introduction passage · rigidity.tex, line 71
Suppose a genuine nonzero relative multiplier has been constructed as a_Δ(v)=α e^2π im^Tve^h(v), m∈ Z^3, h mean zero, and its based ratio is actually identified with U_Γ/U_rm alg. Write h^#(v)=overlineh(overline v). It suffices to establish [source reference] for h-h^#. The theorem gives h-h^#=g-g∘ M. The gauge e^-g/2 replaces the periodic logarithm by (h+h^#)/2, real on the real torus. Since Mp_3=p_3 modulo the lattice, this periodic gauge has equal values at the two endpoints, leaving fraca_Δ(…
Exact mathematical passage
e^{-g/2}
e^{2\pi im^Tv}Introduction passage · 1 passage
Search aliases: e^2π im^Tv pi
Introduction passage · rigidity.tex, line 67
Phase-only compatibility and the missing identification. Suppose a genuine nonzero relative multiplier has been constructed as a_Δ(v)=α e^2π im^Tve^h(v), m∈ Z^3, h mean zero, and its based ratio is actually identified with U_Γ/U_rm alg. Write h^#(v)=overlineh(overline v). It suffices to establish [source reference] for h-h^#. The theorem gives h-h^#=g-g∘ M. The gauge e^-g/2 replaces the periodic logarithm by (h+h^#)/2, real on the real torus. Since Mp_3=p_3 modulo the lattice, this periodic gaug…
Exact mathematical passage
a_\Delta(v)=\alpha e^{2\pi im^Tv}e^{h(v)},\qquad m\in\mathbb Z^3,\quad h\text{ mean zero},
e^{2\pi im_2/3}Introduction passage · 1 passage
Search aliases: e^2π im_2/3 pi
Introduction passage · rigidity.tex, line 72
is actually identified with U_Γ/U_rm alg. Write h^#(v)=overlineh(overline v). It suffices to establish [source reference] for h-h^#. The theorem gives h-h^#=g-g∘ M. The gauge e^-g/2 replaces the periodic logarithm by (h+h^#)/2, real on the real torus. Since Mp_3=p_3 modulo the lattice, this periodic gauge has equal values at the two endpoints, leaving fraca_Δ(p_3)a_Δ(0)=e^2π im_2/3e^real number. After the based identification the exact modulus equality forces that real exponent to vanish. The ra…
Exact mathematical passage
\frac{a_\Delta(p_3)}{a_\Delta(0)}=e^{2\pi im_2/3}e^{\text{real number}}.
e^{h(v)}Introduction passage · 1 passage
Search aliases: e^h(v)
Introduction passage · rigidity.tex, line 67
Phase-only compatibility and the missing identification. Suppose a genuine nonzero relative multiplier has been constructed as a_Δ(v)=α e^2π im^Tve^h(v), m∈ Z^3, h mean zero, and its based ratio is actually identified with U_Γ/U_rm alg. Write h^#(v)=overlineh(overline v). It suffices to establish [source reference] for h-h^#. The theorem gives h-h^#=g-g∘ M. The gauge e^-g/2 replaces the periodic logarithm by (h+h^#)/2, real on the real torus. Since Mp_3=p_3 modulo the lattice, this periodic gaug…
Exact mathematical passage
a_\Delta(v)=\alpha e^{2\pi im^Tv}e^{h(v)},\qquad m\in\mathbb Z^3,\quad h\text{ mean zero},
e^{\text{real number}}Introduction passage · 1 passage
Search aliases: e^real number text
Introduction passage · rigidity.tex, line 72
is actually identified with U_Γ/U_rm alg. Write h^#(v)=overlineh(overline v). It suffices to establish [source reference] for h-h^#. The theorem gives h-h^#=g-g∘ M. The gauge e^-g/2 replaces the periodic logarithm by (h+h^#)/2, real on the real torus. Since Mp_3=p_3 modulo the lattice, this periodic gauge has equal values at the two endpoints, leaving fraca_Δ(p_3)a_Δ(0)=e^2π im_2/3e^real number. After the based identification the exact modulus equality forces that real exponent to vanish. The ra…
Exact mathematical passage
\frac{a_\Delta(p_3)}{a_\Delta(0)}=e^{2\pi im_2/3}e^{\text{real number}}.
gIntroduction passage · 42 passages
Search aliases: g
Introduction passage · polarization.tex, line 30
The actual arithmetic candidate U=U_rm alg is defined by the retained rational formulas in nolinkurlprior/ARITHMETIC_INPUT.md. This block starts from those formulas. No gamma value is used to find or fit coefficients.
Exact mathematical passage
U=U_{\rm alg}
hIntroduction passage · 26 passages
Search aliases: h
Introduction passage · rigidity.tex, line 67
Phase-only compatibility and the missing identification. Suppose a genuine nonzero relative multiplier has been constructed as a_Δ(v)=α e^2π im^Tve^h(v), m∈ Z^3, h mean zero, and its based ratio is actually identified with U_Γ/U_rm alg. Write h^#(v)=overlineh(overline v). It suffices to establish [source reference] for h-h^#. The theorem gives h-h^#=g-g∘ M. The gauge e^-g/2 replaces the periodic logarithm by (h+h^#)/2, real on the real torus. Since Mp_3=p_3 modulo the lattice, this periodic gaug…
Exact mathematical passage
a_\Delta(v)=\alpha e^{2\pi im^Tv}e^{h(v)},\qquad m\in\mathbb Z^3,\quad h\text{ mean zero},
h^\#Introduction passage · 4 passages
Search aliases: h^#
Introduction passage · rigidity.tex, line 71
Phase-only compatibility and the missing identification. Suppose a genuine nonzero relative multiplier has been constructed as a_Δ(v)=α e^2π im^Tve^h(v), m∈ Z^3, h mean zero, and its based ratio is actually identified with U_Γ/U_rm alg. Write h^#(v)=overlineh(overline v). It suffices to establish [source reference] for h-h^#. The theorem gives h-h^#=g-g∘ M. The gauge e^-g/2 replaces the periodic logarithm by (h+h^#)/2, real on the real torus. Since Mp_3=p_3 modulo the lattice, this periodic gaug…
Exact mathematical passage
h^\#(v)=\overline{h(\overline v)}
iIntroduction passage · 32 passages
Search aliases: i
Introduction passage · rigidity.tex, line 67
Phase-only compatibility and the missing identification. Suppose a genuine nonzero relative multiplier has been constructed as a_Δ(v)=α e^2π im^Tve^h(v), m∈ Z^3, h mean zero, and its based ratio is actually identified with U_Γ/U_rm alg. Write h^#(v)=overlineh(overline v). It suffices to establish [source reference] for h-h^#. The theorem gives h-h^#=g-g∘ M. The gauge e^-g/2 replaces the periodic logarithm by (h+h^#)/2, real on the real torus. Since Mp_3=p_3 modulo the lattice, this periodic gaug…
Exact mathematical passage
a_\Delta(v)=\alpha e^{2\pi im^Tv}e^{h(v)},\qquad m\in\mathbb Z^3,\quad h\text{ mean zero},
lIntroduction passage · 37 passages
Search aliases: l
Introduction passage · polarization.tex, line 30
The actual arithmetic candidate U=U_rm alg is defined by the retained rational formulas in nolinkurlprior/ARITHMETIC_INPUT.md. This block starts from those formulas. No gamma value is used to find or fit coefficients.
Exact mathematical passage
U=U_{\rm alg}
MIntroduction passage · 16 passages
Search aliases: M
Introduction passage · rigidity.tex, line 71
nd the missing identification Suppose a genuine nonzero relative multiplier has been constructed as a_Δ(v)=α e^2π im^Tve^h(v), m∈ Z^3, h mean zero, and its based ratio is actually identified with U_Γ/U_rm alg. Write h^#(v)=overlineh(overline v). It suffices to establish [source reference] for h-h^#. The theorem gives h-h^#=g-g∘ M. The gauge e^-g/2 replaces the periodic logarithm by (h+h^#)/2, real on the real torus. Since Mp_3=p_3 modulo the lattice, this periodic gauge has equal values at the t…
Exact mathematical passage
h-h^\#=g-g\circ M
mIntroduction passage · 29 passages
Search aliases: m
Introduction passage · rigidity.tex, line 67
Phase-only compatibility and the missing identification. Suppose a genuine nonzero relative multiplier has been constructed as a_Δ(v)=α e^2π im^Tve^h(v), m∈ Z^3, h mean zero, and its based ratio is actually identified with U_Γ/U_rm alg. Write h^#(v)=overlineh(overline v). It suffices to establish [source reference] for h-h^#. The theorem gives h-h^#=g-g∘ M. The gauge e^-g/2 replaces the periodic logarithm by (h+h^#)/2, real on the real torus. Since Mp_3=p_3 modulo the lattice, this periodic gaug…
Exact mathematical passage
a_\Delta(v)=\alpha e^{2\pi im^Tv}e^{h(v)},\qquad m\in\mathbb Z^3,\quad h\text{ mean zero},
m^TIntroduction passage · 1 passage
Search aliases: m^T
Introduction passage · rigidity.tex, line 67
Phase-only compatibility and the missing identification. Suppose a genuine nonzero relative multiplier has been constructed as a_Δ(v)=α e^2π im^Tve^h(v), m∈ Z^3, h mean zero, and its based ratio is actually identified with U_Γ/U_rm alg. Write h^#(v)=overlineh(overline v). It suffices to establish [source reference] for h-h^#. The theorem gives h-h^#=g-g∘ M. The gauge e^-g/2 replaces the periodic logarithm by (h+h^#)/2, real on the real torus. Since Mp_3=p_3 modulo the lattice, this periodic gaug…
Exact mathematical passage
a_\Delta(v)=\alpha e^{2\pi im^Tv}e^{h(v)},\qquad m\in\mathbb Z^3,\quad h\text{ mean zero},
m_2Introduction passage · 1 passage
Search aliases: m_2
Introduction passage · rigidity.tex, line 72
is actually identified with U_Γ/U_rm alg. Write h^#(v)=overlineh(overline v). It suffices to establish [source reference] for h-h^#. The theorem gives h-h^#=g-g∘ M. The gauge e^-g/2 replaces the periodic logarithm by (h+h^#)/2, real on the real torus. Since Mp_3=p_3 modulo the lattice, this periodic gauge has equal values at the two endpoints, leaving fraca_Δ(p_3)a_Δ(0)=e^2π im_2/3e^real number. After the based identification the exact modulus equality forces that real exponent to vanish. The ra…
Exact mathematical passage
\frac{a_\Delta(p_3)}{a_\Delta(0)}=e^{2\pi im_2/3}e^{\text{real number}}.
\overline{h(\overline v)}Introduction passage · 1 passage
Search aliases: overlineh(overline v) overline overline
Introduction passage · rigidity.tex, line 71
Phase-only compatibility and the missing identification. Suppose a genuine nonzero relative multiplier has been constructed as a_Δ(v)=α e^2π im^Tve^h(v), m∈ Z^3, h mean zero, and its based ratio is actually identified with U_Γ/U_rm alg. Write h^#(v)=overlineh(overline v). It suffices to establish [source reference] for h-h^#. The theorem gives h-h^#=g-g∘ M. The gauge e^-g/2 replaces the periodic logarithm by (h+h^#)/2, real on the real torus. Since Mp_3=p_3 modulo the lattice, this periodic gaug…
Exact mathematical passage
h^\#(v)=\overline{h(\overline v)}
pIntroduction passage · 24 passages
Search aliases: p
Introduction passage · rigidity.tex, line 71
ha e^2π im^Tve^h(v), m∈ Z^3, h mean zero, and its based ratio is actually identified with U_Γ/U_rm alg. Write h^#(v)=overlineh(overline v). It suffices to establish [source reference] for h-h^#. The theorem gives h-h^#=g-g∘ M. The gauge e^-g/2 replaces the periodic logarithm by (h+h^#)/2, real on the real torus. Since Mp_3=p_3 modulo the lattice, this periodic gauge has equal values at the two endpoints, leaving fraca_Δ(p_3)a_Δ(0)=e^2π im_2/3e^real number. After the based identification the exac…
Exact mathematical passage
Mp_3=p_3
p_3Introduction passage · 5 passages
Search aliases: p_3
Introduction passage · rigidity.tex, line 71
ha e^2π im^Tve^h(v), m∈ Z^3, h mean zero, and its based ratio is actually identified with U_Γ/U_rm alg. Write h^#(v)=overlineh(overline v). It suffices to establish [source reference] for h-h^#. The theorem gives h-h^#=g-g∘ M. The gauge e^-g/2 replaces the periodic logarithm by (h+h^#)/2, real on the real torus. Since Mp_3=p_3 modulo the lattice, this periodic gauge has equal values at the two endpoints, leaving fraca_Δ(p_3)a_Δ(0)=e^2π im_2/3e^real number. After the based identification the exac…
Exact mathematical passage
Mp_3=p_3
\rmIntroduction passage · 41 passages
Search aliases: rm
Introduction passage · polarization.tex, line 30
The actual arithmetic candidate U=U_rm alg is defined by the retained rational formulas in nolinkurlprior/ARITHMETIC_INPUT.md. This block starts from those formulas. No gamma value is used to find or fit coefficients.
Exact mathematical passage
U=U_{\rm alg}
TIntroduction passage · 29 passages
Search aliases: T
Introduction passage · rigidity.tex, line 67
Phase-only compatibility and the missing identification. Suppose a genuine nonzero relative multiplier has been constructed as a_Δ(v)=α e^2π im^Tve^h(v), m∈ Z^3, h mean zero, and its based ratio is actually identified with U_Γ/U_rm alg. Write h^#(v)=overlineh(overline v). It suffices to establish [source reference] for h-h^#. The theorem gives h-h^#=g-g∘ M. The gauge e^-g/2 replaces the periodic logarithm by (h+h^#)/2, real on the real torus. Since Mp_3=p_3 modulo the lattice, this periodic gaug…
Exact mathematical passage
a_\Delta(v)=\alpha e^{2\pi im^Tv}e^{h(v)},\qquad m\in\mathbb Z^3,\quad h\text{ mean zero},
UIntroduction passage · 50 passages
Search aliases: U
Introduction passage · polarization.tex, line 30
The actual arithmetic candidate U=U_rm alg is defined by the retained rational formulas in nolinkurlprior/ARITHMETIC_INPUT.md. This block starts from those formulas. No gamma value is used to find or fit coefficients.
Exact mathematical passage
U=U_{\rm alg}
U_{\rm alg}Introduction passage · 18 passages
Search aliases: U_rm alg rm
Introduction passage · polarization.tex, line 30
The actual arithmetic candidate U=U_rm alg is defined by the retained rational formulas in nolinkurlprior/ARITHMETIC_INPUT.md. This block starts from those formulas. No gamma value is used to find or fit coefficients.
Exact mathematical passage
U=U_{\rm alg}
U_\GammaIntroduction passage · 22 passages
Search aliases: U_Γ Gamma
Introduction passage · rigidity.tex, line 71
Phase-only compatibility and the missing identification. Suppose a genuine nonzero relative multiplier has been constructed as a_Δ(v)=α e^2π im^Tve^h(v), m∈ Z^3, h mean zero, and its based ratio is actually identified with U_Γ/U_rm alg. Write h^#(v)=overlineh(overline v). It suffices to establish [source reference] for h-h^#. The theorem gives h-h^#=g-g∘ M. The gauge e^-g/2 replaces the periodic logarithm by (h+h^#)/2, real on the real torus. Since Mp_3=p_3 modulo the lattice, this periodic gaug…
Exact mathematical passage
U_\Gamma/U_{\rm alg}
vIntroduction passage · 13 passages
Search aliases: v
Introduction passage · rigidity.tex, line 67
Phase-only compatibility and the missing identification. Suppose a genuine nonzero relative multiplier has been constructed as a_Δ(v)=α e^2π im^Tve^h(v), m∈ Z^3, h mean zero, and its based ratio is actually identified with U_Γ/U_rm alg. Write h^#(v)=overlineh(overline v). It suffices to establish [source reference] for h-h^#. The theorem gives h-h^#=g-g∘ M. The gauge e^-g/2 replaces the periodic logarithm by (h+h^#)/2, real on the real torus. Since Mp_3=p_3 modulo the lattice, this periodic gaug…
Exact mathematical passage
a_\Delta(v)=\alpha e^{2\pi im^Tv}e^{h(v)},\qquad m\in\mathbb Z^3,\quad h\text{ mean zero},
\mathbb ZIntroduction passage · 21 passages
Search aliases: Z mathbb
Introduction passage · rigidity.tex, line 67
Phase-only compatibility and the missing identification. Suppose a genuine nonzero relative multiplier has been constructed as a_Δ(v)=α e^2π im^Tve^h(v), m∈ Z^3, h mean zero, and its based ratio is actually identified with U_Γ/U_rm alg. Write h^#(v)=overlineh(overline v). It suffices to establish [source reference] for h-h^#. The theorem gives h-h^#=g-g∘ M. The gauge e^-g/2 replaces the periodic logarithm by (h+h^#)/2, real on the real torus. Since Mp_3=p_3 modulo the lattice, this periodic gaug…
Exact mathematical passage
a_\Delta(v)=\alpha e^{2\pi im^Tv}e^{h(v)},\qquad m\in\mathbb Z^3,\quad h\text{ mean zero},
\mathbb Z^3Introduction passage · 5 passages
Search aliases: Z^3 mathbb
Introduction passage · rigidity.tex, line 67
Phase-only compatibility and the missing identification. Suppose a genuine nonzero relative multiplier has been constructed as a_Δ(v)=α e^2π im^Tve^h(v), m∈ Z^3, h mean zero, and its based ratio is actually identified with U_Γ/U_rm alg. Write h^#(v)=overlineh(overline v). It suffices to establish [source reference] for h-h^#. The theorem gives h-h^#=g-g∘ M. The gauge e^-g/2 replaces the periodic logarithm by (h+h^#)/2, real on the real torus. Since Mp_3=p_3 modulo the lattice, this periodic gaug…
Exact mathematical passage
a_\Delta(v)=\alpha e^{2\pi im^Tv}e^{h(v)},\qquad m\in\mathbb Z^3,\quad h\text{ mean zero},
\alphaIntroduction passage · 20 passages
Search aliases: α alpha
Introduction passage · rigidity.tex, line 67
Phase-only compatibility and the missing identification. Suppose a genuine nonzero relative multiplier has been constructed as a_Δ(v)=α e^2π im^Tve^h(v), m∈ Z^3, h mean zero, and its based ratio is actually identified with U_Γ/U_rm alg. Write h^#(v)=overlineh(overline v). It suffices to establish [source reference] for h-h^#. The theorem gives h-h^#=g-g∘ M. The gauge e^-g/2 replaces the periodic logarithm by (h+h^#)/2, real on the real torus. Since Mp_3=p_3 modulo the lattice, this periodic gaug…
Exact mathematical passage
a_\Delta(v)=\alpha e^{2\pi im^Tv}e^{h(v)},\qquad m\in\mathbb Z^3,\quad h\text{ mean zero},
\GammaIntroduction passage · 39 passages
Search aliases: Γ Gamma
Introduction passage · rigidity.tex, line 71
Phase-only compatibility and the missing identification. Suppose a genuine nonzero relative multiplier has been constructed as a_Δ(v)=α e^2π im^Tve^h(v), m∈ Z^3, h mean zero, and its based ratio is actually identified with U_Γ/U_rm alg. Write h^#(v)=overlineh(overline v). It suffices to establish [source reference] for h-h^#. The theorem gives h-h^#=g-g∘ M. The gauge e^-g/2 replaces the periodic logarithm by (h+h^#)/2, real on the real torus. Since Mp_3=p_3 modulo the lattice, this periodic gaug…
Exact mathematical passage
U_\Gamma/U_{\rm alg}
\DeltaIntroduction passage · 2 passages
Search aliases: Δ Delta
Introduction passage · rigidity.tex, line 67
Phase-only compatibility and the missing identification. Suppose a genuine nonzero relative multiplier has been constructed as a_Δ(v)=α e^2π im^Tve^h(v), m∈ Z^3, h mean zero, and its based ratio is actually identified with U_Γ/U_rm alg. Write h^#(v)=overlineh(overline v). It suffices to establish [source reference] for h-h^#. The theorem gives h-h^#=g-g∘ M. The gauge e^-g/2 replaces the periodic logarithm by (h+h^#)/2, real on the real torus. Since Mp_3=p_3 modulo the lattice, this periodic gaug…
Exact mathematical passage
a_\Delta(v)=\alpha e^{2\pi im^Tv}e^{h(v)},\qquad m\in\mathbb Z^3,\quad h\text{ mean zero},
\kappaIntroduction passage · 9 passages
Search aliases: κ kappa
Introduction passage · stark.tex, line 80
Actual complex conjugation is a different automorphism. Denote it by κ. In the chosen embedding,
Exact mathematical passage
\kappa
\piIntroduction passage · 30 passages
Search aliases: π pi
Introduction passage · rigidity.tex, line 67
Phase-only compatibility and the missing identification. Suppose a genuine nonzero relative multiplier has been constructed as a_Δ(v)=α e^2π im^Tve^h(v), m∈ Z^3, h mean zero, and its based ratio is actually identified with U_Γ/U_rm alg. Write h^#(v)=overlineh(overline v). It suffices to establish [source reference] for h-h^#. The theorem gives h-h^#=g-g∘ M. The gauge e^-g/2 replaces the periodic logarithm by (h+h^#)/2, real on the real torus. Since Mp_3=p_3 modulo the lattice, this periodic gaug…
Exact mathematical passage
a_\Delta(v)=\alpha e^{2\pi im^Tv}e^{h(v)},\qquad m\in\mathbb Z^3,\quad h\text{ mean zero},
\inIntroduction passage · 22 passages
Search aliases: ∈ in
Introduction passage · rigidity.tex, line 67
Phase-only compatibility and the missing identification. Suppose a genuine nonzero relative multiplier has been constructed as a_Δ(v)=α e^2π im^Tve^h(v), m∈ Z^3, h mean zero, and its based ratio is actually identified with U_Γ/U_rm alg. Write h^#(v)=overlineh(overline v). It suffices to establish [source reference] for h-h^#. The theorem gives h-h^#=g-g∘ M. The gauge e^-g/2 replaces the periodic logarithm by (h+h^#)/2, real on the real torus. Since Mp_3=p_3 modulo the lattice, this periodic gaug…
Exact mathematical passage
a_\Delta(v)=\alpha e^{2\pi im^Tv}e^{h(v)},\qquad m\in\mathbb Z^3,\quad h\text{ mean zero},
\circIntroduction passage · 4 passages
Search aliases: ∘ circ
Introduction passage · rigidity.tex, line 71
nd the missing identification Suppose a genuine nonzero relative multiplier has been constructed as a_Δ(v)=α e^2π im^Tve^h(v), m∈ Z^3, h mean zero, and its based ratio is actually identified with U_Γ/U_rm alg. Write h^#(v)=overlineh(overline v). It suffices to establish [source reference] for h-h^#. The theorem gives h-h^#=g-g∘ M. The gauge e^-g/2 replaces the periodic logarithm by (h+h^#)/2, real on the real torus. Since Mp_3=p_3 modulo the lattice, this periodic gauge has equal values at the t…
Exact mathematical passage
h-h^\#=g-g\circ M
*Use in source · 6 passages
Search aliases: *
Use in source · H12_Cubic_Divisor_Traditional.tex, line 99
We use F_0= Q(ω) for the imaginary-quadratic field, to separate it from the scalar gamma function F. The eta quotient is H(a)=η(5a)/η(a)^5; the integer step function H(n) used in the divisor formula is a different object. The symbol R_0 is the scalar anchor, while R_* is a counting cutoff radius. In the arithmetic block ϱ is a ray character and a radical lift, as explicitly indicated there; it is distinct from the contracting period multiplier ρ.
Exact mathematical passage
R_*
\mathbf1Use in source · 2 passages
Search aliases: 1 mathbf
Use in source · principal.tex, line 37
Base divisor and the integral lift. Define H(n)=1_{n>0}, w(c)=4, · 5mid c, -1, · 5nmid c. The function H here is an integer step function, distinct from the eta quotient H and the locally defined CM Green quantity H(z) used elsewhere.
Exact mathematical passage
H(n)=\mathbf1_{\{n>0\}},\qquad w(c)=\begin{cases}4,&5\mid c,\\-1,&5\nmid c.\end{cases}
\mathbf1_{\{1\le b\le15\}}Use in source · 1 passage
Search aliases: 1_{1≤ b≤15} mathbf le le
Use in source · principal.tex, line 120
gives two q_s factors and four reciprocal q_t factors; the 2τ shift disappears from q_t by ellipticity. Finally P_7(z+δ)/P_7(z)=F(z+7δ)/F(z) and the last two identities of [source reference] give [source reference]. Under z↦ z+1 the triple changes to (a,b-15,c-10), leaving p and w(c) unchanged. Formula~[source reference] gives ord_z+1R-ord_zR =-w(c)1_{1≤ b≤15}, the divisor of [source reference]. Under z↦ z+δ the triple becomes (a,b-8,c-5); its digit decreases by one modulo seven, and its pivot i…
Exact mathematical passage
\operatorname{ord}_{z+1}R-\operatorname{ord}_zR =-w(c)\mathbf1_{\{1\le b\le15\}},
\mathbf1_{\{n>0\}}Use in source · 1 passage
Search aliases: 1_{n>0} mathbf
Use in source · principal.tex, line 37
Base divisor and the integral lift. Define H(n)=1_{n>0}, w(c)=4, · 5mid c, -1, · 5nmid c. The function H here is an integer step function, distinct from the eta quotient H and the locally defined CM Green quantity H(z) used elsewhere.
Exact mathematical passage
H(n)=\mathbf1_{\{n>0\}},\qquad w(c)=\begin{cases}4,&5\mid c,\\-1,&5\nmid c.\end{cases}
AUse in source · 18 passages
Search aliases: A
Use in source · norms.tex, line 45
The exact information lost by doubling. Quotient nonzero meromorphic elliptic functions by constants and translation coboundaries A(w+s)/A(w). Set e=[q_t], p=[ψ] in additive notation. These generate a free Z^2: the divisor charges on the disjoint infinite translation orbits of 0 and -S/2 are respectively (4,4) and (0,-4). A finite divisor coboundary has charge zero on every orbit by telescoping, so a relation ae+bp=0 forces b=0 then a=0.
Exact mathematical passage
A(w+s)/A(w)
\mathcal AUse in source · 9 passages
Search aliases: A mathcal
Use in source · polarization.tex, line 117
A=a_0a_1^3=fraca_1^2a_2, B=a_0^2a_1^4=(fraca_1a_2)^2.
Exact mathematical passage
\mathcal A=a_0a_1^3=\frac{a_1^2}{a_2},\qquad \mathcal B=a_0^2a_1^4=\left(\frac{a_1}{a_2}\right)^2.
\mathfrak aUse in source · 1 passage
Search aliases: a mathfrak
Use in source · stark.tex, line 632
|Γ_ a,h( f b^-1)|^6 =|(σ_ bW_rm St)^N( a)-σ_ a|. tagA7.4
Exact mathematical passage
|\mathbf\Gamma_{\mathfrak a,h}(\mathfrak f\mathfrak b^{-1})|^6 =\left|(\sigma_{\mathfrak b}W_{\rm St})^{N(\mathfrak a)-\sigma_{\mathfrak a}}\right|. \tag{A7.4}
a^{-1/2}Use in source · 1 passage
Search aliases: a^-1/2
Use in source · rigidity.tex, line 15
The cubic characteristic polynomial (x-2)^3+7 gives a stable real eigenvalue a=2-7^1/3∈(0,1) and a conjugate unstable pair of modulus a^-1/2>1. Every nonzero integer A-orbit escapes in both directions: otherwise an integer vector would tend to zero along positive or negative iterates. Fourier orbit sums are therefore absolutely convergent for smooth coefficients. The dual degree-seven image is exactly k∈ B Z^3 Longleftrightarrow ℓ(k):=k_0+2k_1+4k_2≡0pmod7. Indeed k-ℓ(k)e_0=B(-k_1,k_2,-8k_2)^T an…
Exact mathematical passage
a^{-1/2}>1
a^2Use in source · 1 passage
Search aliases: a^2
Use in source · stark.tex, line 311
where initially 0≤ a<1 and B_2(a)=a^2-a+1/6. This is the absolute value of the usual Siegel function. Its value is independent of the representative of z modulo Λ_0.
Exact mathematical passage
B_2(a)=a^2-a+1/6
A^3Use in source · 1 passage
Search aliases: A^3
Use in source · theta.tex, line 108
c=A^3/B. tagN2.2
Exact mathematical passage
\boxed{c=A^3/B.} \tag{N2.2}
A_0Use in source · 2 passages
Search aliases: A_0
Use in source · product_proof.tex, line 3
Full seven-product series proof. For positive imaginary periods t,s,d with t+7d=2s+4, put P=e^2π it, Q=e^2π is, D=e^2π id and Z=e^2π iz. Then PD^7=Q^2. In a common convergence region the fifth-smoothed logarithms are log S_5(z;a,b) · =5∑_substackn≥1 5nmid n fracZ^n-(A_0B_0/Z)^nn(1-A_0^n)(1-B_0^n), log F_t,s(z) · =-5∑_substackn≥1 5nmid n fracP^nZ^n-Q^nZ^-nn(1-P^n)(1-Q^n), where A_0=e^2π ia,B_0=e^2π ib in the first line. At one surviving frequency suppress the index n. Summing seven shifts multipl…
Exact mathematical passage
\log S_5(z;a,b)&=5\sum_{\substack{n\ge1\\5\nmid n}} \frac{Z^n-(A_0B_0/Z)^n}{n(1-A_0^n)(1-B_0^n)},\\ \log F_{t,s}(z)&=-5\sum_{\substack{n\ge1\\5\nmid n}} \frac{P^nZ^n-Q^nZ^{-n}}{n(1-P^n)(1-Q^n)},
a_0Use in source · 8 passages
Search aliases: a_0
Use in source · polarization.tex, line 37
a_0=(1+ω)ξ^2-ξ+1, a_1=-ξ^2+(1+ω)ξ+1, a_2=-ωξ^2-ωξ+1.
Exact mathematical passage
a_0=(1+\omega)\xi^2-\xi+1,\quad a_1=-\xi^2+(1+\omega)\xi+1,\quad a_2=-\omega\xi^2-\omega\xi+1.
a_0^2Use in source · 1 passage
Search aliases: a_0^2
Use in source · polarization.tex, line 117
A=a_0a_1^3=fraca_1^2a_2, B=a_0^2a_1^4=(fraca_1a_2)^2.
Exact mathematical passage
\mathcal A=a_0a_1^3=\frac{a_1^2}{a_2},\qquad \mathcal B=a_0^2a_1^4=\left(\frac{a_1}{a_2}\right)^2.
A_0^nUse in source · 1 passage
Search aliases: A_0^n
Use in source · product_proof.tex, line 3
Full seven-product series proof. For positive imaginary periods t,s,d with t+7d=2s+4, put P=e^2π it, Q=e^2π is, D=e^2π id and Z=e^2π iz. Then PD^7=Q^2. In a common convergence region the fifth-smoothed logarithms are log S_5(z;a,b) · =5∑_substackn≥1 5nmid n fracZ^n-(A_0B_0/Z)^nn(1-A_0^n)(1-B_0^n), log F_t,s(z) · =-5∑_substackn≥1 5nmid n fracP^nZ^n-Q^nZ^-nn(1-P^n)(1-Q^n), where A_0=e^2π ia,B_0=e^2π ib in the first line. At one surviving frequency suppress the index n. Summing seven shifts multipl…
Exact mathematical passage
\log S_5(z;a,b)&=5\sum_{\substack{n\ge1\\5\nmid n}} \frac{Z^n-(A_0B_0/Z)^n}{n(1-A_0^n)(1-B_0^n)},\\ \log F_{t,s}(z)&=-5\sum_{\substack{n\ge1\\5\nmid n}} \frac{P^nZ^n-Q^nZ^{-n}}{n(1-P^n)(1-Q^n)},
a_1Use in source · 12 passages
Search aliases: a_1
Use in source · polarization.tex, line 37
a_0=(1+ω)ξ^2-ξ+1, a_1=-ξ^2+(1+ω)ξ+1, a_2=-ωξ^2-ωξ+1.
Exact mathematical passage
a_0=(1+\omega)\xi^2-\xi+1,\quad a_1=-\xi^2+(1+\omega)\xi+1,\quad a_2=-\omega\xi^2-\omega\xi+1.
a_1^2Use in source · 3 passages
Search aliases: a_1^2
Use in source · polarization.tex, line 117
A=a_0a_1^3=fraca_1^2a_2, B=a_0^2a_1^4=(fraca_1a_2)^2.
Exact mathematical passage
\mathcal A=a_0a_1^3=\frac{a_1^2}{a_2},\qquad \mathcal B=a_0^2a_1^4=\left(\frac{a_1}{a_2}\right)^2.
a_1^3Use in source · 1 passage
Search aliases: a_1^3
Use in source · polarization.tex, line 117
A=a_0a_1^3=fraca_1^2a_2, B=a_0^2a_1^4=(fraca_1a_2)^2.
Exact mathematical passage
\mathcal A=a_0a_1^3=\frac{a_1^2}{a_2},\qquad \mathcal B=a_0^2a_1^4=\left(\frac{a_1}{a_2}\right)^2.
a_1^4Use in source · 1 passage
Search aliases: a_1^4
Use in source · polarization.tex, line 117
A=a_0a_1^3=fraca_1^2a_2, B=a_0^2a_1^4=(fraca_1a_2)^2.
Exact mathematical passage
\mathcal A=a_0a_1^3=\frac{a_1^2}{a_2},\qquad \mathcal B=a_0^2a_1^4=\left(\frac{a_1}{a_2}\right)^2.
a_1^5Use in source · 4 passages
Search aliases: a_1^5
Use in source · polarization.tex, line 269
|U_Γ|^3=|ϵ^3a_2a_1^5|.
Exact mathematical passage
|U_\Gamma|^3=|\epsilon^3a_2a_1^5|.
a_2Use in source · 15 passages
Search aliases: a_2
Use in source · polarization.tex, line 37
a_0=(1+ω)ξ^2-ξ+1, a_1=-ξ^2+(1+ω)ξ+1, a_2=-ωξ^2-ωξ+1.
Exact mathematical passage
a_0=(1+\omega)\xi^2-\xi+1,\quad a_1=-\xi^2+(1+\omega)\xi+1,\quad a_2=-\omega\xi^2-\omega\xi+1.
a_2^3Use in source · 2 passages
Search aliases: a_2^3
Use in source · polarization.tex, line 230
C=fracϵ a_1a_2^3ω
Exact mathematical passage
\boxed{C=\frac{\epsilon a_1a_2^3}{\omega}}
a_{h+1}Use in source · 1 passage
Search aliases: a_h+1
Use in source · stark.tex, line 520
log|σ^h W|=(-1)^h(frac R2+2log|a_h+1|) =-3ζ_ f'(σ^h+5,0). tagA6.3
Exact mathematical passage
\log|\sigma^h W|=(-1)^h\left(\frac{\mathcal R}{2}+2\log|a_{h+1}|\right) =-3\zeta_{\mathfrak f}'(\sigma^{h+5},0). \tag{A6.3}
a_{h+1}^2Use in source · 1 passage
Search aliases: a_h+1^2
Use in source · stark.tex, line 499
N_κ(σ^hW) =N_κ(ϵ a_h+1^2)^(-1)^h, 0≤ h<6. tagA6.2
Exact mathematical passage
\boxed{N_\kappa(\sigma^hW) =N_\kappa(\epsilon a_{h+1}^2)^{(-1)^h},\quad 0\le h<6.} \tag{A6.2}
a_iUse in source · 1 passage
Search aliases: a_i
Use in source · stark.tex, line 444
a_0+a_1+a_2=3, ∑_i<ja_i a_j=6ω, a_0a_1a_2=1.
Exact mathematical passage
a_0+a_1+a_2=3,\qquad \sum_{i<j}a_i a_j=6\omega,\qquad a_0a_1a_2=1.
a_jUse in source · 4 passages
Search aliases: a_j
Use in source · polarization.tex, line 45
a_0a_1a_2=1, a_j^3-3a_j^2+6ω a_j-1=0.
Exact mathematical passage
a_0a_1a_2=1,\qquad a_j^3-3a_j^2+6\omega a_j-1=0.
a_{j+1}Use in source · 1 passage
Search aliases: a_j+1
Use in source · polarization.tex, line 74
κ(a_j)=ι(a_j+1), jpmod3. tagQ1
Exact mathematical passage
\boxed{\kappa(a_j)=\iota(a_{j+1})},\qquad j\pmod3. \tag{Q1}
a_j^2Use in source · 1 passage
Search aliases: a_j^2
Use in source · polarization.tex, line 45
a_0a_1a_2=1, a_j^3-3a_j^2+6ω a_j-1=0.
Exact mathematical passage
a_0a_1a_2=1,\qquad a_j^3-3a_j^2+6\omega a_j-1=0.
a_j^3Use in source · 1 passage
Search aliases: a_j^3
Use in source · polarization.tex, line 45
a_0a_1a_2=1, a_j^3-3a_j^2+6ω a_j-1=0.
Exact mathematical passage
a_0a_1a_2=1,\qquad a_j^3-3a_j^2+6\omega a_j-1=0.
a_{k-1}Use in source · 1 passage
Search aliases: a_k-1
Use in source · stark.tex, line 467
ζ_ f'(c^e s_0^k,0) =(-1)^e(frac R6+frac23log|a_k-1|). tagA5.10
Exact mathematical passage
\boxed{\zeta_{\mathfrak f}'(c^e s_0^k,0) =(-1)^e\left(\frac{\mathcal R}{6}+\frac23\log|a_{k-1}|\right).} \tag{A5.10}
a_mUse in source · 4 passages
Search aliases: a_m
Use in source · stark.tex, line 425
V_m^3=|a_m|, tagA5.6
Exact mathematical passage
\boxed{V_m^3=|a_m|,} \tag{A5.6}
a_m^2Use in source · 1 passage
Search aliases: a_m^2
Use in source · stark.tex, line 451
a_m^3-3a_m^2+6ω a_m-1=0. tagA5.8
Exact mathematical passage
a_m^3-3a_m^2+6\omega a_m-1=0. \tag{A5.8}
a_m^3Use in source · 1 passage
Search aliases: a_m^3
Use in source · stark.tex, line 451
a_m^3-3a_m^2+6ω a_m-1=0. tagA5.8
Exact mathematical passage
a_m^3-3a_m^2+6\omega a_m-1=0. \tag{A5.8}
a_sUse in source · 1 passage
Search aliases: a_s
Use in source · theta.tex, line 368
a_s=ρ^2a_t, b_s=ρ^2b_t, c_s=ρ^4c_t,
Exact mathematical passage
a_s=\rho^2a_t,\quad b_s=\rho^2b_t,\quad c_s=\rho^4c_t,
a_tUse in source · 1 passage
Search aliases: a_t
Use in source · theta.tex, line 368
a_s=ρ^2a_t, b_s=ρ^2b_t, c_s=ρ^4c_t,
Exact mathematical passage
a_s=\rho^2a_t,\quad b_s=\rho^2b_t,\quad c_s=\rho^4c_t,
A_vUse in source · 2 passages
Search aliases: A_v
Use in source · theta.tex, line 397
F(ρ w_v)=F(w_Tv)B_v, B_v=∏_j=0^A_v-1q_t(w_T^2v+js), A_v=5+4b-2c+7a.
Exact mathematical passage
F(\rho w_v)=F(w_{Tv})B_v,\qquad B_v=\prod_{j=0}^{A_v-1}q_t(w_{T^2v}+js), \quad A_v=5+4b-2c+7a.
A_WUse in source · 2 passages
Search aliases: A_W
Use in source · stark.tex, line 485
W=(A_W+ω B_W)/3,
Exact mathematical passage
W=(A_W+\omega B_W)/3,
A_\ellUse in source · 1 passage
Search aliases: A_ℓ ell
Use in source · reconstruction.tex, line 50
The precise finite-theta obstruction. [Fixed affine finite-theta class] Fix finitely many symmetric period matrices Ω_ℓ with positive-definite imaginary parts ImΩ_ℓ, affine maps z↦ A_ℓ z+b_ℓ, and finite derivatives of their Riemann theta restrictions. Let P,Q be nonzero entire functions formed by finite polynomial expressions in these restrictions, derivatives and polynomials in z. Let u be a single global nowhere-zero entire function. Every meromorphic function u(z)fracP(z)Q(z) has zero and pol…
Exact mathematical passage
z\mapsto A_\ell z+b_\ell
BUse in source · 23 passages
Search aliases: B
Use in source · norms.tex, line 34
ks)^4, · 5nmid n, n=2k. For odd n, doubling permutes E_t[n]. For n=2k, it maps E_t[n] to E_t[k] with four lifts per point, so each denominator norm is a fourth power. Apply [source reference] and cancel the numerator. If 5mid n then all these norms are one, including the even case since 5mid k. For n=2k+1 prime to five, B_k(w)=∏_j=0^k-1q_t(2w+(2j+1)s) gives N_2k+1ψ(w)=ψ(w)fracB_k(w)B_k(w+s). For n=2k prime to five, C_k(w)=∏_j=0^k-1q_t(w+js)^4 gives N_2kψ(w)=q_t(w)^-6fracC_k(w)C_k(w+s). Finite pr…
Exact mathematical passage
B_k(w)=\prod_{j=0}^{k-1}q_t(2w+(2j+1)s)
\mathcal BUse in source · 3 passages
Search aliases: B mathcal
Use in source · polarization.tex, line 117
A=a_0a_1^3=fraca_1^2a_2, B=a_0^2a_1^4=(fraca_1a_2)^2.
Exact mathematical passage
\mathcal A=a_0a_1^3=\frac{a_1^2}{a_2},\qquad \mathcal B=a_0^2a_1^4=\left(\frac{a_1}{a_2}\right)^2.
\mathfrak bUse in source · 1 passage
Search aliases: b mathfrak
Use in source · stark.tex, line 632
|Γ_ a,h( f b^-1)|^6 =|(σ_ bW_rm St)^N( a)-σ_ a|. tagA7.4
Exact mathematical passage
|\mathbf\Gamma_{\mathfrak a,h}(\mathfrak f\mathfrak b^{-1})|^6 =\left|(\sigma_{\mathfrak b}W_{\rm St})^{N(\mathfrak a)-\sigma_{\mathfrak a}}\right|. \tag{A7.4}
bUse in source · 67 passages
Search aliases: b
Use in source · H12_Cubic_Divisor_Traditional.tex, line 72
For positive imaginary periods use the additive product convention θ_0(z;a) · =∏_n≥0(1-e^2π i(z+na))(1-e^2π i((n+1)a-z)), Γ_rm ell(z;a,b) · =∏_j,k≥0 frac1-e^2π i((j+1)a+(k+1)b-z)1-e^2π i(ja+kb+z). The negative-first-period continuation is Γ_rm ell(z;-t,s)=Γ_rm ell(z+t;t,s)^-1. Consequently the main scalar function is F(z)=fracΓ_rm ell(z;τ,s)^5Γ_rm ell(5z;5τ,5s) =fracΓ_rm ell(5(z+t);5t,5s)Γ_rm ell(z+t;t,s)^5, U_Γ=F(1/3). This convention agrees with the checked primary authors' copy of Bergeron--C…
Exact mathematical passage
\theta_0(z;a)&=\prod_{n\ge0}(1-e^{2\pi i(z+na)})(1-e^{2\pi i((n+1)a-z)}),\label{eq:theta-product}\\ \Gamma_{\rm ell}(z;a,b)&=\prod_{j,k\ge0} \frac{1-e^{2\pi i((j+1)a+(k+1)b-z)}}{1-e^{2\pi i(ja+kb+z)}}.\label{eq:gamma-product}
\mathfrak b^{-1}Use in source · 1 passage
Search aliases: b^-1 mathfrak
Use in source · stark.tex, line 632
|Γ_ a,h( f b^-1)|^6 =|(σ_ bW_rm St)^N( a)-σ_ a|. tagA7.4
Exact mathematical passage
|\mathbf\Gamma_{\mathfrak a,h}(\mathfrak f\mathfrak b^{-1})|^6 =\left|(\sigma_{\mathfrak b}W_{\rm St})^{N(\mathfrak a)-\sigma_{\mathfrak a}}\right|. \tag{A7.4}
B^3Use in source · 1 passage
Search aliases: B^3
Use in source · rigidity.tex, line 53
ta_Mr, then h=δ_Mg for a smooth mean-zero g, with convergent formula g=-frac17∑_j≥1L_N^jr. If h,r extend holomorphically to all of X, so does g. Apply L_N/7 to [source reference]. The trace relation gives L_Nh-h=(1/7)δ_M L_Nr, and induction yields L_N^nh-h=frac17δ_M∑_j=1^nL_N^jr. Because B^3=7I, L_N^3nf=∑_k≠0343^nc_7^nke_k for a mean-zero f. For any C^q norm choose s>max(q,3). Smooth Fourier coefficients have finite weighted ℓ^1 norms of every order, and the displayed expression is bounded, up t…
Exact mathematical passage
B^3=7I
B_0Use in source · 2 passages
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Use in source · product_proof.tex, line 3
Full seven-product series proof. For positive imaginary periods t,s,d with t+7d=2s+4, put P=e^2π it, Q=e^2π is, D=e^2π id and Z=e^2π iz. Then PD^7=Q^2. In a common convergence region the fifth-smoothed logarithms are log S_5(z;a,b) · =5∑_substackn≥1 5nmid n fracZ^n-(A_0B_0/Z)^nn(1-A_0^n)(1-B_0^n), log F_t,s(z) · =-5∑_substackn≥1 5nmid n fracP^nZ^n-Q^nZ^-nn(1-P^n)(1-Q^n), where A_0=e^2π ia,B_0=e^2π ib in the first line. At one surviving frequency suppress the index n. Summing seven shifts multipl…
Exact mathematical passage
\log S_5(z;a,b)&=5\sum_{\substack{n\ge1\\5\nmid n}} \frac{Z^n-(A_0B_0/Z)^n}{n(1-A_0^n)(1-B_0^n)},\\ \log F_{t,s}(z)&=-5\sum_{\substack{n\ge1\\5\nmid n}} \frac{P^nZ^n-Q^nZ^{-n}}{n(1-P^n)(1-Q^n)},
B_0^nUse in source · 1 passage
Search aliases: B_0^n
Use in source · product_proof.tex, line 3
Full seven-product series proof. For positive imaginary periods t,s,d with t+7d=2s+4, put P=e^2π it, Q=e^2π is, D=e^2π id and Z=e^2π iz. Then PD^7=Q^2. In a common convergence region the fifth-smoothed logarithms are log S_5(z;a,b) · =5∑_substackn≥1 5nmid n fracZ^n-(A_0B_0/Z)^nn(1-A_0^n)(1-B_0^n), log F_t,s(z) · =-5∑_substackn≥1 5nmid n fracP^nZ^n-Q^nZ^-nn(1-P^n)(1-Q^n), where A_0=e^2π ia,B_0=e^2π ib in the first line. At one surviving frequency suppress the index n. Summing seven shifts multipl…
Exact mathematical passage
\log S_5(z;a,b)&=5\sum_{\substack{n\ge1\\5\nmid n}} \frac{Z^n-(A_0B_0/Z)^n}{n(1-A_0^n)(1-B_0^n)},\\ \log F_{t,s}(z)&=-5\sum_{\substack{n\ge1\\5\nmid n}} \frac{P^nZ^n-Q^nZ^{-n}}{n(1-P^n)(1-Q^n)},
B_2Use in source · 4 passages
Search aliases: B_2
Use in source · stark.tex, line 304
H(z)=e^-πIm(ω)B_2(a) |1-e^2π iz| ∏_n≥1|(1-e^2π i(nω+z))(1-e^2π i(nω-z))|, tagA4.1
Exact mathematical passage
H(z)=e^{-\pi\operatorname{Im}(\omega)B_2(a)} |1-e^{2\pi iz}| \prod_{n\ge1}|(1-e^{2\pi i(n\omega+z)})(1-e^{2\pi i(n\omega-z)})|, \tag{A4.1}
b_2Use in source · 2 passages
Search aliases: b_2
Use in source · theta.tex, line 331
Writing b_2=1-6u+u^2, b_4=u^2-u, the derivative and curve equation imply
Exact mathematical passage
b_2=1-6u+u^2
b_2^2Use in source · 1 passage
Search aliases: b_2^2
Use in source · theta.tex, line 334
g_2=λ^4(b_2^2/12-2b_4), g_3=λ^6(-b_2^3/216+b_2b_4/6-u^2).
Exact mathematical passage
g_2=\lambda^4(b_2^2/12-2b_4),\quad g_3=\lambda^6(-b_2^3/216+b_2b_4/6-u^2).
b_2^3Use in source · 1 passage
Search aliases: b_2^3
Use in source · theta.tex, line 334
g_2=λ^4(b_2^2/12-2b_4), g_3=λ^6(-b_2^3/216+b_2b_4/6-u^2).
Exact mathematical passage
g_2=\lambda^4(b_2^2/12-2b_4),\quad g_3=\lambda^6(-b_2^3/216+b_2b_4/6-u^2).
b_4Use in source · 2 passages
Search aliases: b_4
Use in source · theta.tex, line 331
Writing b_2=1-6u+u^2, b_4=u^2-u, the derivative and curve equation imply
Exact mathematical passage
b_4=u^2-u
B_DUse in source · 3 passages
Search aliases: B_D
Use in source · reconstruction.tex, line 166
The period-one transition fixes the cubic coefficient. The points 0,1,1/3 are regular and nonzero for R and W_D. For 1, [source reference] has nonzero factors at zero. For 1/3, β/3=-5s-2/3, whose rational coordinate is not a rational fifth; rational independence excludes a singular lattice point. Define B_D(1)=(log W_D)''(1) =∑_λ∈ Dm_λ[-frac1(λ-1)^2 +frac1λ^2+frac2λ^3]. The combined summand is O(|λ|^-4), so this is an absolutely convergent sum. Its terms must remain combined; the separately dive…
Exact mathematical passage
\label{eq:BD} B_D(1)=(\log W_D)''(1) =\sum_{\lambda\in D}m_\lambda\left[-\frac1{(\lambda-1)^2} +\frac1{\lambda^2}+\frac2{\lambda^3}\right].
B_kUse in source · 2 passages
Search aliases: B_k
Use in source · norms.tex, line 34
ks)^4, · 5nmid n, n=2k. For odd n, doubling permutes E_t[n]. For n=2k, it maps E_t[n] to E_t[k] with four lifts per point, so each denominator norm is a fourth power. Apply [source reference] and cancel the numerator. If 5mid n then all these norms are one, including the even case since 5mid k. For n=2k+1 prime to five, B_k(w)=∏_j=0^k-1q_t(2w+(2j+1)s) gives N_2k+1ψ(w)=ψ(w)fracB_k(w)B_k(w+s). For n=2k prime to five, C_k(w)=∏_j=0^k-1q_t(w+js)^4 gives N_2kψ(w)=q_t(w)^-6fracC_k(w)C_k(w+s). Finite pr…
Exact mathematical passage
B_k(w)=\prod_{j=0}^{k-1}q_t(2w+(2j+1)s)
b_sUse in source · 1 passage
Search aliases: b_s
Use in source · theta.tex, line 368
a_s=ρ^2a_t, b_s=ρ^2b_t, c_s=ρ^4c_t,
Exact mathematical passage
a_s=\rho^2a_t,\quad b_s=\rho^2b_t,\quad c_s=\rho^4c_t,
b_tUse in source · 1 passage
Search aliases: b_t
Use in source · theta.tex, line 368
a_s=ρ^2a_t, b_s=ρ^2b_t, c_s=ρ^4c_t,
Exact mathematical passage
a_s=\rho^2a_t,\quad b_s=\rho^2b_t,\quad c_s=\rho^4c_t,
B_vUse in source · 2 passages
Search aliases: B_v
Use in source · theta.tex, line 397
F(ρ w_v)=F(w_Tv)B_v, B_v=∏_j=0^A_v-1q_t(w_T^2v+js), A_v=5+4b-2c+7a.
Exact mathematical passage
F(\rho w_v)=F(w_{Tv})B_v,\qquad B_v=\prod_{j=0}^{A_v-1}q_t(w_{T^2v}+js), \quad A_v=5+4b-2c+7a.
B_WUse in source · 2 passages
Search aliases: B_W
Use in source · stark.tex, line 485
W=(A_W+ω B_W)/3,
Exact mathematical passage
W=(A_W+\omega B_W)/3,
b_\ellUse in source · 1 passage
Search aliases: b_ℓ ell
Use in source · reconstruction.tex, line 50
The precise finite-theta obstruction. [Fixed affine finite-theta class] Fix finitely many symmetric period matrices Ω_ℓ with positive-definite imaginary parts ImΩ_ℓ, affine maps z↦ A_ℓ z+b_ℓ, and finite derivatives of their Riemann theta restrictions. Let P,Q be nonzero entire functions formed by finite polynomial expressions in these restrictions, derivatives and polynomials in z. Let u be a single global nowhere-zero entire function. Every meromorphic function u(z)fracP(z)Q(z) has zero and pol…
Exact mathematical passage
z\mapsto A_\ell z+b_\ell
\bar JUse in source · 4 passages
Search aliases: bar J bar
Use in source · stark.tex, line 9
ω=e^2π i/3, β=7^1/3ω^2, J=2+3ω, bar J=-1-3ω, Jbar J=7.
Exact mathematical passage
\omega=e^{2\pi i/3},\quad \beta=7^{1/3}\omega^2,\quad J=2+3\omega,\quad \bar J=-1-3\omega,\quad J\bar J=7.
\bar sUse in source · 2 passages
Search aliases: bar s bar
Use in source · polarization.tex, line 348
overlineF_t,s(z)=F_-bar t,-bar s(-bar z).
Exact mathematical passage
\overline{F_{t,s}(z)}=F_{-\bar t,-\bar s}(-\bar z).
\bar tUse in source · 2 passages
Search aliases: bar t bar
Use in source · polarization.tex, line 348
overlineF_t,s(z)=F_-bar t,-bar s(-bar z).
Exact mathematical passage
\overline{F_{t,s}(z)}=F_{-\bar t,-\bar s}(-\bar z).
\bar zUse in source · 1 passage
Search aliases: bar z bar
Use in source · polarization.tex, line 348
overlineF_t,s(z)=F_-bar t,-bar s(-bar z).
Exact mathematical passage
\overline{F_{t,s}(z)}=F_{-\bar t,-\bar s}(-\bar z).
\bar\betaUse in source · 1 passage
Search aliases: barβ bar beta
Use in source · stark.tex, line 88
For the last formula, the conjugate Gauss resolvent is 7/G, and barβ=ω^2β. Formula (A1.5) follows directly. It must not be replaced by the involution σ^3, which fixes K.
Exact mathematical passage
\bar\beta=\omega^2\beta
\bmodUse in source · 1 passage
Search aliases: bmod
Use in source · norms.tex, line 16
The point S=sbmodΛ_t has infinite order by rational independence. Put ψ(w)=fracq_t(w)^2q_t(2w)q_t(2w+s). [All finite norms of the return kernel] For every integer n≥1, N_nψ(w)= 1, · 5mid n,
Exact mathematical passage
S=s\bmod\Lambda_t
CUse in source · 28 passages
Search aliases: C
Use in source · norms.tex, line 38
maps E_t[n] to E_t[k] with four lifts per point, so each denominator norm is a fourth power. Apply [source reference] and cancel the numerator. If 5mid n then all these norms are one, including the even case since 5mid k. For n=2k+1 prime to five, B_k(w)=∏_j=0^k-1q_t(2w+(2j+1)s) gives N_2k+1ψ(w)=ψ(w)fracB_k(w)B_k(w+s). For n=2k prime to five, C_k(w)=∏_j=0^k-1q_t(w+js)^4 gives N_2kψ(w)=q_t(w)^-6fracC_k(w)C_k(w+s). Finite products therefore retain explicit endpoint multipliers. Closing a permutati…
Exact mathematical passage
C_k(w)=\prod_{j=0}^{k-1}q_t(w+js)^4
\mathbb CUse in source · 7 passages
Search aliases: C mathbb
Use in source · history.tex, line 28
Three geometries must stay distinct. The scalar plane carries the meromorphic correction and its canonical divisor product. The intermediate cover X= C^3/ Z^3 carries periodic multipliers and a degree-seven principal cover. The compact CM threefold has lattice Z[ω]^3 and degree 49 for the same matrix. Finite projective representations or oriented torsion sectors can encode cocycle data in the compact model. They do not alone provide a scalar meromorphic section with the divisor of R, a periodic …
Exact mathematical passage
X=\mathbb C^3/\mathbb Z^3
cUse in source · 59 passages
Search aliases: c
Use in source · history.tex, line 25
The scalar missing from a closed finite orbit. The earlier index-23 and index-10 orbit decompositions are 529=1+528 and 100=1+3+8·12. They are exact finite permutation data. For values z_j on a cycle of length d, replacing every value by cz_j leaves all adjacent ratios unchanged, including their product one, but multiplies the cycle product by c^d. An anchor on another fixed-point orbit does not remove this freedom. Thus orbit closure alone does not evaluate a primitive analytic sector or fix it…
Exact mathematical passage
cz_j
C'Use in source · 1 passage
Search aliases: C'
Use in source · reconstruction.tex, line 61
on. Every meromorphic function u(z)fracP(z)Q(z) has zero and pole counts O(r^2). Thus R is not representable in this class. For Y=ImΩ>0, the defining theta series has terms bounded by exp{-π n^TYn+2π n Im(Az+b) }. Completing the square, and absorbing half of the positive quadratic form into a convergent Gaussian sum, gives Cexp(C'|z|^2) for a fixed affine restriction. Cauchy estimates on fixed-radius circles give the same form for each fixed derivative. Finite sums, products and polynomial facto…
Exact mathematical passage
C\exp(C'|z|^2)
\mathbb C^*Use in source · 1 passage
Search aliases: C^* mathbb
Use in source · rigidity.tex, line 5
The actual two commuting actions. On X= C^3/ Z^3≃( C^*)^3 and its real torus T^3, use M=0 · 1 · 0
Exact mathematical passage
X=\mathbb C^3/\mathbb Z^3\simeq(\mathbb C^*)^3
C^3Use in source · 1 passage
Search aliases: C^3
Use in source · stark.tex, line 422
Multiplication and (A4.3) give C^3|X^-3+1|=1. Cubing each equation in (A5.5) now removes the unknown constant exactly:
Exact mathematical passage
C^3|X^{-3}+1|=1
\mathbb C^3Use in source · 2 passages
Search aliases: C^3 mathbb
Use in source · history.tex, line 28
Three geometries must stay distinct. The scalar plane carries the meromorphic correction and its canonical divisor product. The intermediate cover X= C^3/ Z^3 carries periodic multipliers and a degree-seven principal cover. The compact CM threefold has lattice Z[ω]^3 and degree 49 for the same matrix. Finite projective representations or oriented torsion sectors can encode cocycle data in the compact model. They do not alone provide a scalar meromorphic section with the divisor of R, a periodic …
Exact mathematical passage
X=\mathbb C^3/\mathbb Z^3
c^3Use in source · 1 passage
Search aliases: c^3
Use in source · theta.tex, line 106
More generally, if c^3=A and c^8=B with c≠0, then
Exact mathematical passage
c^3=A
c^8Use in source · 1 passage
Search aliases: c^8
Use in source · theta.tex, line 106
More generally, if c^3=A and c^8=B with c≠0, then
Exact mathematical passage
c^8=B
c^dUse in source · 1 passage
Search aliases: c^d
Use in source · history.tex, line 25
The scalar missing from a closed finite orbit. The earlier index-23 and index-10 orbit decompositions are 529=1+528 and 100=1+3+8·12. They are exact finite permutation data. For values z_j on a cycle of length d, replacing every value by cz_j leaves all adjacent ratios unchanged, including their product one, but multiplies the cycle product by c^d. An anchor on another fixed-point orbit does not remove this freedom. Thus orbit closure alone does not evaluate a primitive analytic sector or fix it…
Exact mathematical passage
c^d
c^eUse in source · 2 passages
Search aliases: c^e
Use in source · stark.tex, line 274
ζ_ f'(c^e s_0^k,0) =(-1)^e(frac R6-ℓ_k-1). tagA3.5
Exact mathematical passage
\boxed{\zeta_{\mathfrak f}'(c^e s_0^k,0) =(-1)^e\left(\frac{\mathcal R}{6}-\ell_{k-1}\right).} \tag{A3.5}
C^qUse in source · 2 passages
Search aliases: C^q
Use in source · rigidity.tex, line 57
formula g=-frac17∑_j≥1L_N^jr. If h,r extend holomorphically to all of X, so does g. Apply L_N/7 to [source reference]. The trace relation gives L_Nh-h=(1/7)δ_M L_Nr, and induction yields L_N^nh-h=frac17δ_M∑_j=1^nL_N^jr. Because B^3=7I, L_N^3nf=∑_k≠0343^nc_7^nke_k for a mean-zero f. For any C^q norm choose s>max(q,3). Smooth Fourier coefficients have finite weighted ℓ^1 norms of every order, and the displayed expression is bounded, up to a derivative convention constant, by 7^n(3-s)∑_k≠0 k ^s|c_k…
Exact mathematical passage
C^q
C_{18}Use in source · 1 passage
Search aliases: C_18
Use in source · stark.tex, line 565
For t^6=W, the lift ϱ(t)=t^5/U_rm alg commutes with t↦ c_6t, and satisfies ϱ^6(t)=c_6^4t. Irreducibility is checked at the residue assignment (β,v,ω)=(32,99,48) over F_181, where W=23 and T^6-23 is irreducible. The earlier local-integrality argument validates this reduction. The resulting abelian group is C_2× C_18.
Exact mathematical passage
C_2\times C_{18}
C_2Use in source · 3 passages
Search aliases: C_2
Use in source · stark.tex, line 132
Ind_C_2^S_3(sgn) =sgn⊕Std, Std=Ind_C_3^S_3(ϱ)
Exact mathematical passage
\operatorname{Ind}_{C_2}^{S_3}(\operatorname{sgn}) =\operatorname{sgn}\oplus\operatorname{Std}, \qquad \operatorname{Std}=\operatorname{Ind}_{C_3}^{S_3}(\varrho)
C_3Use in source · 3 passages
Search aliases: C_3
Use in source · stark.tex, line 100
Gal(E/ Q)=S_3× C_3. tagA2.1
Exact mathematical passage
\operatorname{Gal}(E/\mathbb Q)=S_3\times C_3. \tag{A2.1}
C_3^2Use in source · 2 passages
Search aliases: C_3^2
Use in source · polarization.tex, line 25
E= Q(ω,β,ξ), F_0= Q(ω), Gal(E/F_0)=C_3^2.
Exact mathematical passage
E=\mathbb Q(\omega,\beta,\xi),\quad F_0=\mathbb Q(\omega), \qquad \operatorname{Gal}(E/F_0)=C_3^2.
c_6Use in source · 3 passages
Search aliases: c_6
Use in source · stark.tex, line 554
and, for c_6=-ω^2,
Exact mathematical passage
c_6=-\omega^2
c_6^4Use in source · 1 passage
Search aliases: c_6^4
Use in source · stark.tex, line 561
For t^6=W, the lift ϱ(t)=t^5/U_rm alg commutes with t↦ c_6t, and satisfies ϱ^6(t)=c_6^4t. Irreducibility is checked at the residue assignment (β,v,ω)=(32,99,48) over F_181, where W=23 and T^6-23 is irreducible. The earlier local-integrality argument validates this reduction. The resulting abelian group is C_2× C_18.
Exact mathematical passage
\varrho^6(t)=c_6^4t
c_{7^nk}Use in source · 1 passage
Search aliases: c_7^nk
Use in source · rigidity.tex, line 54
equation then h=δ_Mg for a smooth mean-zero g, with convergent formula g=-frac17∑_j≥1L_N^jr. If h,r extend holomorphically to all of X, so does g. Apply L_N/7 to [source reference]. The trace relation gives L_Nh-h=(1/7)δ_M L_Nr, and induction yields L_N^nh-h=frac17δ_M∑_j=1^nL_N^jr. Because B^3=7I, L_N^3nf=∑_k≠0343^nc_7^nke_k for a mean-zero f. For any C^q norm choose s>max(q,3). Smooth Fourier coefficients have finite weighted ℓ^1 norms of every order, and the displayed expression is bounded, up…
Exact mathematical passage
L_N^{3n}f=\sum_{k\ne0}343^nc_{7^nk}e_k
c_{Bk}Use in source · 1 passage
Search aliases: c_Bk
Use in source · rigidity.tex, line 30
The convergent transfer proof. For functions on T^3, let S_Nh(v)=∑_κ∈ker Nh(v+κ), C_Nh(v)=h(Nv), L_Nh(v)=∑_Nw=vh(w), δ_Mh=h-h∘ M. The transfer L_N is unnormalized. For h=∑ c_ke_k, S_Nh=∑_k∈ B Z^37c_ke_k, L_Nh=∑_k7c_Bke_k, S_Nh=(L_Nh)∘ N, L_NC_Nh=7h. Since M permutes ker N and commutes with N, the required operators commute with composition by M.
Exact mathematical passage
S_Nh=\sum_{k\in B\mathbb Z^3}7c_ke_k,\qquad L_Nh=\sum_k7c_{Bk}e_k,\qquad S_Nh=(L_Nh)\circ N, \qquad L_NC_Nh=7h.
C_kUse in source · 2 passages
Search aliases: C_k
Use in source · norms.tex, line 38
maps E_t[n] to E_t[k] with four lifts per point, so each denominator norm is a fourth power. Apply [source reference] and cancel the numerator. If 5mid n then all these norms are one, including the even case since 5mid k. For n=2k+1 prime to five, B_k(w)=∏_j=0^k-1q_t(2w+(2j+1)s) gives N_2k+1ψ(w)=ψ(w)fracB_k(w)B_k(w+s). For n=2k prime to five, C_k(w)=∏_j=0^k-1q_t(w+js)^4 gives N_2kψ(w)=q_t(w)^-6fracC_k(w)C_k(w+s). Finite products therefore retain explicit endpoint multipliers. Closing a permutati…
Exact mathematical passage
C_k(w)=\prod_{j=0}^{k-1}q_t(w+js)^4
c_kUse in source · 3 passages
Search aliases: c_k
Use in source · rigidity.tex, line 29
The convergent transfer proof. For functions on T^3, let S_Nh(v)=∑_κ∈ker Nh(v+κ), C_Nh(v)=h(Nv), L_Nh(v)=∑_Nw=vh(w), δ_Mh=h-h∘ M. The transfer L_N is unnormalized. For h=∑ c_ke_k, S_Nh=∑_k∈ B Z^37c_ke_k, L_Nh=∑_k7c_Bke_k, S_Nh=(L_Nh)∘ N, L_NC_Nh=7h. Since M permutes ker N and commutes with N, the required operators commute with composition by M.
Exact mathematical passage
h=\sum c_ke_k
C_NUse in source · 2 passages
Search aliases: C_N
Use in source · rigidity.tex, line 24
The convergent transfer proof. For functions on T^3, let S_Nh(v)=∑_κ∈ker Nh(v+κ), C_Nh(v)=h(Nv), L_Nh(v)=∑_Nw=vh(w), δ_Mh=h-h∘ M. The transfer L_N is unnormalized. For h=∑ c_ke_k, S_Nh=∑_k∈ B Z^37c_ke_k, L_Nh=∑_k7c_Bke_k, S_Nh=(L_Nh)∘ N, L_NC_Nh=7h. Since M permutes ker N and commutes with N, the required operators commute with composition by M.
Exact mathematical passage
S_Nh(v)=\sum_{\kappa\in\ker N}h(v+\kappa),\quad C_Nh(v)=h(Nv),\quad L_Nh(v)=\sum_{Nw=v}h(w),\quad \delta_Mh=h-h\circ M.
C_{\rm pol}Use in source · 4 passages
Search aliases: C_rm pol rm
Use in source · remaining.tex, line 3
The exact remaining arithmetic comparison. The fixed arithmetic presentation gives, with a_0,a_1,a_2 as in the polarized block, C_rm pol=fracϵ a_1a_2^3ω, C_rm polU_rm alg^3=R_+:=|ϵ|^4|a_1|^6|a_2|^4>0. The modulus theorem gives |C_rm polU_Γ^3|=R_+. The exact statement still missing is Im(C_rm polU_Γ^3)=0. Reality together with a valid positive-real-part enclosure would give C_rm polU_Γ^3=R_+, hence U_Γ^3=U_rm alg^3. A valid independent bound |U_Γ/U_rm alg-1|<1/2 would select the root U_Γ=U_rm alg…
Exact mathematical passage
C_{\rm pol}=\frac{\epsilon a_1a_2^3}{\omega},\qquad C_{\rm pol}U_{\rm alg}^3=R_+:=|\epsilon|^4|a_1|^6|a_2|^4>0.
c_sUse in source · 1 passage
Search aliases: c_s
Use in source · theta.tex, line 368
a_s=ρ^2a_t, b_s=ρ^2b_t, c_s=ρ^4c_t,
Exact mathematical passage
a_s=\rho^2a_t,\quad b_s=\rho^2b_t,\quad c_s=\rho^4c_t,
c_tUse in source · 7 passages
Search aliases: c_t
Use in source · theta.tex, line 51
For the smoothing denominator, multiplication by five permutes the real residues modulo n; the same two steps apply with period 5t. The numerator becomes θ_0(nz;t)^5 and the denominator becomes θ_0(5nz;5t). This proves (N1.1) with no omitted multiplier and without using the unknown scalar c_t.
Exact mathematical passage
c_t
c_t^{-36}Use in source · 1 passage
Search aliases: c_t^-36
Use in source · theta.tex, line 432
c_t^-36=d^-72. tagN8.2
Exact mathematical passage
c_t^{-36}=d^{-72}. \tag{N8.2}
c_t^3Use in source · 2 passages
Search aliases: c_t^3
Use in source · theta.tex, line 211
c_t^3=16∏_η≠0Q_5(x(η))=f(a)f(b)=d^6. tagN4.2
Exact mathematical passage
c_t^3=16\prod_{\eta\ne0}Q_5(x(\eta))=f(a)f(b)=d^6. \tag{N4.2}
c_t^8Use in source · 2 passages
Search aliases: c_t^8
Use in source · theta.tex, line 231
c_t^8=(ψ_3(a)ψ_3(b))^2=d^16. tagN4.3
Exact mathematical passage
c_t^8=(\psi_3(a)\psi_3(b))^2=d^{16}. \tag{N4.3}
\cosUse in source · 1 passage
Search aliases: cos
Use in source · stark.tex, line 35
f=(3,β-1), E=K(ω,θ), θ=2cos(6π/7).
Exact mathematical passage
\mathfrak f=(3,\beta-1),\qquad E=K(\omega,\theta),\qquad \theta=2\cos(6\pi/7).
DUse in source · 36 passages
Search aliases: D
Use in source · product_proof.tex, line 2
Full seven-product series proof. For positive imaginary periods t,s,d with t+7d=2s+4, put P=e^2π it, Q=e^2π is, D=e^2π id and Z=e^2π iz. Then PD^7=Q^2. In a common convergence region the fifth-smoothed logarithms are log S_5(z;a,b) · =5∑_substackn≥1
Exact mathematical passage
D=e^{2\pi id}
dUse in source · 29 passages
Search aliases: d
Use in source · history.tex, line 25
The scalar missing from a closed finite orbit. The earlier index-23 and index-10 orbit decompositions are 529=1+528 and 100=1+3+8·12. They are exact finite permutation data. For values z_j on a cycle of length d, replacing every value by cz_j leaves all adjacent ratios unchanged, including their product one, but multiplies the cycle product by c^d. An anchor on another fixed-point orbit does not remove this freedom. Thus orbit closure alone does not evaluate a primitive analytic sector or fix it…
Exact mathematical passage
d
d^{-72}Use in source · 1 passage
Search aliases: d^-72
Use in source · theta.tex, line 432
c_t^-36=d^-72. tagN8.2
Exact mathematical passage
c_t^{-36}=d^{-72}. \tag{N8.2}
D^{-k}Use in source · 2 passages
Search aliases: D^-k
Use in source · product_proof.tex, line 9
5nmid n fracP^nZ^n-Q^nZ^-nn(1-P^n)(1-Q^n), where A_0=e^2π ia,B_0=e^2π ib in the first line. At one surviving frequency suppress the index n. Summing seven shifts multiplies the two coefficients by ∑_k=0^6D^k and ∑_k=0^6D^-k. The exact identities are fracP∑_k=0^6D^k(1-P)(1-Q) · =frac1+Q(1-P)(1-D)-frac1(1-Q)(1-D),
Exact mathematical passage
\sum_{k=0}^{6}D^{-k}
d^{16}Use in source · 2 passages
Search aliases: d^16
Use in source · theta.tex, line 231
c_t^8=(ψ_3(a)ψ_3(b))^2=d^16. tagN4.3
Exact mathematical passage
c_t^8=(\psi_3(a)\psi_3(b))^2=d^{16}. \tag{N4.3}
d^{18}Use in source · 1 passage
Search aliases: d^18
Use in source · theta.tex, line 237
c_t=(c_t^3)^3/c_t^8=d^18/d^16=d^2. tagN4.4
Exact mathematical passage
\boxed{c_t=(c_t^3)^3/c_t^8=d^{18}/d^{16}=d^2.} \tag{N4.4}
d^2Use in source · 3 passages
Search aliases: d^2
Use in source · theta.tex, line 138
4y_ay_b=(ℓ^2-m^2)d^2=4d^3.
Exact mathematical passage
4y_ay_b=(\ell^2-m^2)d^2=4d^3.
d^3Use in source · 2 passages
Search aliases: d^3
Use in source · theta.tex, line 138
4y_ay_b=(ℓ^2-m^2)d^2=4d^3.
Exact mathematical passage
4y_ay_b=(\ell^2-m^2)d^2=4d^3.
d^4Use in source · 1 passage
Search aliases: d^4
Use in source · reconstruction.tex, line 120
beginequation R(z)=R_0exp(α_1z+α_2z^2+α_3z^3)W_D(z), α_j=frac(log R)^(j)(0)j!. The logarithmic derivatives are branch independent. The notation (log R)^(4) means the third derivative of R'/R, a single-valued meromorphic function. The sine partial fractions [source reference] give fracd^4dw^4log(1-e^2π iw) =-6∑_n∈ Z(w-n)^-4. The linear exponential factor relating 1-e^2π iw to sinπ w has zero fourth logarithmic derivative. Apply this identity to each factor of the defining double gamma products. T…
Exact mathematical passage
\frac{d^4}{dw^4}\log(1-e^{2\pi iw}) =-6\sum_{n\in\mathbb Z}(w-n)^{-4}.
d^6Use in source · 2 passages
Search aliases: d^6
Use in source · theta.tex, line 144
y_ay_b=d^3, f(a)f(b)=d^6. tagN3.2
Exact mathematical passage
\boxed{y_ay_b=d^3,\qquad f(a)f(b)=d^6.} \tag{N3.2}
D^7Use in source · 2 passages
Search aliases: D^7
Use in source · product_proof.tex, line 2
Full seven-product series proof. For positive imaginary periods t,s,d with t+7d=2s+4, put P=e^2π it, Q=e^2π is, D=e^2π id and Z=e^2π iz. Then PD^7=Q^2. In a common convergence region the fifth-smoothed logarithms are log S_5(z;a,b) · =5∑_substackn≥1
Exact mathematical passage
PD^7=Q^2
d^8Use in source · 1 passage
Search aliases: d^8
Use in source · theta.tex, line 180
ψ_3(a)ψ_3(b)=-d^8. tagN3.3
Exact mathematical passage
\boxed{\psi_3(a)\psi_3(b)=-d^8.} \tag{N3.3}
D^kUse in source · 2 passages
Search aliases: D^k
Use in source · product_proof.tex, line 9
5nmid n fracP^nZ^n-Q^nZ^-nn(1-P^n)(1-Q^n), where A_0=e^2π ia,B_0=e^2π ib in the first line. At one surviving frequency suppress the index n. Summing seven shifts multiplies the two coefficients by ∑_k=0^6D^k and ∑_k=0^6D^-k. The exact identities are fracP∑_k=0^6D^k(1-P)(1-Q) · =frac1+Q(1-P)(1-D)-frac1(1-Q)(1-D),
Exact mathematical passage
\sum_{k=0}^{6}D^k
D_0Use in source · 1 passage
Search aliases: D_0
Use in source · stark.tex, line 162
D_0=L(0,χ_-3)=1/3, D_1=L(0,χ_-3χ_7)=-2ω, D_2=-2ω^2. tagA2.3
Exact mathematical passage
D_0=L(0,\chi_{-3})=1/3,\qquad D_1=L(0,\chi_{-3}\chi_7)=-2\omega,\qquad D_2=-2\omega^2. \tag{A2.3}
D_1Use in source · 1 passage
Search aliases: D_1
Use in source · stark.tex, line 162
D_0=L(0,χ_-3)=1/3, D_1=L(0,χ_-3χ_7)=-2ω, D_2=-2ω^2. tagA2.3
Exact mathematical passage
D_0=L(0,\chi_{-3})=1/3,\qquad D_1=L(0,\chi_{-3}\chi_7)=-2\omega,\qquad D_2=-2\omega^2. \tag{A2.3}
D_2Use in source · 1 passage
Search aliases: D_2
Use in source · stark.tex, line 162
D_0=L(0,χ_-3)=1/3, D_1=L(0,χ_-3χ_7)=-2ω, D_2=-2ω^2. tagA2.3
Exact mathematical passage
D_0=L(0,\chi_{-3})=1/3,\qquad D_1=L(0,\chi_{-3}\chi_7)=-2\omega,\qquad D_2=-2\omega^2. \tag{A2.3}
\detUse in source · 3 passages
Search aliases: det
Use in source · principal.tex, line 60
Multiplication by β in the ordered basis (τ,s,1/5) is N_5=2 · -1 · 0 4 · 0 · -3 5 · 0 · -2, N_5^3=7I, det N_5=7, N_5^-1=frac170 · -2 · 3 -7 · -4 · 6 0 · -5 · 4. Its quotient generator is (1,2,20)/7, representing δ. Thus β^-1Λ_5=Λ_5+ Zδ. Write a potential singular point as λ(a,b,c)=fracaτ+bs+c/5β, p=lceilfrac3c-2b7rceil, k=7p+2b-3c∈{0,…,6}. Exactly one argument λ+kδ lies in Λ_5. Its integral coordinates are (widehat a,widehat b,widehat c)=(p,2p-a,20p+5b-8c). Indeed N_5(widehat a,widehat b,widehat…
Exact mathematical passage
N_5=\begin{pmatrix}2&-1&0\\4&0&-3\\5&0&-2\end{pmatrix},\qquad N_5^3=7I,\quad\det N_5=7,\quad N_5^{-1}=\frac17\begin{pmatrix}0&-2&3\\-7&-4&6\\0&-5&4\end{pmatrix}.
\operatorname{disc}Use in source · 1 passage
Search aliases: disc operatorname
Use in source · theta.tex, line 249
q_t(z)=frac(a-b)^2(x(z)-a)(x(z)-b) =fracdisc(Q_5)Q_5(x(z)). tagN5.1
Exact mathematical passage
\boxed{ q_t(z)=\frac{(a-b)^2}{(x(z)-a)(x(z)-b)} =\frac{\operatorname{disc}(Q_5)}{Q_5(x(z))}. } \tag{N5.1}
\operatorname{div}Use in source · 3 passages
Search aliases: div operatorname
Use in source · principal.tex, line 143
The conjugation divisor cannot be discarded. Put R^#(z)=overlineR(overline z). The cubic fields Q(β) and Q(overlineβ) are distinct: equality would put β/overlineβ=ω in a cubic field. Their intersection is Q. The support of divR lies in the first, and that of R^# in the second. Since τ is irrational (indeed β=2-1/(4+15τ)), ord_τ(R/R^#)=4. Thus this naive conjugation quotient is not a holomorphic unit. A constant polarization, root of unity or nowhere-zero holomorphic gauge cannot change this orde…
Exact mathematical passage
\operatorname{div}R
EUse in source · 29 passages
Search aliases: E
Use in source · H12_Cubic_Divisor_Traditional.tex, line 107
The arithmetic candidate is fixed independently. Let J=2+3ω and choose the Gauss branch X=β^-1∑_a=1^6χ_7(a)e^6π ia/7, (χ_7(1),…,χ_7(6))=(1,ω,ω^2,ω^2,ω,1), X^3=J. Then E= Q(ω,β,X) has degree eighteen. The retained arithmetic construction defines ν · =(X+β/X-β^2)/3, · v · =-β^2ν-2ν^2,
Exact mathematical passage
E=\mathbb Q(\omega,\beta,X)
\mathcal EUse in source · 5 passages
Search aliases: E mathcal
Use in source · stark.tex, line 317
E_z(s)=∑_m,n∈ Z|m+nω+z|^-2s,
Exact mathematical passage
\mathcal E_z(s)=\sum_{m,n\in\mathbb Z}|m+n\omega+z|^{-2s},
e^{-\alpha_2-\alpha_3}Use in source · 1 passage
Search aliases: e^-α_2-α_3 alpha alpha
Use in source · reconstruction.tex, line 180
gent lower-power sums are not substituted. Taking two logarithmic derivatives of [source reference] at the transition [source reference] gives (log K)''(0)=6α_3+B_D(1). Since L_t is even, α_3=-frac16[β^2∑_r=1^15L_t(rs)+B_D(1)]. Evaluating the period-one ratio also determines e^α_1=fracK(0)e^-α_2-α_3W_D(1). This evaluates the exponential, without choosing an unjustified principal logarithm.
Exact mathematical passage
\label{eq:expalpha1} e^{\alpha_1}=\frac{K(0)e^{-\alpha_2-\alpha_3}}{W_D(1)}.
e^{-\pi(\Im z)^2/y}Use in source · 1 passage
Search aliases: e^-π(Im z)^2/y pi Im
Use in source · stark.tex, line 677
H_τ(z)=e^-π(Im z)^2/y |fracθ_1(π z;e^π iτ)η_rm Ded(τ)|.
Exact mathematical passage
H_\tau(z)=e^{-\pi(\Im z)^2/y} \left|\frac{\theta_1(\pi z;e^{\pi i\tau})}{\eta_{\rm Ded}(\tau)}\right|.
e^{-\pi\operatorname{Im}(\omega)B_2(a)}Use in source · 1 passage
Search aliases: e^-πIm(ω)B_2(a) pi operatorname omega
Use in source · stark.tex, line 304
H(z)=e^-πIm(ω)B_2(a) |1-e^2π iz| ∏_n≥1|(1-e^2π i(nω+z))(1-e^2π i(nω-z))|, tagA4.1
Exact mathematical passage
H(z)=e^{-\pi\operatorname{Im}(\omega)B_2(a)} |1-e^{2\pi iz}| \prod_{n\ge1}|(1-e^{2\pi i(n\omega+z)})(1-e^{2\pi i(n\omega-z)})|, \tag{A4.1}
e^{2\pi i((j+1)a+(k+1)b-z)}Use in source · 1 passage
Search aliases: e^2π i((j+1)a+(k+1)b-z) pi
Use in source · H12_Cubic_Divisor_Traditional.tex, line 72
For positive imaginary periods use the additive product convention θ_0(z;a) · =∏_n≥0(1-e^2π i(z+na))(1-e^2π i((n+1)a-z)), Γ_rm ell(z;a,b) · =∏_j,k≥0 frac1-e^2π i((j+1)a+(k+1)b-z)1-e^2π i(ja+kb+z). The negative-first-period continuation is Γ_rm ell(z;-t,s)=Γ_rm ell(z+t;t,s)^-1. Consequently the main scalar function is F(z)=fracΓ_rm ell(z;τ,s)^5Γ_rm ell(5z;5τ,5s) =fracΓ_rm ell(5(z+t);5t,5s)Γ_rm ell(z+t;t,s)^5, U_Γ=F(1/3). This convention agrees with the checked primary authors' copy of Bergeron--C…
Exact mathematical passage
\theta_0(z;a)&=\prod_{n\ge0}(1-e^{2\pi i(z+na)})(1-e^{2\pi i((n+1)a-z)}),\label{eq:theta-product}\\ \Gamma_{\rm ell}(z;a,b)&=\prod_{j,k\ge0} \frac{1-e^{2\pi i((j+1)a+(k+1)b-z)}}{1-e^{2\pi i(ja+kb+z)}}.\label{eq:gamma-product}
e^{2\pi i((n+1)a-z)}Use in source · 1 passage
Search aliases: e^2π i((n+1)a-z) pi
Use in source · H12_Cubic_Divisor_Traditional.tex, line 72
For positive imaginary periods use the additive product convention θ_0(z;a) · =∏_n≥0(1-e^2π i(z+na))(1-e^2π i((n+1)a-z)), Γ_rm ell(z;a,b) · =∏_j,k≥0 frac1-e^2π i((j+1)a+(k+1)b-z)1-e^2π i(ja+kb+z). The negative-first-period continuation is Γ_rm ell(z;-t,s)=Γ_rm ell(z+t;t,s)^-1. Consequently the main scalar function is F(z)=fracΓ_rm ell(z;τ,s)^5Γ_rm ell(5z;5τ,5s) =fracΓ_rm ell(5(z+t);5t,5s)Γ_rm ell(z+t;t,s)^5, U_Γ=F(1/3). This convention agrees with the checked primary authors' copy of Bergeron--C…
Exact mathematical passage
\theta_0(z;a)&=\prod_{n\ge0}(1-e^{2\pi i(z+na)})(1-e^{2\pi i((n+1)a-z)}),\label{eq:theta-product}\\ \Gamma_{\rm ell}(z;a,b)&=\prod_{j,k\ge0} \frac{1-e^{2\pi i((j+1)a+(k+1)b-z)}}{1-e^{2\pi i(ja+kb+z)}}.\label{eq:gamma-product}
e^{2\pi i((n+1)t-z)}Use in source · 1 passage
Search aliases: e^2π i((n+1)t-z) pi
Use in source · theta.tex, line 11
θ_0(z;t)=∏_n≥0(1-e^2π i(z+nt))(1-e^2π i((n+1)t-z)).
Exact mathematical passage
\theta_0(z;t)=\prod_{n\ge0}(1-e^{2\pi i(z+nt)})(1-e^{2\pi i((n+1)t-z)}).
e^{2\pi i(ja+kb+z)}Use in source · 1 passage
Search aliases: e^2π i(ja+kb+z) pi
Use in source · H12_Cubic_Divisor_Traditional.tex, line 72
For positive imaginary periods use the additive product convention θ_0(z;a) · =∏_n≥0(1-e^2π i(z+na))(1-e^2π i((n+1)a-z)), Γ_rm ell(z;a,b) · =∏_j,k≥0 frac1-e^2π i((j+1)a+(k+1)b-z)1-e^2π i(ja+kb+z). The negative-first-period continuation is Γ_rm ell(z;-t,s)=Γ_rm ell(z+t;t,s)^-1. Consequently the main scalar function is F(z)=fracΓ_rm ell(z;τ,s)^5Γ_rm ell(5z;5τ,5s) =fracΓ_rm ell(5(z+t);5t,5s)Γ_rm ell(z+t;t,s)^5, U_Γ=F(1/3). This convention agrees with the checked primary authors' copy of Bergeron--C…
Exact mathematical passage
\theta_0(z;a)&=\prod_{n\ge0}(1-e^{2\pi i(z+na)})(1-e^{2\pi i((n+1)a-z)}),\label{eq:theta-product}\\ \Gamma_{\rm ell}(z;a,b)&=\prod_{j,k\ge0} \frac{1-e^{2\pi i((j+1)a+(k+1)b-z)}}{1-e^{2\pi i(ja+kb+z)}}.\label{eq:gamma-product}
e^{2\pi i(n\tau+z)}Use in source · 1 passage
Search aliases: e^2π i(nτ+z) pi tau
Use in source · stark.tex, line 669
-2log|1-e^2π iz| -2∑_n≥1log|(1-e^2π i(nτ+z))(1-e^2π i(nτ-z))|.
Exact mathematical passage
-2\log|1-e^{2\pi iz}| -2\sum_{n\ge1}\log|(1-e^{2\pi i(n\tau+z)})(1-e^{2\pi i(n\tau-z)})|.
e^{2\pi i(n\tau-z)}Use in source · 1 passage
Search aliases: e^2π i(nτ-z) pi tau
Use in source · stark.tex, line 669
-2log|1-e^2π iz| -2∑_n≥1log|(1-e^2π i(nτ+z))(1-e^2π i(nτ-z))|.
Exact mathematical passage
-2\log|1-e^{2\pi iz}| -2\sum_{n\ge1}\log|(1-e^{2\pi i(n\tau+z)})(1-e^{2\pi i(n\tau-z)})|.
e^{2\pi i(n\omega+z)}Use in source · 1 passage
Search aliases: e^2π i(nω+z) pi omega
Use in source · stark.tex, line 304
H(z)=e^-πIm(ω)B_2(a) |1-e^2π iz| ∏_n≥1|(1-e^2π i(nω+z))(1-e^2π i(nω-z))|, tagA4.1
Exact mathematical passage
H(z)=e^{-\pi\operatorname{Im}(\omega)B_2(a)} |1-e^{2\pi iz}| \prod_{n\ge1}|(1-e^{2\pi i(n\omega+z)})(1-e^{2\pi i(n\omega-z)})|, \tag{A4.1}
e^{2\pi i(n\omega-z)}Use in source · 1 passage
Search aliases: e^2π i(nω-z) pi omega
Use in source · stark.tex, line 304
H(z)=e^-πIm(ω)B_2(a) |1-e^2π iz| ∏_n≥1|(1-e^2π i(nω+z))(1-e^2π i(nω-z))|, tagA4.1
Exact mathematical passage
H(z)=e^{-\pi\operatorname{Im}(\omega)B_2(a)} |1-e^{2\pi iz}| \prod_{n\ge1}|(1-e^{2\pi i(n\omega+z)})(1-e^{2\pi i(n\omega-z)})|, \tag{A4.1}
e^{2\pi i(z+na)}Use in source · 1 passage
Search aliases: e^2π i(z+na) pi
Use in source · H12_Cubic_Divisor_Traditional.tex, line 72
For positive imaginary periods use the additive product convention θ_0(z;a) · =∏_n≥0(1-e^2π i(z+na))(1-e^2π i((n+1)a-z)), Γ_rm ell(z;a,b) · =∏_j,k≥0 frac1-e^2π i((j+1)a+(k+1)b-z)1-e^2π i(ja+kb+z). The negative-first-period continuation is Γ_rm ell(z;-t,s)=Γ_rm ell(z+t;t,s)^-1. Consequently the main scalar function is F(z)=fracΓ_rm ell(z;τ,s)^5Γ_rm ell(5z;5τ,5s) =fracΓ_rm ell(5(z+t);5t,5s)Γ_rm ell(z+t;t,s)^5, U_Γ=F(1/3). This convention agrees with the checked primary authors' copy of Bergeron--C…
Exact mathematical passage
\theta_0(z;a)&=\prod_{n\ge0}(1-e^{2\pi i(z+na)})(1-e^{2\pi i((n+1)a-z)}),\label{eq:theta-product}\\ \Gamma_{\rm ell}(z;a,b)&=\prod_{j,k\ge0} \frac{1-e^{2\pi i((j+1)a+(k+1)b-z)}}{1-e^{2\pi i(ja+kb+z)}}.\label{eq:gamma-product}
e^{2\pi i(z+nt)}Use in source · 1 passage
Search aliases: e^2π i(z+nt) pi
Use in source · theta.tex, line 11
θ_0(z;t)=∏_n≥0(1-e^2π i(z+nt))(1-e^2π i((n+1)t-z)).
Exact mathematical passage
\theta_0(z;t)=\prod_{n\ge0}(1-e^{2\pi i(z+nt)})(1-e^{2\pi i((n+1)t-z)}).
e^{2\pi i/3}Use in source · 3 passages
Search aliases: e^2π i/3 pi
Use in source · H12_Cubic_Divisor_Traditional.tex, line 61
Fixed data, conventions, and the claim boundary. Fix throughout ω · =e^2π i/3, · β · =7^1/3ω^2, · ϵ · =2-β, · ρ · =ϵ^-1, τ · =(β^2+2β)/15, · t · =-τ, · s · =-(β+2)/15, · δ · =(τ+2s+4)/7. The real cube root 7^1/3 is positive; this fixes the lower complex embedding of β. We have Im t,Im s,Imδ>0, Imτ<0 and |ρ|<1. The last follows from N(ϵ)=1 and 0<2-7^1/3<1 at the real embedding. The numbers 1,t,s are rationally independent: a rational linear relation becomes a polynomial of degree at most two in β…
Exact mathematical passage
\label{eq:fixed-data} \begin{aligned} \omega&=e^{2\pi i/3},&\beta&=7^{1/3}\omega^2,&\epsilon&=2-\beta,&\rho&=\epsilon^{-1},\\ \tau&=(\beta^2+2\beta)/15,&t&=-\tau,&s&=-(\beta+2)/15,& \delta&=(\tau+2s+4)/7. \end{aligned}
e^{2\pi ia}Use in source · 1 passage
Search aliases: e^2π ia pi
Use in source · product_proof.tex, line 9
5nmid n fracP^nZ^n-Q^nZ^-nn(1-P^n)(1-Q^n), where A_0=e^2π ia,B_0=e^2π ib in the first line. At one surviving frequency suppress the index n. Summing seven shifts multiplies the two coefficients by ∑_k=0^6D^k and ∑_k=0^6D^-k. The exact identities are fracP∑_k=0^6D^k(1-P)(1-Q) · =frac1+Q(1-P)(1-D)-frac1(1-Q)(1-D),
Exact mathematical passage
A_0=e^{2\pi ia},B_0=e^{2\pi ib}
e^{2\pi ib}Use in source · 1 passage
Search aliases: e^2π ib pi
Use in source · product_proof.tex, line 9
5nmid n fracP^nZ^n-Q^nZ^-nn(1-P^n)(1-Q^n), where A_0=e^2π ia,B_0=e^2π ib in the first line. At one surviving frequency suppress the index n. Summing seven shifts multiplies the two coefficients by ∑_k=0^6D^k and ∑_k=0^6D^-k. The exact identities are fracP∑_k=0^6D^k(1-P)(1-Q) · =frac1+Q(1-P)(1-D)-frac1(1-Q)(1-D),
Exact mathematical passage
A_0=e^{2\pi ia},B_0=e^{2\pi ib}
e^{2\pi id}Use in source · 1 passage
Search aliases: e^2π id pi
Use in source · product_proof.tex, line 2
Full seven-product series proof. For positive imaginary periods t,s,d with t+7d=2s+4, put P=e^2π it, Q=e^2π is, D=e^2π id and Z=e^2π iz. Then PD^7=Q^2. In a common convergence region the fifth-smoothed logarithms are log S_5(z;a,b) · =5∑_substackn≥1
Exact mathematical passage
D=e^{2\pi id}
e^{2\pi ik^Tv}Use in source · 1 passage
Search aliases: e^2π ik^Tv pi
Use in source · rigidity.tex, line 13
The kernel on X has representatives j(1,2,4)^T/7, 0≤ j≤6. The conductor point p_3=e_2/3 obeys Mp_3-p_3=(0,5,1)^T and Np_3-p_3=(0,-5,-1)^T. The dual character e_k(v)=e^2π ik^Tv satisfies e_k(Mv)=e_Ak(v). Frequency transport is distinct from the scalar return sequence F(ρ^m/3).
Exact mathematical passage
e_k(v)=e^{2\pi ik^Tv}
e^{2\pi is}Use in source · 1 passage
Search aliases: e^2π is pi
Use in source · product_proof.tex, line 2
Full seven-product series proof. For positive imaginary periods t,s,d with t+7d=2s+4, put P=e^2π it, Q=e^2π is, D=e^2π id and Z=e^2π iz. Then PD^7=Q^2. In a common convergence region the fifth-smoothed logarithms are log S_5(z;a,b) · =5∑_substackn≥1
Exact mathematical passage
Q=e^{2\pi is}
e^{2\pi it}Use in source · 1 passage
Search aliases: e^2π it pi
Use in source · product_proof.tex, line 2
Full seven-product series proof. For positive imaginary periods t,s,d with t+7d=2s+4, put P=e^2π it, Q=e^2π is, D=e^2π id and Z=e^2π iz. Then PD^7=Q^2. In a common convergence region the fifth-smoothed logarithms are log S_5(z;a,b) · =5∑_substackn≥1
Exact mathematical passage
P=e^{2\pi it}
e^{2\pi iw}Use in source · 2 passages
Search aliases: e^2π iw pi
Use in source · reconstruction.tex, line 120
beginequation R(z)=R_0exp(α_1z+α_2z^2+α_3z^3)W_D(z), α_j=frac(log R)^(j)(0)j!. The logarithmic derivatives are branch independent. The notation (log R)^(4) means the third derivative of R'/R, a single-valued meromorphic function. The sine partial fractions [source reference] give fracd^4dw^4log(1-e^2π iw) =-6∑_n∈ Z(w-n)^-4. The linear exponential factor relating 1-e^2π iw to sinπ w has zero fourth logarithmic derivative. Apply this identity to each factor of the defining double gamma products. T…
Exact mathematical passage
\frac{d^4}{dw^4}\log(1-e^{2\pi iw}) =-6\sum_{n\in\mathbb Z}(w-n)^{-4}.
e^{2\pi iz}Use in source · 3 passages
Search aliases: e^2π iz pi
Use in source · product_proof.tex, line 2
Full seven-product series proof. For positive imaginary periods t,s,d with t+7d=2s+4, put P=e^2π it, Q=e^2π is, D=e^2π id and Z=e^2π iz. Then PD^7=Q^2. In a common convergence region the fifth-smoothed logarithms are log S_5(z;a,b) · =5∑_substackn≥1
Exact mathematical passage
Z=e^{2\pi iz}
e^{6\pi i k/7}Use in source · 1 passage
Search aliases: e^6π i k/7 pi
Use in source · polarization.tex, line 18
ξ=β^-1∑_k=1^6χ_7(k)e^6π i k/7, (χ_7(1),…,χ_7(6))=(1,ω,ω^2,ω^2,ω,1).
Exact mathematical passage
\xi=\beta^{-1}\sum_{k=1}^6\chi_7(k)e^{6\pi i k/7}, \qquad (\chi_7(1),\ldots,\chi_7(6))=(1,\omega,\omega^2,\omega^2,\omega,1).
e^{6\pi ia/7}Use in source · 2 passages
Search aliases: e^6π ia/7 pi
Use in source · H12_Cubic_Divisor_Traditional.tex, line 103
The arithmetic candidate is fixed independently. Let J=2+3ω and choose the Gauss branch X=β^-1∑_a=1^6χ_7(a)e^6π ia/7, (χ_7(1),…,χ_7(6))=(1,ω,ω^2,ω^2,ω,1), X^3=J. Then E= Q(ω,β,X) has degree eighteen. The retained arithmetic construction defines ν · =(X+β/X-β^2)/3, · v · =-β^2ν-2ν^2,
Exact mathematical passage
X=\beta^{-1}\sum_{a=1}^{6}\chi_7(a)e^{6\pi ia/7},\qquad (\chi_7(1),\ldots,\chi_7(6))=(1,\omega,\omega^2,\omega^2,\omega,1),\quad X^3=J.
E^HUse in source · 1 passage
Search aliases: E^H
Use in source · polarization.tex, line 103
∏_H a lineN_E/E^H(z) =z^3N_E/F_0(z). tagQ2
Exact mathematical passage
\boxed{ \prod_{H\text{ a line}}N_{E/E^H}(z) =z^3N_{E/F_0}(z). } \tag{Q2}
e^{h(z)}Use in source · 1 passage
Search aliases: e^h(z)
Use in source · reconstruction.tex, line 128
that of R. Thus R/W_D extends to a nowhere-zero entire function, and on the simply connected plane has an entire logarithm h. The fourth derivative of h vanishes, hence h is a polynomial of degree at most three. Its value at zero is fixed by R_0, and the vanished canonical jets identify the coefficients in [source reference]. Divisor matching alone would give e^h(z) for an arbitrary entire h. The fourth-log identity is the extra analytic statement forcing a cubic polynomial; no unproved finite-o…
Exact mathematical passage
e^{h(z)}
e^{R\|k\|}Use in source · 1 passage
Search aliases: e^R k
Use in source · rigidity.tex, line 61
is summable. The other two residue classes are obtained by the fixed continuous operators L_N,L_N^2. The series for g converges in every C^q norm; taking the limit gives h=δ_Mg. For a global holomorphic function on X, its Laurent coefficients decay faster than every fixed exponential. On each compact product annulus the decay at 7^nk dominates both 343^n and the annular factor e^R k , proving normal convergence there and a global holomorphic g. For a finite Laurent r of coordinate height at most…
Exact mathematical passage
e^{R\|k\|}
e^{w+w^2/2+w^3/3}Use in source · 1 passage
Search aliases: e^w+w^2/2+w^3/3
Use in source · reconstruction.tex, line 67
The genus-three divisor product. Let D={λ:m_λ≠0}; zero is not in D. Define E_3(w)=(1-w)e^w+w^2/2+w^3/3, W_D(z)=∏_λ∈ DE_3(z/λ)^m_λ. Negative exponents mean the quotient of the separate positive and negative entire products. [Canonical regularizer and absolute genus] The products defining W_D converge normally on compact sets away from their poles, and divW_D=divR, W_D(0)=1, (log W_D)^(j)(0)=0 (1≤ j≤3). Genus three is sufficient and minimal for this ungrouped absolute construction. Special signed …
Exact mathematical passage
\label{eq:WD} E_3(w)=(1-w)e^{w+w^2/2+w^3/3},\qquad W_D(z)=\prod_{\lambda\in D}E_3(z/\lambda)^{m_\lambda}.
e^{z^3}Use in source · 1 passage
Search aliases: e^z^3
Use in source · reconstruction.tex, line 63
en applied about a point where P is nonzero, using an outer circle of twice the radius, bounds its zero count by O(r^2); the same holds for Q. Cancellation only decreases their divisor counts. The factor u contributes no finite divisor. Theorem~[source reference] contradicts this bound for R. The single global unit in this grammar cannot be moved inside arbitrary sums: 1+e^z^3 has cubically many zeros and would invalidate that enlarged claim. The theorem concerns fixed finite dimensions and affi…
Exact mathematical passage
1+e^{z^3}
e^{\alpha_1}Use in source · 1 passage
Search aliases: e^α_1 alpha
Use in source · reconstruction.tex, line 180
gent lower-power sums are not substituted. Taking two logarithmic derivatives of [source reference] at the transition [source reference] gives (log K)''(0)=6α_3+B_D(1). Since L_t is even, α_3=-frac16[β^2∑_r=1^15L_t(rs)+B_D(1)]. Evaluating the period-one ratio also determines e^α_1=fracK(0)e^-α_2-α_3W_D(1). This evaluates the exponential, without choosing an unjustified principal logarithm.
Exact mathematical passage
\label{eq:expalpha1} e^{\alpha_1}=\frac{K(0)e^{-\alpha_2-\alpha_3}}{W_D(1)}.
e^{\alpha_1+\alpha_2/3+\alpha_3/9}Use in source · 1 passage
Search aliases: e^α_1+α_2/3+α_3/9 alpha alpha alpha
Use in source · reconstruction.tex, line 194
fracW_D(1/3)^3W_D(1) exp(-frac23α_2-frac89α_3). The right side consists of specified theta values and derivatives, algebraic period data and absolutely convergent genus-three products and sums. It contains no gamma value, arbitrary normalization scalar or new root choice. Cube [source reference] at 1/3 to get R_0^3e^α_1+α_2/3+α_3/9W_D(1/3)^3, then substitute [source reference]. The resulting exponents are -2α_2/3 and -8α_3/9. This is an analytic representation by an infinite transcendental produ…
Exact mathematical passage
R_0^3e^{\alpha_1+\alpha_2/3+\alpha_3/9}W_D(1/3)^3
e^{\pi i/6}Use in source · 1 passage
Search aliases: e^π i/6 pi
Use in source · stark.tex, line 60
X=fracΩ^2wp_Λ_0(1/J), Λ_0= Z+ Zω, Ω=e^π i/6fracΓ_rm Euler(1/3)^32π. tagA1.3
Exact mathematical passage
X=\frac{\Omega^2}{\wp_{\Lambda_0}(1/J)},\qquad \Lambda_0=\mathbb Z+\mathbb Z\omega,\qquad \Omega=e^{\pi i/6}\frac{\Gamma_{\rm Euler}(1/3)^3}{2\pi}. \tag{A1.3}
e^{\pi i\tau}Use in source · 1 passage
Search aliases: e^π iτ pi tau
Use in source · stark.tex, line 677
H_τ(z)=e^-π(Im z)^2/y |fracθ_1(π z;e^π iτ)η_rm Ded(τ)|.
Exact mathematical passage
H_\tau(z)=e^{-\pi(\Im z)^2/y} \left|\frac{\theta_1(\pi z;e^{\pi i\tau})}{\eta_{\rm Ded}(\tau)}\right|.
E_0Use in source · 1 passage
Search aliases: E_0
Use in source · stark.tex, line 371
On the normalized curve E_0:y^2+y=x^3, set
Exact mathematical passage
E_0:y^2+y=x^3
e_0Use in source · 2 passages
Search aliases: e_0
Use in source · rigidity.tex, line 20
stable pair of modulus a^-1/2>1. Every nonzero integer A-orbit escapes in both directions: otherwise an integer vector would tend to zero along positive or negative iterates. Fourier orbit sums are therefore absolutely convergent for smooth coefficients. The dual degree-seven image is exactly k∈ B Z^3 Longleftrightarrow ℓ(k):=k_0+2k_1+4k_2≡0pmod7. Indeed k-ℓ(k)e_0=B(-k_1,k_2,-8k_2)^T and B(0,-2,15)^T=7e_0. If ℓ(k)=7r its explicit preimage is (-k_1,k_2-2r,15r-8k_2)^T.
Exact mathematical passage
k-\ell(k)e_0=B(-k_1,k_2,-8k_2)^T
E_2Use in source · 1 passage
Search aliases: E_2
Use in source · reconstruction.tex, line 88
The lower bound n_D(r)≥ c r^3 yields divergence of ∑_D|m_λ|/|λ|^3: partial summation contains 3∫ n_D(r)r^-4 dr, which diverges logarithmically. For a fixed z≠0, log E_2(z/λ)=-fracz^33λ^3+O(|λ|^-4). Its weighted absolute logarithm sum diverges, because the fourth-power remainder is summable. This proves failure of ungrouped absolute genus two, without a claim about cancellations in specially grouped signed products. [Explicit analytic tail] If n_D(r)=∑_|λ|≤ r|m_λ|≤ Ar^3 for all r≥ R_*, then for T…
Exact mathematical passage
\log E_2(z/\lambda)=-\frac{z^3}{3\lambda^3}+O(|\lambda|^{-4}).
e_2Use in source · 1 passage
Search aliases: e_2
Use in source · rigidity.tex, line 13
The kernel on X has representatives j(1,2,4)^T/7, 0≤ j≤6. The conductor point p_3=e_2/3 obeys Mp_3-p_3=(0,5,1)^T and Np_3-p_3=(0,-5,-1)^T. The dual character e_k(v)=e^2π ik^Tv satisfies e_k(Mv)=e_Ak(v). Frequency transport is distinct from the scalar return sequence F(ρ^m/3).
Exact mathematical passage
p_3=e_2/3
E_3Use in source · 4 passages
Search aliases: E_3
Use in source · reconstruction.tex, line 67
The genus-three divisor product. Let D={λ:m_λ≠0}; zero is not in D. Define E_3(w)=(1-w)e^w+w^2/2+w^3/3, W_D(z)=∏_λ∈ DE_3(z/λ)^m_λ. Negative exponents mean the quotient of the separate positive and negative entire products. [Canonical regularizer and absolute genus] The products defining W_D converge normally on compact sets away from their poles, and divW_D=divR, W_D(0)=1, (log W_D)^(j)(0)=0 (1≤ j≤3). Genus three is sufficient and minimal for this ungrouped absolute construction. Special signed …
Exact mathematical passage
\label{eq:WD} E_3(w)=(1-w)e^{w+w^2/2+w^3/3},\qquad W_D(z)=\prod_{\lambda\in D}E_3(z/\lambda)^{m_\lambda}.
e_{Ak}Use in source · 1 passage
Search aliases: e_Ak
Use in source · rigidity.tex, line 13
The kernel on X has representatives j(1,2,4)^T/7, 0≤ j≤6. The conductor point p_3=e_2/3 obeys Mp_3-p_3=(0,5,1)^T and Np_3-p_3=(0,-5,-1)^T. The dual character e_k(v)=e^2π ik^Tv satisfies e_k(Mv)=e_Ak(v). Frequency transport is distinct from the scalar return sequence F(ρ^m/3).
Exact mathematical passage
e_k(Mv)=e_{Ak}(v)
e_kUse in source · 5 passages
Search aliases: e_k
Use in source · rigidity.tex, line 13
The kernel on X has representatives j(1,2,4)^T/7, 0≤ j≤6. The conductor point p_3=e_2/3 obeys Mp_3-p_3=(0,5,1)^T and Np_3-p_3=(0,-5,-1)^T. The dual character e_k(v)=e^2π ik^Tv satisfies e_k(Mv)=e_Ak(v). Frequency transport is distinct from the scalar return sequence F(ρ^m/3).
Exact mathematical passage
e_k(v)=e^{2\pi ik^Tv}
E_tUse in source · 10 passages
Search aliases: E_t
Use in source · norms.tex, line 5
Exactly normalized elliptic norms. On E_t= C/Λ_t define N_nf(w)=∏_nz=wf(z) for integers n≥1, with all n^2 preimages counted. The product is independent of the representative lift. Root-of-unity averaging and vertical residue splitting in [source reference] give N_nq_t(w)=q_t(w), · 5nmid n,
Exact mathematical passage
E_t=\mathbb C/\Lambda_t
\mathcal E_zUse in source · 4 passages
Search aliases: E_z mathcal
Use in source · stark.tex, line 317
E_z(s)=∑_m,n∈ Z|m+nω+z|^-2s,
Exact mathematical passage
\mathcal E_z(s)=\sum_{m,n\in\mathbb Z}|m+n\omega+z|^{-2s},
\mathcal E_z'Use in source · 1 passage
Search aliases: E_z' mathcal
Use in source · stark.tex, line 323
E_z(0)=0, E_z'(0)=-2log H(z). tagA4.2
Exact mathematical passage
\mathcal E_z(0)=0,\qquad \mathcal E_z'(0)=-2\log H(z). \tag{A4.2}
\mathcal E_{\alpha/(3J)}Use in source · 1 passage
Search aliases: E_α/(3J) mathcal alpha
Use in source · stark.tex, line 335
63^-s E_α/(3J)(s).
Exact mathematical passage
63^{-s}\mathcal E_{\alpha/(3J)}(s).
\expUse in source · 5 passages
Search aliases: exp
Use in source · reconstruction.tex, line 58
functions formed by finite polynomial expressions in these restrictions, derivatives and polynomials in z. Let u be a single global nowhere-zero entire function. Every meromorphic function u(z)fracP(z)Q(z) has zero and pole counts O(r^2). Thus R is not representable in this class. For Y=ImΩ>0, the defining theta series has terms bounded by exp{-π n^TYn+2π n Im(Az+b) }. Completing the square, and absorbing half of the positive quadratic form into a convergent Gaussian sum, gives Cexp(C'|z|^2) for…
Exact mathematical passage
\exp\{-\pi n^TYn+2\pi\|n\|\,\|\Im(Az+b)\|\}.
FUse in source · 47 passages
Search aliases: F
Use in source · H12_Cubic_Divisor_Traditional.tex, line 82
Γ_rm ell(z;a,b) · =∏_j,k≥0 frac1-e^2π i((j+1)a+(k+1)b-z)1-e^2π i(ja+kb+z). The negative-first-period continuation is Γ_rm ell(z;-t,s)=Γ_rm ell(z+t;t,s)^-1. Consequently the main scalar function is F(z)=fracΓ_rm ell(z;τ,s)^5Γ_rm ell(5z;5τ,5s) =fracΓ_rm ell(5(z+t);5t,5s)Γ_rm ell(z+t;t,s)^5, U_Γ=F(1/3). This convention agrees with the checked primary authors' copy of Bergeron--Charollois--Garc ia [source reference]. All analytic identities at removable or polar points mean meromorphic identities. P…
Exact mathematical passage
F(z)=\frac{\Gamma_{\rm ell}(z;\tau,s)^5}{\Gamma_{\rm ell}(5z;5\tau,5s)} =\frac{\Gamma_{\rm ell}(5(z+t);5t,5s)}{\Gamma_{\rm ell}(z+t;t,s)^5}, \qquad U_\Gamma=F(1/3).
\mathbb FUse in source · 2 passages
Search aliases: F mathbb
Use in source · stark.tex, line 228
Reduction modulo J sends ω to 4∈ F_7. Formula (A3.2) groups the representatives as follows:
Exact mathematical passage
4\in\mathbb F_7
\mathfrak fUse in source · 10 passages
Search aliases: f mathfrak
Use in source · stark.tex, line 35
f=(3,β-1), E=K(ω,θ), θ=2cos(6π/7).
Exact mathematical passage
\mathfrak f=(3,\beta-1),\qquad E=K(\omega,\theta),\qquad \theta=2\cos(6\pi/7).
fUse in source · 12 passages
Search aliases: f
Use in source · norms.tex, line 5
Exactly normalized elliptic norms. On E_t= C/Λ_t define N_nf(w)=∏_nz=wf(z) for integers n≥1, with all n^2 preimages counted. The product is independent of the representative lift. Root-of-unity averaging and vertical residue splitting in [source reference] give N_nq_t(w)=q_t(w), · 5nmid n,
Exact mathematical passage
N_nf(w)=\prod_{nz=w}f(z)
f^\#Use in source · 1 passage
Search aliases: f^#
Use in source · reconstruction.tex, line 200
The repaired conjugation quotient and its boundary. For f^#(z)=overlinef(overline z), the canonical product W_D^# has the conjugate divisor. Hence fracR/R^#W_D/W_D^# =fracR_0overlineR_0 exp(∑_j=1^3(α_j-overlineα_j)z^j). This is entire and nowhere zero, with modulus one on the real axis. It cancels all unmatched divisor orders by a canonical meromorphic factor. That factor is not a holomorphic-unit gauge or a finite theta section. Regularity on the scalar plane
Exact mathematical passage
f^\#(z)=\overline{f(\overline z)}
F_{-\bar t,-\bar s}Use in source · 2 passages
Search aliases: F_-bar t,-bar s bar bar
Use in source · polarization.tex, line 348
overlineF_t,s(z)=F_-bar t,-bar s(-bar z).
Exact mathematical passage
\overline{F_{t,s}(z)}=F_{-\bar t,-\bar s}(-\bar z).
F_0Use in source · 22 passages
Search aliases: F_0
Use in source · H12_Cubic_Divisor_Traditional.tex, line 99
We use F_0= Q(ω) for the imaginary-quadratic field, to separate it from the scalar gamma function F. The eta quotient is H(a)=η(5a)/η(a)^5; the integer step function H(n) used in the divisor formula is a different object. The symbol R_0 is the scalar anchor, while R_* is a counting cutoff radius. In the arithmetic block ϱ is a ray character and a radical lift, as explicitly indicated there; it is distinct from the contracting period multiplier ρ.
Exact mathematical passage
F_0=\mathbb Q(\omega)
\mathbb F_{181}Use in source · 1 passage
Search aliases: F_181 mathbb
Use in source · stark.tex, line 563
For t^6=W, the lift ϱ(t)=t^5/U_rm alg commutes with t↦ c_6t, and satisfies ϱ^6(t)=c_6^4t. Irreducibility is checked at the residue assignment (β,v,ω)=(32,99,48) over F_181, where W=23 and T^6-23 is irreducible. The earlier local-integrality argument validates this reduction. The resulting abelian group is C_2× C_18.
Exact mathematical passage
\mathbb F_{181}
\mathbb F_7Use in source · 1 passage
Search aliases: F_7 mathbb
Use in source · stark.tex, line 228
Reduction modulo J sends ω to 4∈ F_7. Formula (A3.2) groups the representatives as follows:
Exact mathematical passage
4\in\mathbb F_7
F_{t,s}Use in source · 1 passage
Search aliases: F_t,s
Use in source · product_proof.tex, line 3
Full seven-product series proof. For positive imaginary periods t,s,d with t+7d=2s+4, put P=e^2π it, Q=e^2π is, D=e^2π id and Z=e^2π iz. Then PD^7=Q^2. In a common convergence region the fifth-smoothed logarithms are log S_5(z;a,b) · =5∑_substackn≥1 5nmid n fracZ^n-(A_0B_0/Z)^nn(1-A_0^n)(1-B_0^n), log F_t,s(z) · =-5∑_substackn≥1 5nmid n fracP^nZ^n-Q^nZ^-nn(1-P^n)(1-Q^n), where A_0=e^2π ia,B_0=e^2π ib in the first line. At one surviving frequency suppress the index n. Summing seven shifts multipl…
Exact mathematical passage
\log S_5(z;a,b)&=5\sum_{\substack{n\ge1\\5\nmid n}} \frac{Z^n-(A_0B_0/Z)^n}{n(1-A_0^n)(1-B_0^n)},\\ \log F_{t,s}(z)&=-5\sum_{\substack{n\ge1\\5\nmid n}} \frac{P^nZ^n-Q^nZ^{-n}}{n(1-P^n)(1-Q^n)},
GUse in source · 1 passage
Search aliases: G
Use in source · stark.tex, line 87
For the last formula, the conjugate Gauss resolvent is 7/G, and barβ=ω^2β. Formula (A1.5) follows directly. It must not be replaced by the involution σ^3, which fixes K.
Exact mathematical passage
7/G
g_2Use in source · 4 passages
Search aliases: g_2
Use in source · theta.tex, line 55
x(z)=wp_t(z), y(z)=wp_t'(z), f(X)=4X^3-g_2X-g_3.
Exact mathematical passage
x(z)=\wp_t(z),\quad y(z)=\wp_t'(z),\quad f(X)=4X^3-g_2X-g_3.
g_2^2Use in source · 2 passages
Search aliases: g_2^2
Use in source · theta.tex, line 155
ψ_3(X)=3X^4-tfrac32g_2X^2-3g_3X-tfrac116g_2^2,
Exact mathematical passage
\psi_3(X)=3X^4-\tfrac32g_2X^2-3g_3X-\tfrac1{16}g_2^2,
g_3Use in source · 6 passages
Search aliases: g_3
Use in source · stark.tex, line 392
For completeness, T is nonzero three-torsion and its normalized x-coordinate is a root of 3x(x^3+1). The lattice is invariant under ordinary conjugation, so wp(1/3) and its derivative are real. Since g_3(Λ_0)>0, the real differential equation wp'^2=4wp^3-g_3 implies wp(1/3)>0. The argument of Ω^2 is π/3, selecting -ω among the three roots of x^3=-1.
Exact mathematical passage
g_3(\Lambda_0)>0
\operatorname{Gal}Use in source · 4 passages
Search aliases: Gal operatorname
Use in source · polarization.tex, line 25
E= Q(ω,β,ξ), F_0= Q(ω), Gal(E/F_0)=C_3^2.
Exact mathematical passage
E=\mathbb Q(\omega,\beta,\xi),\quad F_0=\mathbb Q(\omega), \qquad \operatorname{Gal}(E/F_0)=C_3^2.
HUse in source · 21 passages
Search aliases: H
Use in source · H12_Cubic_Divisor_Traditional.tex, line 99
We use F_0= Q(ω) for the imaginary-quadratic field, to separate it from the scalar gamma function F. The eta quotient is H(a)=η(5a)/η(a)^5; the integer step function H(n) used in the divisor formula is a different object. The symbol R_0 is the scalar anchor, while R_* is a counting cutoff radius. In the arithmetic block ϱ is a ray character and a radical lift, as explicitly indicated there; it is distinct from the contracting period multiplier ρ.
Exact mathematical passage
H(n)
\mathcal HUse in source · 6 passages
Search aliases: H mathcal
Use in source · H12_Cubic_Divisor_Traditional.tex, line 99
We use F_0= Q(ω) for the imaginary-quadratic field, to separate it from the scalar gamma function F. The eta quotient is H(a)=η(5a)/η(a)^5; the integer step function H(n) used in the divisor formula is a different object. The symbol R_0 is the scalar anchor, while R_* is a counting cutoff radius. In the arithmetic block ϱ is a ray character and a radical lift, as explicitly indicated there; it is distinct from the contracting period multiplier ρ.
Exact mathematical passage
\mathcal H(a)=\eta(5a)/\eta(a)^5
h_KUse in source · 1 passage
Search aliases: h_K
Use in source · stark.tex, line 177
The analytic class number formula, h_K=3, and ζ_ Q(0)=-1/2 give
Exact mathematical passage
h_K=3
H_\tauUse in source · 1 passage
Search aliases: H_τ tau
Use in source · stark.tex, line 677
H_τ(z)=e^-π(Im z)^2/y |fracθ_1(π z;e^π iτ)η_rm Ded(τ)|.
Exact mathematical passage
H_\tau(z)=e^{-\pi(\Im z)^2/y} \left|\frac{\theta_1(\pi z;e^{\pi i\tau})}{\eta_{\rm Ded}(\tau)}\right|.
IUse in source · 5 passages
Search aliases: I
Use in source · norms.tex, line 48
The norm matrices on this free subgroup are N_n=I (n odd,5nmid n), N_n=1 · -6 0 · 0 (n even,5nmid n), N_n=0 (5mid n). The middle matrix is a split integral projector with image Z(1,0) and kernel Z(6,1). Its primitive kernel class 6e+p is nonzero, as its two charges are 28,-4. Thus repeated full doubling cannot recover the deleted component. The residue-ten matrix law is multiplicative; its finite coefficient statement belongs to the inherited formal core, whereas the application of divisors to a…
Exact mathematical passage
N_n=I\quad(n\text{ odd},5\nmid n),\qquad N_n=\begin{pmatrix}1&-6\\0&0\end{pmatrix}\quad(n\text{ even},5\nmid n),\qquad N_n=0\quad(5\mid n).
\ImUse in source · 13 passages
Search aliases: Im
Use in source · H12_Cubic_Divisor_Traditional.tex, line 68
τ · =(β^2+2β)/15, · t · =-τ, · s · =-(β+2)/15, · δ · =(τ+2s+4)/7. The real cube root 7^1/3 is positive; this fixes the lower complex embedding of β. We have Im t,Im s,Imδ>0, Imτ<0 and |ρ|<1. The last follows from N(ϵ)=1 and 0<2-7^1/3<1 at the real embedding. The numbers 1,t,s are rationally independent: a rational linear relation becomes a polynomial of degree at most two in β, contradicting its irreducible cubic minimal polynomial. [htbp]centeringincludegraphics[width=.96linewidth]figures/01_pe…
Exact mathematical passage
\Im t,\Im s,\Im\delta>0
\operatorname{Im}Use in source · 1 passage
Search aliases: Im operatorname
Use in source · stark.tex, line 304
H(z)=e^-πIm(ω)B_2(a) |1-e^2π iz| ∏_n≥1|(1-e^2π i(nω+z))(1-e^2π i(nω-z))|, tagA4.1
Exact mathematical passage
H(z)=e^{-\pi\operatorname{Im}(\omega)B_2(a)} |1-e^{2\pi iz}| \prod_{n\ge1}|(1-e^{2\pi i(n\omega+z)})(1-e^{2\pi i(n\omega-z)})|, \tag{A4.1}
\operatorname{Ind}Use in source · 1 passage
Search aliases: Ind operatorname
Use in source · stark.tex, line 132
Ind_C_2^S_3(sgn) =sgn⊕Std, Std=Ind_C_3^S_3(ϱ)
Exact mathematical passage
\operatorname{Ind}_{C_2}^{S_3}(\operatorname{sgn}) =\operatorname{sgn}\oplus\operatorname{Std}, \qquad \operatorname{Std}=\operatorname{Ind}_{C_3}^{S_3}(\varrho)
\operatorname{Ind}_{C_2}^{S_3}Use in source · 1 passage
Search aliases: Ind_C_2^S_3 operatorname
Use in source · stark.tex, line 132
Ind_C_2^S_3(sgn) =sgn⊕Std, Std=Ind_C_3^S_3(ϱ)
Exact mathematical passage
\operatorname{Ind}_{C_2}^{S_3}(\operatorname{sgn}) =\operatorname{sgn}\oplus\operatorname{Std}, \qquad \operatorname{Std}=\operatorname{Ind}_{C_3}^{S_3}(\varrho)
\operatorname{Ind}_{C_3}^{S_3}Use in source · 1 passage
Search aliases: Ind_C_3^S_3 operatorname
Use in source · stark.tex, line 132
Ind_C_2^S_3(sgn) =sgn⊕Std, Std=Ind_C_3^S_3(ϱ)
Exact mathematical passage
\operatorname{Ind}_{C_2}^{S_3}(\operatorname{sgn}) =\operatorname{sgn}\oplus\operatorname{Std}, \qquad \operatorname{Std}=\operatorname{Ind}_{C_3}^{S_3}(\varrho)
JUse in source · 26 passages
Search aliases: J
Use in source · H12_Cubic_Divisor_Traditional.tex, line 102
The arithmetic candidate is fixed independently. Let J=2+3ω and choose the Gauss branch X=β^-1∑_a=1^6χ_7(a)e^6π ia/7, (χ_7(1),…,χ_7(6))=(1,ω,ω^2,ω^2,ω,1), X^3=J. Then E= Q(ω,β,X) has degree eighteen. The retained arithmetic construction defines ν · =(X+β/X-β^2)/3, · v · =-β^2ν-2ν^2,
Exact mathematical passage
J=2+3\omega
jUse in source · 38 passages
Search aliases: j
Use in source · H12_Cubic_Divisor_Traditional.tex, line 72
For positive imaginary periods use the additive product convention θ_0(z;a) · =∏_n≥0(1-e^2π i(z+na))(1-e^2π i((n+1)a-z)), Γ_rm ell(z;a,b) · =∏_j,k≥0 frac1-e^2π i((j+1)a+(k+1)b-z)1-e^2π i(ja+kb+z). The negative-first-period continuation is Γ_rm ell(z;-t,s)=Γ_rm ell(z+t;t,s)^-1. Consequently the main scalar function is F(z)=fracΓ_rm ell(z;τ,s)^5Γ_rm ell(5z;5τ,5s) =fracΓ_rm ell(5(z+t);5t,5s)Γ_rm ell(z+t;t,s)^5, U_Γ=F(1/3). This convention agrees with the checked primary authors' copy of Bergeron--C…
Exact mathematical passage
\theta_0(z;a)&=\prod_{n\ge0}(1-e^{2\pi i(z+na)})(1-e^{2\pi i((n+1)a-z)}),\label{eq:theta-product}\\ \Gamma_{\rm ell}(z;a,b)&=\prod_{j,k\ge0} \frac{1-e^{2\pi i((j+1)a+(k+1)b-z)}}{1-e^{2\pi i(ja+kb+z)}}.\label{eq:gamma-product}
J_{n,v}Use in source · 1 passage
Search aliases: J_n,v
Use in source · theta.tex, line 440
The same substitution works for every oriented three-step return product J_n,v from the predecessor. Consequently ALL seven nontrivial signed-channel limits have finite rational elliptic expressions without a separately supplied eta constant. The first-order limits still have only their guaranteed first-order local cancellation; this normalization change does not upgrade their convergence rate.
Exact mathematical passage
J_{n,v}
KUse in source · 23 passages
Search aliases: K
Use in source · principal.tex, line 106
All transition multipliers and the scalar anchor. [Theta transitions] The following hold meromorphically: K(z):=fracR(z+1)R(z) · =∏_r=1^15q_t(β z-rs)^-1, fracR(z+s)R(z) · =fracq_s(β z-τ)∏_k=0^6q_t(z+kδ), fracR(z+τ)R(z) · = frac∏_k=0^6q_s(z+kδ)∏_r=0^3q_t(β z+rs) q_s(β z)q_s(β z+τ), fracR(z+δ)R(z) · = fracq_s(z)q_t(z)q_t(z+s)∏_r=1^8q_t(β z-rs). Use F(z+1)=F(z), F(z+τ)/F(z)=q_s(z) and F(z+s)/F(z)=q_t(z)^-1. For [source reference], P_7 is period one and β=-15s-2, producing fifteen negative s shifts …
Exact mathematical passage
K(z):=\frac{R(z+1)}{R(z)}&=\prod_{r=1}^{15}q_t(\beta z-rs)^{-1},\label{eq:K}\\ \frac{R(z+s)}{R(z)}&=\frac{q_s(\beta z-\tau)}{\prod_{k=0}^{6}q_t(z+k\delta)},\label{eq:Rs}\\ \frac{R(z+\tau)}{R(z)}&= \frac{\prod_{k=0}^{6}q_s(z+k\delta)\prod_{r=0}^{3}q_t(\beta z+rs)} {q_s(\beta z)q_s(\beta z+\tau)},\label{eq:Rtau}\\ \frac{R(z+\delta)}{R(z)}&= \frac{q_s(z)}{q_t(z)q_t(z+s)\prod_{r=1}^{8}q_t(\beta z-rs)}.\label{eq:Rdelta}
kUse in source · 57 passages
Search aliases: k
Use in source · H12_Cubic_Divisor_Traditional.tex, line 72
For positive imaginary periods use the additive product convention θ_0(z;a) · =∏_n≥0(1-e^2π i(z+na))(1-e^2π i((n+1)a-z)), Γ_rm ell(z;a,b) · =∏_j,k≥0 frac1-e^2π i((j+1)a+(k+1)b-z)1-e^2π i(ja+kb+z). The negative-first-period continuation is Γ_rm ell(z;-t,s)=Γ_rm ell(z+t;t,s)^-1. Consequently the main scalar function is F(z)=fracΓ_rm ell(z;τ,s)^5Γ_rm ell(5z;5τ,5s) =fracΓ_rm ell(5(z+t);5t,5s)Γ_rm ell(z+t;t,s)^5, U_Γ=F(1/3). This convention agrees with the checked primary authors' copy of Bergeron--C…
Exact mathematical passage
\theta_0(z;a)&=\prod_{n\ge0}(1-e^{2\pi i(z+na)})(1-e^{2\pi i((n+1)a-z)}),\label{eq:theta-product}\\ \Gamma_{\rm ell}(z;a,b)&=\prod_{j,k\ge0} \frac{1-e^{2\pi i((j+1)a+(k+1)b-z)}}{1-e^{2\pi i(ja+kb+z)}}.\label{eq:gamma-product}
k'Use in source · 1 passage
Search aliases: k'
Use in source · reconstruction.tex, line 10
Z_i,j,k^(n)=λ(-i,j,5(n+k)), P_i,j,k,r^(n)=λ(-i,j,5(n+k)-r), 1≤ r≤4. Every such triple has a<0, b>0, 3c>2b. Hence p>0 and the bracket in [source reference] is one. The Z points have order four and the P points have order minus one. Rational independence makes the parameter map injective. The pole indices are distinct because 5(k-k')=r-r' forces both sides to vanish; the zero and pole families are disjoint by c modulo five. [Two-sided cubic growth] Let n_+(r) and n_-(r) count the zeros and poles o…
Exact mathematical passage
5(k-k')=r-r'
k^TUse in source · 1 passage
Search aliases: k^T
Use in source · rigidity.tex, line 13
The kernel on X has representatives j(1,2,4)^T/7, 0≤ j≤6. The conductor point p_3=e_2/3 obeys Mp_3-p_3=(0,5,1)^T and Np_3-p_3=(0,-5,-1)^T. The dual character e_k(v)=e^2π ik^Tv satisfies e_k(Mv)=e_Ak(v). Frequency transport is distinct from the scalar return sequence F(ρ^m/3).
Exact mathematical passage
e_k(v)=e^{2\pi ik^Tv}
K_0Use in source · 4 passages
Search aliases: K_0
Use in source · reconstruction.tex, line 17
e distinct because 5(k-k')=r-r' forces both sides to vanish; the zero and pole families are disjoint by c modulo five. [Two-sided cubic growth] Let n_+(r) and n_-(r) count the zeros and poles of R in |z|≤ r, with multiplicity. Then n_+(r)=Θ(r^3), n_-(r)=Θ(r^3), n_D(r):=n_+(r)+n_-(r)=Θ(r^3). In particular, for K_0=(|τ|+|s|+2)/|β|, n_+(K_0n)≥4n^3, n_-(K_0n)≥4n^3. These are unsigned counts; no cubic lower bound for a signed zero-minus-pole mass is asserted. In [source reference], |a|,b≤ n and 0<c≤1…
Exact mathematical passage
K_0=(|\tau|+|s|+2)/|\beta|
k_0Use in source · 1 passage
Search aliases: k_0
Use in source · rigidity.tex, line 16
tic polynomial (x-2)^3+7 gives a stable real eigenvalue a=2-7^1/3∈(0,1) and a conjugate unstable pair of modulus a^-1/2>1. Every nonzero integer A-orbit escapes in both directions: otherwise an integer vector would tend to zero along positive or negative iterates. Fourier orbit sums are therefore absolutely convergent for smooth coefficients. The dual degree-seven image is exactly k∈ B Z^3 Longleftrightarrow ℓ(k):=k_0+2k_1+4k_2≡0pmod7. Indeed k-ℓ(k)e_0=B(-k_1,k_2,-8k_2)^T and B(0,-2,15)^T=7e_0. …
Exact mathematical passage
k\in B\mathbb Z^3\quad\Longleftrightarrow\quad \ell(k):=k_0+2k_1+4k_2\equiv0\pmod7.
k_1Use in source · 3 passages
Search aliases: k_1
Use in source · rigidity.tex, line 16
tic polynomial (x-2)^3+7 gives a stable real eigenvalue a=2-7^1/3∈(0,1) and a conjugate unstable pair of modulus a^-1/2>1. Every nonzero integer A-orbit escapes in both directions: otherwise an integer vector would tend to zero along positive or negative iterates. Fourier orbit sums are therefore absolutely convergent for smooth coefficients. The dual degree-seven image is exactly k∈ B Z^3 Longleftrightarrow ℓ(k):=k_0+2k_1+4k_2≡0pmod7. Indeed k-ℓ(k)e_0=B(-k_1,k_2,-8k_2)^T and B(0,-2,15)^T=7e_0. …
Exact mathematical passage
k\in B\mathbb Z^3\quad\Longleftrightarrow\quad \ell(k):=k_0+2k_1+4k_2\equiv0\pmod7.
k_2Use in source · 3 passages
Search aliases: k_2
Use in source · rigidity.tex, line 16
tic polynomial (x-2)^3+7 gives a stable real eigenvalue a=2-7^1/3∈(0,1) and a conjugate unstable pair of modulus a^-1/2>1. Every nonzero integer A-orbit escapes in both directions: otherwise an integer vector would tend to zero along positive or negative iterates. Fourier orbit sums are therefore absolutely convergent for smooth coefficients. The dual degree-seven image is exactly k∈ B Z^3 Longleftrightarrow ℓ(k):=k_0+2k_1+4k_2≡0pmod7. Indeed k-ℓ(k)e_0=B(-k_1,k_2,-8k_2)^T and B(0,-2,15)^T=7e_0. …
Exact mathematical passage
k\in B\mathbb Z^3\quad\Longleftrightarrow\quad \ell(k):=k_0+2k_1+4k_2\equiv0\pmod7.
\kerUse in source · 2 passages
Search aliases: ker
Use in source · rigidity.tex, line 24
The convergent transfer proof. For functions on T^3, let S_Nh(v)=∑_κ∈ker Nh(v+κ), C_Nh(v)=h(Nv), L_Nh(v)=∑_Nw=vh(w), δ_Mh=h-h∘ M. The transfer L_N is unnormalized. For h=∑ c_ke_k, S_Nh=∑_k∈ B Z^37c_ke_k, L_Nh=∑_k7c_Bke_k, S_Nh=(L_Nh)∘ N, L_NC_Nh=7h. Since M permutes ker N and commutes with N, the required operators commute with composition by M.
Exact mathematical passage
S_Nh(v)=\sum_{\kappa\in\ker N}h(v+\kappa),\quad C_Nh(v)=h(Nv),\quad L_Nh(v)=\sum_{Nw=v}h(w),\quad \delta_Mh=h-h\circ M.
LUse in source · 30 passages
Search aliases: L
Use in source · bibliography.tex, line 7
its bytes with the inaccessible v2 artifact. Product/continuation conventions, Theorem~3.2 and the fixed example in Section~5.2 are the controlling uses. bibitemMilne J.~S.~Milne, Class Field Theory, version~4.03, August~6, 2020. Cubic reciprocity and Artin induction in Chapter~VIII. https://www.jmilne.org/math/CourseNotes/CFTc.pdf. bibitemCogdell J.~W.~Cogdell, On Artin L-functions, author's PDF, Section~2.2, local inertia-invariant Euler factors and induction compatibility. https://math.osu.ed…
Exact mathematical passage
L
L_{\mathbb Q}Use in source · 1 passage
Search aliases: L_ Q mathbb
Use in source · stark.tex, line 141
L_K(s,χ_j) =L_ Q(s,χ_-3χ_7^j) L_F_0(s,ϱη^j). tagA2.2
Exact mathematical passage
\boxed{L_K(s,\chi_j) =L_{\mathbb Q}(s,\chi_{-3}\chi_7^j)\, L_{F_0}(s,\varrho\eta^j).} \tag{A2.2}
L_{F_0}Use in source · 3 passages
Search aliases: L_F_0
Use in source · stark.tex, line 141
L_K(s,χ_j) =L_ Q(s,χ_-3χ_7^j) L_F_0(s,ϱη^j). tagA2.2
Exact mathematical passage
\boxed{L_K(s,\chi_j) =L_{\mathbb Q}(s,\chi_{-3}\chi_7^j)\, L_{F_0}(s,\varrho\eta^j).} \tag{A2.2}
L_{F_0}'Use in source · 3 passages
Search aliases: L_F_0'
Use in source · stark.tex, line 180
L_F_0'(0,ϱ)=3 R. tagA2.4
Exact mathematical passage
L_{F_0}'(0,\varrho)=3\mathcal R. \tag{A2.4}
L_KUse in source · 1 passage
Search aliases: L_K
Use in source · stark.tex, line 141
L_K(s,χ_j) =L_ Q(s,χ_-3χ_7^j) L_F_0(s,ϱη^j). tagA2.2
Exact mathematical passage
\boxed{L_K(s,\chi_j) =L_{\mathbb Q}(s,\chi_{-3}\chi_7^j)\, L_{F_0}(s,\varrho\eta^j).} \tag{A2.2}
L_NUse in source · 6 passages
Search aliases: L_N
Use in source · rigidity.tex, line 24
The convergent transfer proof. For functions on T^3, let S_Nh(v)=∑_κ∈ker Nh(v+κ), C_Nh(v)=h(Nv), L_Nh(v)=∑_Nw=vh(w), δ_Mh=h-h∘ M. The transfer L_N is unnormalized. For h=∑ c_ke_k, S_Nh=∑_k∈ B Z^37c_ke_k, L_Nh=∑_k7c_Bke_k, S_Nh=(L_Nh)∘ N, L_NC_Nh=7h. Since M permutes ker N and commutes with N, the required operators commute with composition by M.
Exact mathematical passage
S_Nh(v)=\sum_{\kappa\in\ker N}h(v+\kappa),\quad C_Nh(v)=h(Nv),\quad L_Nh(v)=\sum_{Nw=v}h(w),\quad \delta_Mh=h-h\circ M.
L_N^2Use in source · 1 passage
Search aliases: L_N^2
Use in source · rigidity.tex, line 61
k for a mean-zero f. For any C^q norm choose s>max(q,3). Smooth Fourier coefficients have finite weighted ℓ^1 norms of every order, and the displayed expression is bounded, up to a derivative convention constant, by 7^n(3-s)∑_k≠0 k ^s|c_k|. It tends geometrically to zero and is summable. The other two residue classes are obtained by the fixed continuous operators L_N,L_N^2. The series for g converges in every C^q norm; taking the limit gives h=δ_Mg. For a global holomorphic function on X, its La…
Exact mathematical passage
L_N,L_N^2
L_N^{3n}Use in source · 2 passages
Search aliases: L_N^3n
Use in source · rigidity.tex, line 54
equation then h=δ_Mg for a smooth mean-zero g, with convergent formula g=-frac17∑_j≥1L_N^jr. If h,r extend holomorphically to all of X, so does g. Apply L_N/7 to [source reference]. The trace relation gives L_Nh-h=(1/7)δ_M L_Nr, and induction yields L_N^nh-h=frac17δ_M∑_j=1^nL_N^jr. Because B^3=7I, L_N^3nf=∑_k≠0343^nc_7^nke_k for a mean-zero f. For any C^q norm choose s>max(q,3). Smooth Fourier coefficients have finite weighted ℓ^1 norms of every order, and the displayed expression is bounded, up…
Exact mathematical passage
L_N^{3n}f=\sum_{k\ne0}343^nc_{7^nk}e_k
L_N^jUse in source · 2 passages
Search aliases: L_N^j
Use in source · rigidity.tex, line 43
[Principal-norm rigidity] Let h,r be smooth mean-zero complex functions on T^3. If S_Nh-h∘ N=δ_Mr, then h=δ_Mg for a smooth mean-zero g, with convergent formula g=-frac17∑_j≥1L_N^jr. If h,r extend holomorphically to all of X, so does g. Apply L_N/7 to [source reference]. The trace relation gives L_Nh-h=(1/7)δ_M L_Nr, and induction yields L_N^nh-h=frac17δ_M∑_j=1^nL_N^jr. Because B^3=7I, L_N^3nf=∑_k≠0343^nc_7^nke_k for a mean-zero f. For any C^q norm choose s>max(q,3). Smooth Fourier coefficients …
Exact mathematical passage
g=-\frac17\sum_{j\ge1}L_N^jr.
L_N^nUse in source · 1 passage
Search aliases: L_N^n
Use in source · rigidity.tex, line 50
eginequation S_Nh-h∘ N=δ_Mr, then h=δ_Mg for a smooth mean-zero g, with convergent formula g=-frac17∑_j≥1L_N^jr. If h,r extend holomorphically to all of X, so does g. Apply L_N/7 to [source reference]. The trace relation gives L_Nh-h=(1/7)δ_M L_Nr, and induction yields L_N^nh-h=frac17δ_M∑_j=1^nL_N^jr. Because B^3=7I, L_N^3nf=∑_k≠0343^nc_7^nke_k for a mean-zero f. For any C^q norm choose s>max(q,3). Smooth Fourier coefficients have finite weighted ℓ^1 norms of every order, and the displayed expre…
Exact mathematical passage
L_N^nh-h=\frac17\delta_M\sum_{j=1}^{n}L_N^jr.
L_tUse in source · 12 passages
Search aliases: L_t
Use in source · norms.tex, line 56
The fifth-isogeny differential retains its constant. Let a=wp_t(1/5), b=wp_t(2/5) and L_t=(q_t'/q_t)'. The enlarged lattice is Λ_t^+=tfrac15 Z+ Zt, with wp_+(z)=25wp_5t(5z). The degree-five quotient satisfies L_t(z)=wp_+(z)-5wp_t(z)+2(a+b). At zero L_t has principal part -4/z^2; at the four nonzero kernel points it has +1/(z-P_j)^2, with no simple-pole terms. Thus L_t-wp_++5wp_t is a constant elliptic function. The canonical formula gives q_t(z)=(a-b)^2z^4[1+(a+b)z^2+O(z^4)], fixing that
Exact mathematical passage
L_t=(q_t'/q_t)'
\lceilUse in source · 2 passages
Search aliases: lceil
Use in source · principal.tex, line 70
0 · -5 · 4. Its quotient generator is (1,2,20)/7, representing δ. Thus β^-1Λ_5=Λ_5+ Zδ. Write a potential singular point as λ(a,b,c)=fracaτ+bs+c/5β, p=lceilfrac3c-2b7rceil, k=7p+2b-3c∈{0,…,6}. Exactly one argument λ+kδ lies in Λ_5. Its integral coordinates are (widehat a,widehat b,widehat c)=(p,2p-a,20p+5b-8c). Indeed N_5(widehat a,widehat b,widehat c)^T=(a,b-8k,c-5k)^T, which expresses β(λ+kδ)=βλ+k(-8s-1). The other six digits fail integrality. Also widehat c-2c=5(4p+b-2c), w(widehat c)=w(c). T…
Exact mathematical passage
\label{eq:lambda} \lambda(a,b,c)=\frac{a\tau+bs+c/5}{\beta},\qquad p=\left\lceil\frac{3c-2b}{7}\right\rceil,\qquad k=7p+2b-3c\in\{0,\ldots,6\}.
\lfloorUse in source · 1 passage
Search aliases: lfloor
Use in source · reconstruction.tex, line 30
A bounded complex radius bounds |a| and |b| by O(r); the real part then bounds |c| by O(r). There are O((1+r)^3) triples, and absolute weights are at most four. The seven translated numerator factors and the scaled denominator in [source reference] give a finite union of such cone supports and cannot increase this order. Combining the upper bound with the explicit lower families, choosing n=lfloor r/K_0rfloor, proves the assertion for all sufficiently large r.
Exact mathematical passage
n=\lfloor r/K_0\rfloor
\logUse in source · 27 passages
Search aliases: log
Use in source · product_proof.tex, line 3
Full seven-product series proof. For positive imaginary periods t,s,d with t+7d=2s+4, put P=e^2π it, Q=e^2π is, D=e^2π id and Z=e^2π iz. Then PD^7=Q^2. In a common convergence region the fifth-smoothed logarithms are log S_5(z;a,b) · =5∑_substackn≥1 5nmid n fracZ^n-(A_0B_0/Z)^nn(1-A_0^n)(1-B_0^n), log F_t,s(z) · =-5∑_substackn≥1 5nmid n fracP^nZ^n-Q^nZ^-nn(1-P^n)(1-Q^n), where A_0=e^2π ia,B_0=e^2π ib in the first line. At one surviving frequency suppress the index n. Summing seven shifts multipl…
Exact mathematical passage
\log S_5(z;a,b)&=5\sum_{\substack{n\ge1\\5\nmid n}} \frac{Z^n-(A_0B_0/Z)^n}{n(1-A_0^n)(1-B_0^n)},\\ \log F_{t,s}(z)&=-5\sum_{\substack{n\ge1\\5\nmid n}} \frac{P^nZ^n-Q^nZ^{-n}}{n(1-P^n)(1-Q^n)},
\LongleftrightarrowUse in source · 1 passage
Search aliases: Longleftrightarrow
Use in source · rigidity.tex, line 16
tic polynomial (x-2)^3+7 gives a stable real eigenvalue a=2-7^1/3∈(0,1) and a conjugate unstable pair of modulus a^-1/2>1. Every nonzero integer A-orbit escapes in both directions: otherwise an integer vector would tend to zero along positive or negative iterates. Fourier orbit sums are therefore absolutely convergent for smooth coefficients. The dual degree-seven image is exactly k∈ B Z^3 Longleftrightarrow ℓ(k):=k_0+2k_1+4k_2≡0pmod7. Indeed k-ℓ(k)e_0=B(-k_1,k_2,-8k_2)^T and B(0,-2,15)^T=7e_0. …
Exact mathematical passage
k\in B\mathbb Z^3\quad\Longleftrightarrow\quad \ell(k):=k_0+2k_1+4k_2\equiv0\pmod7.
\mathfrak mUse in source · 1 passage
Search aliases: m mathfrak
Use in source · stark.tex, line 195
m=3Jmathscr O_F_0. tagA3.1
Exact mathematical passage
\mathfrak m=3J\mathscr O_{F_0}. \tag{A3.1}
m^2Use in source · 2 passages
Search aliases: m^2
Use in source · theta.tex, line 128
ℓ^2=4(2a+b), m^2=4(a+2b),
Exact mathematical passage
\ell^2=4(2a+b),\quad m^2=4(a+2b),
M^TUse in source · 1 passage
Search aliases: M^T
Use in source · rigidity.tex, line 6
The actual two commuting actions. On X= C^3/ Z^3≃( C^*)^3 and its real torus T^3, use M=0 · 1 · 0 -4 · 2 · 15 -1 · 0 · 4, N=2I-M=2 · -1 · 0 4 · 0 · -15 1 · 0 · -2, A=M^T, B=N^T. The period row (τ,s,1) satisfies (τ,s,1)M=ϵ(τ,s,1) and (τ,s,1)N=β(τ,s,1). We have MN=NM, det M=1, det N=7 and N^3=7I. Thus N is a degree-seven cover on X. On the compact lattice Z[ω]^3 its degree is 49; these refer to different lattices.
Exact mathematical passage
M=\begin{pmatrix}0&1&0\\-4&2&15\\-1&0&4\end{pmatrix},\quad N=2I-M=\begin{pmatrix}2&-1&0\\4&0&-15\\1&0&-2\end{pmatrix},\quad A=M^T,\quad B=N^T.
m_\lambdaUse in source · 10 passages
Search aliases: m_λ lambda
Use in source · principal.tex, line 87
[Complete ceiling divisor] There are no zeros or poles of R outside β^-1Λ_5. At [source reference], m_λ:=ord_λR =w(c) [H(p)-H(2p-a)-H(a)+H(b)]∈{-4,-1,0,1,4}. Apply Theorem~[source reference] at the unique integral numerator lift [source reference] and subtract the denominator order at βλ. The weights agree. To bound the bracket, when a≤0,p>0 it is H(b); when a>0,p≤0 it is H(b)-1. In the other cases it is a difference of two zero-one values. Thus it lies in {-1,0,1}. In particular, ord_nτR=4 (n≥1…
Exact mathematical passage
\label{eq:complete-divisor} m_\lambda:=\operatorname{ord}_{\lambda}R =w(c)\,[H(p)-H(2p-a)-H(a)+H(b)]\in\{-4,-1,0,1,4\}.
\mathscrUse in source · 3 passages
Search aliases: mathscr
Use in source · stark.tex, line 195
m=3Jmathscr O_F_0. tagA3.1
Exact mathematical passage
\mathfrak m=3J\mathscr O_{F_0}. \tag{A3.1}
\maxUse in source · 1 passage
Search aliases: max
Use in source · rigidity.tex, line 57
ac17∑_j≥1L_N^jr. If h,r extend holomorphically to all of X, so does g. Apply L_N/7 to [source reference]. The trace relation gives L_Nh-h=(1/7)δ_M L_Nr, and induction yields L_N^nh-h=frac17δ_M∑_j=1^nL_N^jr. Because B^3=7I, L_N^3nf=∑_k≠0343^nc_7^nke_k for a mean-zero f. For any C^q norm choose s>max(q,3). Smooth Fourier coefficients have finite weighted ℓ^1 norms of every order, and the displayed expression is bounded, up to a derivative convention constant, by 7^n(3-s)∑_k≠0 k ^s|c_k|. It tends g…
Exact mathematical passage
s>\max(q,3)
\midUse in source · 8 passages
Search aliases: mid
Use in source · norms.tex, line 6
Exactly normalized elliptic norms. On E_t= C/Λ_t define N_nf(w)=∏_nz=wf(z) for integers n≥1, with all n^2 preimages counted. The product is independent of the representative lift. Root-of-unity averaging and vertical residue splitting in [source reference] give N_nq_t(w)=q_t(w), · 5nmid n, 1, · 5mid n. In the prime-to-five case the two product identities are ∏_a=0^n-1θ_0(z+a/n;t)=θ_0(nz;nt), ∏_b=0^n-1θ_0(nz+bt;nt)=θ_0(nz;t). Multiplication by five permutes the real residues in the smoothing deno…
Exact mathematical passage
\label{eq:qnorm} N_nq_t(w)=\begin{cases}q_t(w),&5\nmid n,\\1,&5\mid n.\end{cases}
NUse in source · 35 passages
Search aliases: N
Use in source · H12_Cubic_Divisor_Traditional.tex, line 68
τ · =(β^2+2β)/15, · t · =-τ, · s · =-(β+2)/15, · δ · =(τ+2s+4)/7. The real cube root 7^1/3 is positive; this fixes the lower complex embedding of β. We have Im t,Im s,Imδ>0, Imτ<0 and |ρ|<1. The last follows from N(ϵ)=1 and 0<2-7^1/3<1 at the real embedding. The numbers 1,t,s are rationally independent: a rational linear relation becomes a polynomial of degree at most two in β, contradicting its irreducible cubic minimal polynomial. [htbp]centeringincludegraphics[width=.96linewidth]figures/01_pe…
Exact mathematical passage
N(\epsilon)=1
nUse in source · 71 passages
Search aliases: n
Use in source · H12_Cubic_Divisor_Traditional.tex, line 72
For positive imaginary periods use the additive product convention θ_0(z;a) · =∏_n≥0(1-e^2π i(z+na))(1-e^2π i((n+1)a-z)), Γ_rm ell(z;a,b) · =∏_j,k≥0 frac1-e^2π i((j+1)a+(k+1)b-z)1-e^2π i(ja+kb+z). The negative-first-period continuation is Γ_rm ell(z;-t,s)=Γ_rm ell(z+t;t,s)^-1. Consequently the main scalar function is F(z)=fracΓ_rm ell(z;τ,s)^5Γ_rm ell(5z;5τ,5s) =fracΓ_rm ell(5(z+t);5t,5s)Γ_rm ell(z+t;t,s)^5, U_Γ=F(1/3). This convention agrees with the checked primary authors' copy of Bergeron--C…
Exact mathematical passage
\theta_0(z;a)&=\prod_{n\ge0}(1-e^{2\pi i(z+na)})(1-e^{2\pi i((n+1)a-z)}),\label{eq:theta-product}\\ \Gamma_{\rm ell}(z;a,b)&=\prod_{j,k\ge0} \frac{1-e^{2\pi i((j+1)a+(k+1)b-z)}}{1-e^{2\pi i(ja+kb+z)}}.\label{eq:gamma-product}
n^2Use in source · 1 passage
Search aliases: n^2
Use in source · norms.tex, line 5
Exactly normalized elliptic norms. On E_t= C/Λ_t define N_nf(w)=∏_nz=wf(z) for integers n≥1, with all n^2 preimages counted. The product is independent of the representative lift. Root-of-unity averaging and vertical residue splitting in [source reference] give N_nq_t(w)=q_t(w), · 5nmid n,
Exact mathematical passage
n^2
N^3Use in source · 1 passage
Search aliases: N^3
Use in source · rigidity.tex, line 11
1 · 0 · -2, A=M^T, B=N^T. The period row (τ,s,1) satisfies (τ,s,1)M=ϵ(τ,s,1) and (τ,s,1)N=β(τ,s,1). We have MN=NM, det M=1, det N=7 and N^3=7I. Thus N is a degree-seven cover on X. On the compact lattice Z[ω]^3 its degree is 49; these refer to different lattices.
Exact mathematical passage
N^3=7I
n^3Use in source · 3 passages
Search aliases: n^3
Use in source · reconstruction.tex, line 18
' forces both sides to vanish; the zero and pole families are disjoint by c modulo five. [Two-sided cubic growth] Let n_+(r) and n_-(r) count the zeros and poles of R in |z|≤ r, with multiplicity. Then n_+(r)=Θ(r^3), n_-(r)=Θ(r^3), n_D(r):=n_+(r)+n_-(r)=Θ(r^3). In particular, for K_0=(|τ|+|s|+2)/|β|, n_+(K_0n)≥4n^3, n_-(K_0n)≥4n^3. These are unsigned counts; no cubic lower bound for a signed zero-minus-pole mass is asserted. In [source reference], |a|,b≤ n and 0<c≤10n. The triangle inequality pl…
Exact mathematical passage
n_+(K_0n)\ge4n^3,\qquad n_-(K_0n)\ge4n^3.
n^4Use in source · 1 passage
Search aliases: n^4
Use in source · theta.tex, line 193
∏_η∈ E_t[n]∖{0}q_t(η)=n^4 (5nmid n). tagN4.1
Exact mathematical passage
\boxed{\prod_{\eta\in E_t[n]\setminus\{0\}}q_t(\eta)=n^4} \quad(5\nmid n). \tag{N4.1}
N^TUse in source · 1 passage
Search aliases: N^T
Use in source · rigidity.tex, line 6
The actual two commuting actions. On X= C^3/ Z^3≃( C^*)^3 and its real torus T^3, use M=0 · 1 · 0 -4 · 2 · 15 -1 · 0 · 4, N=2I-M=2 · -1 · 0 4 · 0 · -15 1 · 0 · -2, A=M^T, B=N^T. The period row (τ,s,1) satisfies (τ,s,1)M=ϵ(τ,s,1) and (τ,s,1)N=β(τ,s,1). We have MN=NM, det M=1, det N=7 and N^3=7I. Thus N is a degree-seven cover on X. On the compact lattice Z[ω]^3 its degree is 49; these refer to different lattices.
Exact mathematical passage
M=\begin{pmatrix}0&1&0\\-4&2&15\\-1&0&4\end{pmatrix},\quad N=2I-M=\begin{pmatrix}2&-1&0\\4&0&-15\\1&0&-2\end{pmatrix},\quad A=M^T,\quad B=N^T.
n^TUse in source · 1 passage
Search aliases: n^T
Use in source · reconstruction.tex, line 58
functions formed by finite polynomial expressions in these restrictions, derivatives and polynomials in z. Let u be a single global nowhere-zero entire function. Every meromorphic function u(z)fracP(z)Q(z) has zero and pole counts O(r^2). Thus R is not representable in this class. For Y=ImΩ>0, the defining theta series has terms bounded by exp{-π n^TYn+2π n Im(Az+b) }. Completing the square, and absorbing half of the positive quadratic form into a convergent Gaussian sum, gives Cexp(C'|z|^2) for…
Exact mathematical passage
\exp\{-\pi n^TYn+2\pi\|n\|\,\|\Im(Az+b)\|\}.
n_+Use in source · 3 passages
Search aliases: n_+
Use in source · reconstruction.tex, line 12
p>0 and the bracket in [source reference] is one. The Z points have order four and the P points have order minus one. Rational independence makes the parameter map injective. The pole indices are distinct because 5(k-k')=r-r' forces both sides to vanish; the zero and pole families are disjoint by c modulo five. [Two-sided cubic growth] Let n_+(r) and n_-(r) count the zeros and poles of R in |z|≤ r, with multiplicity. Then n_+(r)=Θ(r^3), n_-(r)=Θ(r^3), n_D(r):=n_+(r)+n_-(r)=Θ(r^3). In particular,…
Exact mathematical passage
n_+(r)
n_-Use in source · 3 passages
Search aliases: n_-
Use in source · reconstruction.tex, line 12
bracket in [source reference] is one. The Z points have order four and the P points have order minus one. Rational independence makes the parameter map injective. The pole indices are distinct because 5(k-k')=r-r' forces both sides to vanish; the zero and pole families are disjoint by c modulo five. [Two-sided cubic growth] Let n_+(r) and n_-(r) count the zeros and poles of R in |z|≤ r, with multiplicity. Then n_+(r)=Θ(r^3), n_-(r)=Θ(r^3), n_D(r):=n_+(r)+n_-(r)=Θ(r^3). In particular, for K_0=(|τ…
Exact mathematical passage
n_-(r)
N_{2k}Use in source · 1 passage
Search aliases: N_2k
Use in source · norms.tex, line 39
point, so each denominator norm is a fourth power. Apply [source reference] and cancel the numerator. If 5mid n then all these norms are one, including the even case since 5mid k. For n=2k+1 prime to five, B_k(w)=∏_j=0^k-1q_t(2w+(2j+1)s) gives N_2k+1ψ(w)=ψ(w)fracB_k(w)B_k(w+s). For n=2k prime to five, C_k(w)=∏_j=0^k-1q_t(w+js)^4 gives N_2kψ(w)=q_t(w)^-6fracC_k(w)C_k(w+s). Finite products therefore retain explicit endpoint multipliers. Closing a permutation of labels does not remove them or fix t…
Exact mathematical passage
N_{2k}\psi(w)=q_t(w)^{-6}\frac{C_k(w)}{C_k(w+s)}.
N_{2k+1}Use in source · 1 passage
Search aliases: N_2k+1
Use in source · norms.tex, line 35
position For odd n, doubling permutes E_t[n]. For n=2k, it maps E_t[n] to E_t[k] with four lifts per point, so each denominator norm is a fourth power. Apply [source reference] and cancel the numerator. If 5mid n then all these norms are one, including the even case since 5mid k. For n=2k+1 prime to five, B_k(w)=∏_j=0^k-1q_t(2w+(2j+1)s) gives N_2k+1ψ(w)=ψ(w)fracB_k(w)B_k(w+s). For n=2k prime to five, C_k(w)=∏_j=0^k-1q_t(w+js)^4 gives N_2kψ(w)=q_t(w)^-6fracC_k(w)C_k(w+s). Finite products therefor…
Exact mathematical passage
N_{2k+1}\psi(w)=\psi(w)\frac{B_k(w)}{B_k(w+s)}.
N_5Use in source · 2 passages
Search aliases: N_5
Use in source · principal.tex, line 60
Multiplication by β in the ordered basis (τ,s,1/5) is N_5=2 · -1 · 0 4 · 0 · -3 5 · 0 · -2, N_5^3=7I, det N_5=7, N_5^-1=frac170 · -2 · 3 -7 · -4 · 6 0 · -5 · 4. Its quotient generator is (1,2,20)/7, representing δ. Thus β^-1Λ_5=Λ_5+ Zδ. Write a potential singular point as λ(a,b,c)=fracaτ+bs+c/5β, p=lceilfrac3c-2b7rceil, k=7p+2b-3c∈{0,…,6}. Exactly one argument λ+kδ lies in Λ_5. Its integral coordinates are (widehat a,widehat b,widehat c)=(p,2p-a,20p+5b-8c). Indeed N_5(widehat a,widehat b,widehat…
Exact mathematical passage
N_5=\begin{pmatrix}2&-1&0\\4&0&-3\\5&0&-2\end{pmatrix},\qquad N_5^3=7I,\quad\det N_5=7,\quad N_5^{-1}=\frac17\begin{pmatrix}0&-2&3\\-7&-4&6\\0&-5&4\end{pmatrix}.
N_5^{-1}Use in source · 1 passage
Search aliases: N_5^-1
Use in source · principal.tex, line 60
Multiplication by β in the ordered basis (τ,s,1/5) is N_5=2 · -1 · 0 4 · 0 · -3 5 · 0 · -2, N_5^3=7I, det N_5=7, N_5^-1=frac170 · -2 · 3 -7 · -4 · 6 0 · -5 · 4. Its quotient generator is (1,2,20)/7, representing δ. Thus β^-1Λ_5=Λ_5+ Zδ. Write a potential singular point as λ(a,b,c)=fracaτ+bs+c/5β, p=lceilfrac3c-2b7rceil, k=7p+2b-3c∈{0,…,6}. Exactly one argument λ+kδ lies in Λ_5. Its integral coordinates are (widehat a,widehat b,widehat c)=(p,2p-a,20p+5b-8c). Indeed N_5(widehat a,widehat b,widehat…
Exact mathematical passage
N_5=\begin{pmatrix}2&-1&0\\4&0&-3\\5&0&-2\end{pmatrix},\qquad N_5^3=7I,\quad\det N_5=7,\quad N_5^{-1}=\frac17\begin{pmatrix}0&-2&3\\-7&-4&6\\0&-5&4\end{pmatrix}.
N_5^3Use in source · 1 passage
Search aliases: N_5^3
Use in source · principal.tex, line 60
Multiplication by β in the ordered basis (τ,s,1/5) is N_5=2 · -1 · 0 4 · 0 · -3 5 · 0 · -2, N_5^3=7I, det N_5=7, N_5^-1=frac170 · -2 · 3 -7 · -4 · 6 0 · -5 · 4. Its quotient generator is (1,2,20)/7, representing δ. Thus β^-1Λ_5=Λ_5+ Zδ. Write a potential singular point as λ(a,b,c)=fracaτ+bs+c/5β, p=lceilfrac3c-2b7rceil, k=7p+2b-3c∈{0,…,6}. Exactly one argument λ+kδ lies in Λ_5. Its integral coordinates are (widehat a,widehat b,widehat c)=(p,2p-a,20p+5b-8c). Indeed N_5(widehat a,widehat b,widehat…
Exact mathematical passage
N_5=\begin{pmatrix}2&-1&0\\4&0&-3\\5&0&-2\end{pmatrix},\qquad N_5^3=7I,\quad\det N_5=7,\quad N_5^{-1}=\frac17\begin{pmatrix}0&-2&3\\-7&-4&6\\0&-5&4\end{pmatrix}.
n_DUse in source · 4 passages
Search aliases: n_D
Use in source · reconstruction.tex, line 13
the P points have order minus one. Rational independence makes the parameter map injective. The pole indices are distinct because 5(k-k')=r-r' forces both sides to vanish; the zero and pole families are disjoint by c modulo five. [Two-sided cubic growth] Let n_+(r) and n_-(r) count the zeros and poles of R in |z|≤ r, with multiplicity. Then n_+(r)=Θ(r^3), n_-(r)=Θ(r^3), n_D(r):=n_+(r)+n_-(r)=Θ(r^3). In particular, for K_0=(|τ|+|s|+2)/|β|, n_+(K_0n)≥4n^3, n_-(K_0n)≥4n^3. These are unsigned counts…
Exact mathematical passage
n_+(r)=\Theta(r^3),\qquad n_-(r)=\Theta(r^3),\qquad n_D(r):=n_+(r)+n_-(r)=\Theta(r^3).
N_{E/E^H}Use in source · 1 passage
Search aliases: N_E/E^H
Use in source · polarization.tex, line 103
∏_H a lineN_E/E^H(z) =z^3N_E/F_0(z). tagQ2
Exact mathematical passage
\boxed{ \prod_{H\text{ a line}}N_{E/E^H}(z) =z^3N_{E/F_0}(z). } \tag{Q2}
N_{E/F_0}Use in source · 2 passages
Search aliases: N_E/F_0
Use in source · polarization.tex, line 103
∏_H a lineN_E/E^H(z) =z^3N_E/F_0(z). tagQ2
Exact mathematical passage
\boxed{ \prod_{H\text{ a line}}N_{E/E^H}(z) =z^3N_{E/F_0}(z). } \tag{Q2}
N_{K/\mathbb Q}Use in source · 1 passage
Search aliases: N_K/ Q mathbb
Use in source · stark.tex, line 27
Indeed, N_K/ Q(ϵ)=1 and its real image is 2-7^1/3. The class number is three and ϵ is a fundamental unit. These arithmetic facts are recorded in BCG, section 5.2; the earlier unramified-cubic note also proves the class-number and ray-field identification used here.
Exact mathematical passage
N_{K/\mathbb Q}(\epsilon)=1
N_nUse in source · 4 passages
Search aliases: N_n
Use in source · norms.tex, line 5
Exactly normalized elliptic norms. On E_t= C/Λ_t define N_nf(w)=∏_nz=wf(z) for integers n≥1, with all n^2 preimages counted. The product is independent of the representative lift. Root-of-unity averaging and vertical residue splitting in [source reference] give N_nq_t(w)=q_t(w), · 5nmid n,
Exact mathematical passage
N_nf(w)=\prod_{nz=w}f(z)
N_\kappaUse in source · 2 passages
Search aliases: N_κ kappa
Use in source · stark.tex, line 496
Let N_κ(z)=zκ(z)=|z|^2 in the fixed embedding. The six exact identities needed for the comparison are
Exact mathematical passage
N_\kappa(z)=z\kappa(z)=|z|^2
\nmidUse in source · 6 passages
Search aliases: nmid
Use in source · norms.tex, line 6
Exactly normalized elliptic norms. On E_t= C/Λ_t define N_nf(w)=∏_nz=wf(z) for integers n≥1, with all n^2 preimages counted. The product is independent of the representative lift. Root-of-unity averaging and vertical residue splitting in [source reference] give N_nq_t(w)=q_t(w), · 5nmid n, 1, · 5mid n. In the prime-to-five case the two product identities are ∏_a=0^n-1θ_0(z+a/n;t)=θ_0(nz;nt), ∏_b=0^n-1θ_0(nz+bt;nt)=θ_0(nz;t). Multiplication by five permutes the real residues in the smoothing deno…
Exact mathematical passage
\label{eq:qnorm} N_nq_t(w)=\begin{cases}q_t(w),&5\nmid n,\\1,&5\mid n.\end{cases}
OUse in source · 14 passages
Search aliases: O
Use in source · norms.tex, line 60
mathbb Z+ Zt, with wp_+(z)=25wp_5t(5z). The degree-five quotient satisfies L_t(z)=wp_+(z)-5wp_t(z)+2(a+b). At zero L_t has principal part -4/z^2; at the four nonzero kernel points it has +1/(z-P_j)^2, with no simple-pole terms. Thus L_t-wp_++5wp_t is a constant elliptic function. The canonical formula gives q_t(z)=(a-b)^2z^4[1+(a+b)z^2+O(z^4)], fixing that constant as 2(a+b). The degree-five quotient is equivalently the paired sum wp_+(z)=wp_t(z)+∑_j=1^4[wp_t(z+j/5)-wp_t(j/5)]. The analytic iden…
Exact mathematical passage
q_t(z)=(a-b)^2z^4[1+(a+b)z^2+O(z^4)]
oUse in source · 4 passages
Search aliases: o
Use in source · remaining.tex, line 3
The exact remaining arithmetic comparison. The fixed arithmetic presentation gives, with a_0,a_1,a_2 as in the polarized block, C_rm pol=fracϵ a_1a_2^3ω, C_rm polU_rm alg^3=R_+:=|ϵ|^4|a_1|^6|a_2|^4>0. The modulus theorem gives |C_rm polU_Γ^3|=R_+. The exact statement still missing is Im(C_rm polU_Γ^3)=0. Reality together with a valid positive-real-part enclosure would give C_rm polU_Γ^3=R_+, hence U_Γ^3=U_rm alg^3. A valid independent bound |U_Γ/U_rm alg-1|<1/2 would select the root U_Γ=U_rm alg…
Exact mathematical passage
C_{\rm pol}=\frac{\epsilon a_1a_2^3}{\omega},\qquad C_{\rm pol}U_{\rm alg}^3=R_+:=|\epsilon|^4|a_1|^6|a_2|^4>0.
O_{F_0}Use in source · 3 passages
Search aliases: O_F_0
Use in source · stark.tex, line 195
m=3Jmathscr O_F_0. tagA3.1
Exact mathematical passage
\mathfrak m=3J\mathscr O_{F_0}. \tag{A3.1}
\operatorname{ord}Use in source · 6 passages
Search aliases: ord operatorname
Use in source · principal.tex, line 45
[Base half-space divisor] The divisor of F is supported on Λ_5. At its uniquely parametrized point, ord_aτ+bs+c/5F=w(c) [H(a)-H(b)]. For positive periods t,s, the defining gamma product has a simple zero at jt+ks+ Z for j,k≥1, a simple pole for j,k≤0, and no divisor in the mixed cones. Rational independence of 1,t,s prevents collisions. The argument z+t has coordinates (1-a,b) in the (t,s) basis, giving gamma order H(b)-H(a). In F(z)=fracΓ_rm ell(5(z+t);5t,5s) Γ_rm ell(z+t;t,s)^5, the numerator …
Exact mathematical passage
\label{eq:base-divisor} \operatorname{ord}_{a\tau+bs+c/5}F=w(c)\,[H(a)-H(b)].
\operatorname{ord}_{-n}Use in source · 1 passage
Search aliases: ord_-n operatorname
Use in source · principal.tex, line 96
Apply Theorem~[source reference] at the unique integral numerator lift [source reference] and subtract the denominator order at βλ. The weights agree. To bound the bracket, when a≤0,p>0 it is H(b); when a>0,p≤0 it is H(b)-1. In the other cases it is a difference of two zero-one values. Thus it lies in {-1,0,1}. In particular, ord_nτR=4 (n≥1), ord_nsR=-4 (n∈ Z∖{0}), ord_-nR=4 (n≥1). For these assertions the triples are (2n,4n,5n), (-n,0,0) and (0,15n,10n) respectively. At zero the order is zero. …
Exact mathematical passage
\operatorname{ord}_{n\tau}R=4\quad(n\ge1),\qquad \operatorname{ord}_{ns}R=-4\quad(n\in\mathbb Z\setminus\{0\}),\qquad \operatorname{ord}_{-n}R=4\quad(n\ge1).
\operatorname{ord}_{a\tau+bs+c/5}Use in source · 1 passage
Search aliases: ord_aτ+bs+c/5 operatorname tau
Use in source · principal.tex, line 45
[Base half-space divisor] The divisor of F is supported on Λ_5. At its uniquely parametrized point, ord_aτ+bs+c/5F=w(c) [H(a)-H(b)]. For positive periods t,s, the defining gamma product has a simple zero at jt+ks+ Z for j,k≥1, a simple pole for j,k≤0, and no divisor in the mixed cones. Rational independence of 1,t,s prevents collisions. The argument z+t has coordinates (1-a,b) in the (t,s) basis, giving gamma order H(b)-H(a). In F(z)=fracΓ_rm ell(5(z+t);5t,5s) Γ_rm ell(z+t;t,s)^5, the numerator …
Exact mathematical passage
\label{eq:base-divisor} \operatorname{ord}_{a\tau+bs+c/5}F=w(c)\,[H(a)-H(b)].
\operatorname{ord}_{ns}Use in source · 1 passage
Search aliases: ord_ns operatorname
Use in source · principal.tex, line 96
Apply Theorem~[source reference] at the unique integral numerator lift [source reference] and subtract the denominator order at βλ. The weights agree. To bound the bracket, when a≤0,p>0 it is H(b); when a>0,p≤0 it is H(b)-1. In the other cases it is a difference of two zero-one values. Thus it lies in {-1,0,1}. In particular, ord_nτR=4 (n≥1), ord_nsR=-4 (n∈ Z∖{0}), ord_-nR=4 (n≥1). For these assertions the triples are (2n,4n,5n), (-n,0,0) and (0,15n,10n) respectively. At zero the order is zero. …
Exact mathematical passage
\operatorname{ord}_{n\tau}R=4\quad(n\ge1),\qquad \operatorname{ord}_{ns}R=-4\quad(n\in\mathbb Z\setminus\{0\}),\qquad \operatorname{ord}_{-n}R=4\quad(n\ge1).
\operatorname{ord}_{n\tau}Use in source · 1 passage
Search aliases: ord_nτ operatorname tau
Use in source · principal.tex, line 96
Apply Theorem~[source reference] at the unique integral numerator lift [source reference] and subtract the denominator order at βλ. The weights agree. To bound the bracket, when a≤0,p>0 it is H(b); when a>0,p≤0 it is H(b)-1. In the other cases it is a difference of two zero-one values. Thus it lies in {-1,0,1}. In particular, ord_nτR=4 (n≥1), ord_nsR=-4 (n∈ Z∖{0}), ord_-nR=4 (n≥1). For these assertions the triples are (2n,4n,5n), (-n,0,0) and (0,15n,10n) respectively. At zero the order is zero. …
Exact mathematical passage
\operatorname{ord}_{n\tau}R=4\quad(n\ge1),\qquad \operatorname{ord}_{ns}R=-4\quad(n\in\mathbb Z\setminus\{0\}),\qquad \operatorname{ord}_{-n}R=4\quad(n\ge1).
\operatorname{ord}_zUse in source · 1 passage
Search aliases: ord_z operatorname
Use in source · principal.tex, line 120
gives two q_s factors and four reciprocal q_t factors; the 2τ shift disappears from q_t by ellipticity. Finally P_7(z+δ)/P_7(z)=F(z+7δ)/F(z) and the last two identities of [source reference] give [source reference]. Under z↦ z+1 the triple changes to (a,b-15,c-10), leaving p and w(c) unchanged. Formula~[source reference] gives ord_z+1R-ord_zR =-w(c)1_{1≤ b≤15}, the divisor of [source reference]. Under z↦ z+δ the triple becomes (a,b-8,c-5); its digit decreases by one modulo seven, and its pivot i…
Exact mathematical passage
\operatorname{ord}_{z+1}R-\operatorname{ord}_zR =-w(c)\mathbf1_{\{1\le b\le15\}},
\operatorname{ord}_{z+1}Use in source · 1 passage
Search aliases: ord_z+1 operatorname
Use in source · principal.tex, line 120
gives two q_s factors and four reciprocal q_t factors; the 2τ shift disappears from q_t by ellipticity. Finally P_7(z+δ)/P_7(z)=F(z+7δ)/F(z) and the last two identities of [source reference] give [source reference]. Under z↦ z+1 the triple changes to (a,b-15,c-10), leaving p and w(c) unchanged. Formula~[source reference] gives ord_z+1R-ord_zR =-w(c)1_{1≤ b≤15}, the divisor of [source reference]. Under z↦ z+δ the triple becomes (a,b-8,c-5); its digit decreases by one modulo seven, and its pivot i…
Exact mathematical passage
\operatorname{ord}_{z+1}R-\operatorname{ord}_zR =-w(c)\mathbf1_{\{1\le b\le15\}},
\operatorname{ord}_{\lambda}Use in source · 1 passage
Search aliases: ord_λ operatorname lambda
Use in source · principal.tex, line 87
[Complete ceiling divisor] There are no zeros or poles of R outside β^-1Λ_5. At [source reference], m_λ:=ord_λR =w(c) [H(p)-H(2p-a)-H(a)+H(b)]∈{-4,-1,0,1,4}. Apply Theorem~[source reference] at the unique integral numerator lift [source reference] and subtract the denominator order at βλ. The weights agree. To bound the bracket, when a≤0,p>0 it is H(b); when a>0,p≤0 it is H(b)-1. In the other cases it is a difference of two zero-one values. Thus it lies in {-1,0,1}. In particular, ord_nτR=4 (n≥1…
Exact mathematical passage
\label{eq:complete-divisor} m_\lambda:=\operatorname{ord}_{\lambda}R =w(c)\,[H(p)-H(2p-a)-H(a)+H(b)]\in\{-4,-1,0,1,4\}.
\operatorname{ord}_{\tau}Use in source · 2 passages
Search aliases: ord_τ operatorname tau
Use in source · principal.tex, line 144
r cannot be discarded Put R^#(z)=overlineR(overline z). The cubic fields Q(β) and Q(overlineβ) are distinct: equality would put β/overlineβ=ω in a cubic field. Their intersection is Q. The support of divR lies in the first, and that of R^# in the second. Since τ is irrational (indeed β=2-1/(4+15τ)), ord_τ(R/R^#)=4. Thus this naive conjugation quotient is not a holomorphic unit. A constant polarization, root of unity or nowhere-zero holomorphic gauge cannot change this order. A relative scalar mo…
Exact mathematical passage
\operatorname{ord}_{\tau}(R/R^\#)=4.
\overline CUse in source · 1 passage
Search aliases: overline C overline
Use in source · polarization.tex, line 332
overlineC U_Γ^3=C U_Γ^3, (fracU_ΓoverlineU_Γ)^3 =fracoverline CC. tagQ9
Exact mathematical passage
\overline{C U_\Gamma^3}=C U_\Gamma^3, \qquad \left(\frac{U_\Gamma}{\overline{U_\Gamma}}\right)^3 =\frac{\overline C}{C}. \tag{Q9}
\overline{a_1}Use in source · 3 passages
Search aliases: overlinea_1 overline
Use in source · polarization.tex, line 214
frac Aι A =fraca_1^2overlinea_1^ 3overlinea_2^ 2a_2.
Exact mathematical passage
\frac{\mathcal A}{\iota\mathcal A} =\frac{a_1^2\overline{a_1}^{\,3}\overline{a_2}^{\,2}}{a_2}.
\overline{a_1}^{\,3}Use in source · 2 passages
Search aliases: overlinea_1^ 3 overline
Use in source · polarization.tex, line 214
frac Aι A =fraca_1^2overlinea_1^ 3overlinea_2^ 2a_2.
Exact mathematical passage
\frac{\mathcal A}{\iota\mathcal A} =\frac{a_1^2\overline{a_1}^{\,3}\overline{a_2}^{\,2}}{a_2}.
\overline{a_2}Use in source · 3 passages
Search aliases: overlinea_2 overline
Use in source · polarization.tex, line 214
frac Aι A =fraca_1^2overlinea_1^ 3overlinea_2^ 2a_2.
Exact mathematical passage
\frac{\mathcal A}{\iota\mathcal A} =\frac{a_1^2\overline{a_1}^{\,3}\overline{a_2}^{\,2}}{a_2}.
\overline{a_2}^{\,2}Use in source · 2 passages
Search aliases: overlinea_2^ 2 overline
Use in source · polarization.tex, line 214
frac Aι A =fraca_1^2overlinea_1^ 3overlinea_2^ 2a_2.
Exact mathematical passage
\frac{\mathcal A}{\iota\mathcal A} =\frac{a_1^2\overline{a_1}^{\,3}\overline{a_2}^{\,2}}{a_2}.
\overline{C U_\Gamma^3}Use in source · 1 passage
Search aliases: overlineC U_Γ^3 overline Gamma
Use in source · polarization.tex, line 332
overlineC U_Γ^3=C U_Γ^3, (fracU_ΓoverlineU_Γ)^3 =fracoverline CC. tagQ9
Exact mathematical passage
\overline{C U_\Gamma^3}=C U_\Gamma^3, \qquad \left(\frac{U_\Gamma}{\overline{U_\Gamma}}\right)^3 =\frac{\overline C}{C}. \tag{Q9}
\overline{f(\overline z)}Use in source · 1 passage
Search aliases: overlinef(overline z) overline overline
Use in source · reconstruction.tex, line 200
The repaired conjugation quotient and its boundary. For f^#(z)=overlinef(overline z), the canonical product W_D^# has the conjugate divisor. Hence fracR/R^#W_D/W_D^# =fracR_0overlineR_0 exp(∑_j=1^3(α_j-overlineα_j)z^j). This is entire and nowhere zero, with modulus one on the real axis. It cancels all unmatched divisor orders by a canonical meromorphic factor. That factor is not a holomorphic-unit gauge or a finite theta section. Regularity on the scalar plane
Exact mathematical passage
f^\#(z)=\overline{f(\overline z)}
\overline{F_{t,s}(z)}Use in source · 1 passage
Search aliases: overlineF_t,s(z) overline
Use in source · polarization.tex, line 348
overlineF_t,s(z)=F_-bar t,-bar s(-bar z).
Exact mathematical passage
\overline{F_{t,s}(z)}=F_{-\bar t,-\bar s}(-\bar z).
\overline{R(\overline z)}Use in source · 1 passage
Search aliases: overlineR(overline z) overline overline
Use in source · principal.tex, line 143
The conjugation divisor cannot be discarded. Put R^#(z)=overlineR(overline z). The cubic fields Q(β) and Q(overlineβ) are distinct: equality would put β/overlineβ=ω in a cubic field. Their intersection is Q. The support of divR lies in the first, and that of R^# in the second. Since τ is irrational (indeed β=2-1/(4+15τ)), ord_τ(R/R^#)=4. Thus this naive conjugation quotient is not a holomorphic unit. A constant polarization, root of unity or nowhere-zero
Exact mathematical passage
R^\#(z)=\overline{R(\overline z)}
\overline{R_0}Use in source · 1 passage
Search aliases: overlineR_0 overline
Use in source · reconstruction.tex, line 201
The repaired conjugation quotient and its boundary. For f^#(z)=overlinef(overline z), the canonical product W_D^# has the conjugate divisor. Hence fracR/R^#W_D/W_D^# =fracR_0overlineR_0 exp(∑_j=1^3(α_j-overlineα_j)z^j). This is entire and nowhere zero, with modulus one on the real axis. It cancels all unmatched divisor orders by a canonical meromorphic factor. That factor is not a holomorphic-unit gauge or a finite theta section. Regularity on the scalar plane does not establish periodic descent…
Exact mathematical passage
\label{eq:regularized-conjugation} \frac{R/R^\#}{W_D/W_D^\#} =\frac{R_0}{\overline{R_0}} \exp\left(\sum_{j=1}^{3}(\alpha_j-\overline{\alpha_j})z^j\right).
\overline{U_\Gamma}Use in source · 2 passages
Search aliases: overlineU_Γ overline Gamma
Use in source · polarization.tex, line 332
overlineC U_Γ^3=C U_Γ^3, (fracU_ΓoverlineU_Γ)^3 =fracoverline CC. tagQ9
Exact mathematical passage
\overline{C U_\Gamma^3}=C U_\Gamma^3, \qquad \left(\frac{U_\Gamma}{\overline{U_\Gamma}}\right)^3 =\frac{\overline C}{C}. \tag{Q9}
\overline{U_\Gamma}^{\,-1}Use in source · 1 passage
Search aliases: overlineU_Γ^ -1 overline Gamma
Use in source · polarization.tex, line 354
F_-bar t,-bar s(1/3)=overlineU_Γ^ -1.
Exact mathematical passage
F_{-\bar t,-\bar s}(1/3)=\overline{U_\Gamma}^{\,-1}.
\overline{\alpha_j}Use in source · 1 passage
Search aliases: overlineα_j overline alpha
Use in source · reconstruction.tex, line 201
The repaired conjugation quotient and its boundary. For f^#(z)=overlinef(overline z), the canonical product W_D^# has the conjugate divisor. Hence fracR/R^#W_D/W_D^# =fracR_0overlineR_0 exp(∑_j=1^3(α_j-overlineα_j)z^j). This is entire and nowhere zero, with modulus one on the real axis. It cancels all unmatched divisor orders by a canonical meromorphic factor. That factor is not a holomorphic-unit gauge or a finite theta section. Regularity on the scalar plane does not establish periodic descent…
Exact mathematical passage
\label{eq:regularized-conjugation} \frac{R/R^\#}{W_D/W_D^\#} =\frac{R_0}{\overline{R_0}} \exp\left(\sum_{j=1}^{3}(\alpha_j-\overline{\alpha_j})z^j\right).
\overline\betaUse in source · 2 passages
Search aliases: overlineβ overline beta
Use in source · principal.tex, line 143
The conjugation divisor cannot be discarded. Put R^#(z)=overlineR(overline z). The cubic fields Q(β) and Q(overlineβ) are distinct: equality would put β/overlineβ=ω in a cubic field. Their intersection is Q. The support of divR lies in the first, and that of R^# in the second. Since τ is irrational (indeed β=2-1/(4+15τ)), ord_τ(R/R^#)=4. Thus this naive conjugation quotient is not a holomorphic unit. A constant polarization, root of unity or nowhere-zero holomorphic gauge cannot change this orde…
Exact mathematical passage
\mathbb Q(\overline\beta)
\overline\epsilonUse in source · 4 passages
Search aliases: overlineϵ overline epsilon
Use in source · polarization.tex, line 57
In the fixed embedding ϵ_1=2-7^1/3>0, and ϵ_2=overlineϵ.
Exact mathematical passage
\epsilon_2=\overline\epsilon
\overline\epsilon^{\,2}Use in source · 2 passages
Search aliases: overlineϵ^ 2 overline epsilon
Use in source · polarization.tex, line 219
Moreover ϵ_2/ϵ_1=ϵoverlineϵ^ 2. Equation (Q3) becomes
Exact mathematical passage
\epsilon_2/\epsilon_1=\epsilon\overline\epsilon^{\,2}
PUse in source · 31 passages
Search aliases: P
Use in source · norms.tex, line 60
l retains its constant Let a=wp_t(1/5), b=wp_t(2/5) and L_t=(q_t'/q_t)'. The enlarged lattice is Λ_t^+=tfrac15 Z+ Zt, with wp_+(z)=25wp_5t(5z). The degree-five quotient satisfies L_t(z)=wp_+(z)-5wp_t(z)+2(a+b). At zero L_t has principal part -4/z^2; at the four nonzero kernel points it has +1/(z-P_j)^2, with no simple-pole terms. Thus L_t-wp_++5wp_t is a constant elliptic function. The canonical formula gives q_t(z)=(a-b)^2z^4[1+(a+b)z^2+O(z^4)], fixing that constant as 2(a+b). The degree-five q…
Exact mathematical passage
+1/(z-P_j)^2
P^nUse in source · 1 passage
Search aliases: P^n
Use in source · product_proof.tex, line 3
Full seven-product series proof. For positive imaginary periods t,s,d with t+7d=2s+4, put P=e^2π it, Q=e^2π is, D=e^2π id and Z=e^2π iz. Then PD^7=Q^2. In a common convergence region the fifth-smoothed logarithms are log S_5(z;a,b) · =5∑_substackn≥1 5nmid n fracZ^n-(A_0B_0/Z)^nn(1-A_0^n)(1-B_0^n), log F_t,s(z) · =-5∑_substackn≥1 5nmid n fracP^nZ^n-Q^nZ^-nn(1-P^n)(1-Q^n), where A_0=e^2π ia,B_0=e^2π ib in the first line. At one surviving frequency suppress the index n. Summing seven shifts multipl…
Exact mathematical passage
\log S_5(z;a,b)&=5\sum_{\substack{n\ge1\\5\nmid n}} \frac{Z^n-(A_0B_0/Z)^n}{n(1-A_0^n)(1-B_0^n)},\\ \log F_{t,s}(z)&=-5\sum_{\substack{n\ge1\\5\nmid n}} \frac{P^nZ^n-Q^nZ^{-n}}{n(1-P^n)(1-Q^n)},
P_5Use in source · 2 passages
Search aliases: P_5
Use in source · theta.tex, line 256
Writing S_5=a+b and P_5=ab, the symmetric form is
Exact mathematical passage
P_5=ab
P_7Use in source · 7 passages
Search aliases: P_7
Use in source · principal.tex, line 24
βδ · =-8s-1, · 7δ · =τ+2s+4. The seven-point product and its correction are P_7(z)=∏_k=0^6F(z+kδ) =fracS_5(z;s,δ)S_5(z;t,δ)S_5(z+s;t,δ), R(z)=fracP_7(z)F(β z). The product identity is an actual gamma multiplication identity, not the assertion P_7(z)=F(β z). For completeness, its elementary product proof is retained from v0.12: rewrite t=2s+4-7δ, regard periods modulo integers, and perform real root averaging and vertical residue splitting in the defining gamma products. The resulting split facto…
Exact mathematical passage
\label{eq:p7} P_7(z)=\prod_{k=0}^{6}F(z+k\delta) =\frac{S_5(z;s,\delta)}{S_5(z;t,\delta)S_5(z+s;t,\delta)}, \qquad R(z)=\frac{P_7(z)}{F(\beta z)}.
P_{i,j,k,r}^{(n)}Use in source · 1 passage
Search aliases: P_i,j,k,r^(n)
Use in source · reconstruction.tex, line 6
Explicit three-dimensional families. For integers n≥1, 1≤ i,j,k≤ n, define Z_i,j,k^(n)=λ(-i,j,5(n+k)), P_i,j,k,r^(n)=λ(-i,j,5(n+k)-r), 1≤ r≤4. Every such triple has a<0, b>0, 3c>2b. Hence p>0 and the bracket in [source reference] is one. The Z points have order four and the P points have order minus one. Rational independence makes the parameter map injective. The pole indices are distinct because 5(k-k')=r-r' forces both sides to vanish; the zero and pole families are disjoint by c modulo five.…
Exact mathematical passage
\label{eq:families} Z_{i,j,k}^{(n)}=\lambda(-i,j,5(n+k)),\qquad P_{i,j,k,r}^{(n)}=\lambda(-i,j,5(n+k)-r),\quad 1\le r\le4.
P_jUse in source · 1 passage
Search aliases: P_j
Use in source · norms.tex, line 60
l retains its constant Let a=wp_t(1/5), b=wp_t(2/5) and L_t=(q_t'/q_t)'. The enlarged lattice is Λ_t^+=tfrac15 Z+ Zt, with wp_+(z)=25wp_5t(5z). The degree-five quotient satisfies L_t(z)=wp_+(z)-5wp_t(z)+2(a+b). At zero L_t has principal part -4/z^2; at the four nonzero kernel points it has +1/(z-P_j)^2, with no simple-pole terms. Thus L_t-wp_++5wp_t is a constant elliptic function. The canonical formula gives q_t(z)=(a-b)^2z^4[1+(a+b)z^2+O(z^4)], fixing that constant as 2(a+b). The degree-five q…
Exact mathematical passage
+1/(z-P_j)^2
\pmodUse in source · 7 passages
Search aliases: pmod
Use in source · polarization.tex, line 74
κ(a_j)=ι(a_j+1), jpmod3. tagQ1
Exact mathematical passage
\boxed{\kappa(a_j)=\iota(a_{j+1})},\qquad j\pmod3. \tag{Q1}
QUse in source · 16 passages
Search aliases: Q
Use in source · product_proof.tex, line 2
Full seven-product series proof. For positive imaginary periods t,s,d with t+7d=2s+4, put P=e^2π it, Q=e^2π is, D=e^2π id and Z=e^2π iz. Then PD^7=Q^2. In a common convergence region the fifth-smoothed logarithms are log S_5(z;a,b) · =5∑_substackn≥1
Exact mathematical passage
Q=e^{2\pi is}
\mathbb QUse in source · 19 passages
Search aliases: Q mathbb
Use in source · H12_Cubic_Divisor_Traditional.tex, line 99
We use F_0= Q(ω) for the imaginary-quadratic field, to separate it from the scalar gamma function F. The eta quotient is H(a)=η(5a)/η(a)^5; the integer step function H(n) used in the divisor formula is a different object. The symbol R_0 is the scalar anchor, while R_* is a counting cutoff radius. In the arithmetic block ϱ is a ray character and a radical lift, as explicitly indicated there; it is distinct from the contracting period multiplier ρ.
Exact mathematical passage
F_0=\mathbb Q(\omega)
qUse in source · 38 passages
Search aliases: q
Use in source · H12_Cubic_Divisor_Traditional.tex, line 89
Put q_a(z)=θ_0(z;a)^5/θ_0(5z;5a). Direct product identities and the inherited modular normalization give F(z+1) · =F(z), · fracF(z+τ)F(z) · =q_s(z), · fracF(z+s)F(z) · =q_t(z)^-1,
Exact mathematical passage
q_a(z)=\theta_0(z;a)^5/\theta_0(5z;5a)
Q^2Use in source · 3 passages
Search aliases: Q^2
Use in source · product_proof.tex, line 2
Full seven-product series proof. For positive imaginary periods t,s,d with t+7d=2s+4, put P=e^2π it, Q=e^2π is, D=e^2π id and Z=e^2π iz. Then PD^7=Q^2. In a common convergence region the fifth-smoothed logarithms are log S_5(z;a,b) · =5∑_substackn≥1
Exact mathematical passage
PD^7=Q^2
Q^nUse in source · 1 passage
Search aliases: Q^n
Use in source · product_proof.tex, line 3
Full seven-product series proof. For positive imaginary periods t,s,d with t+7d=2s+4, put P=e^2π it, Q=e^2π is, D=e^2π id and Z=e^2π iz. Then PD^7=Q^2. In a common convergence region the fifth-smoothed logarithms are log S_5(z;a,b) · =5∑_substackn≥1 5nmid n fracZ^n-(A_0B_0/Z)^nn(1-A_0^n)(1-B_0^n), log F_t,s(z) · =-5∑_substackn≥1 5nmid n fracP^nZ^n-Q^nZ^-nn(1-P^n)(1-Q^n), where A_0=e^2π ia,B_0=e^2π ib in the first line. At one surviving frequency suppress the index n. Summing seven shifts multipl…
Exact mathematical passage
\log S_5(z;a,b)&=5\sum_{\substack{n\ge1\\5\nmid n}} \frac{Z^n-(A_0B_0/Z)^n}{n(1-A_0^n)(1-B_0^n)},\\ \log F_{t,s}(z)&=-5\sum_{\substack{n\ge1\\5\nmid n}} \frac{P^nZ^n-Q^nZ^{-n}}{n(1-P^n)(1-Q^n)},
Q_5Use in source · 7 passages
Search aliases: Q_5
Use in source · theta.tex, line 71
Q_5(X)=(X-a)(X-b).
Exact mathematical passage
Q_5(X)=(X-a)(X-b).
q_aUse in source · 1 passage
Search aliases: q_a
Use in source · H12_Cubic_Divisor_Traditional.tex, line 89
Put q_a(z)=θ_0(z;a)^5/θ_0(5z;5a). Direct product identities and the inherited modular normalization give F(z+1) · =F(z), · fracF(z+τ)F(z) · =q_s(z), · fracF(z+s)F(z) · =q_t(z)^-1,
Exact mathematical passage
q_a(z)=\theta_0(z;a)^5/\theta_0(5z;5a)
q_sUse in source · 8 passages
Search aliases: q_s
Use in source · H12_Cubic_Divisor_Traditional.tex, line 90
Put q_a(z)=θ_0(z;a)^5/θ_0(5z;5a). Direct product identities and the inherited modular normalization give F(z+1) · =F(z), · fracF(z+τ)F(z) · =q_s(z), · fracF(z+s)F(z) · =q_t(z)^-1, q_s(z) · =q_t(ρ z), · F(0) · =ϵ^2, · F(1/3)F(2/3) · =1. The two elliptic lattices are homothetic, ρΛ_s=Λ_t, with Λ_a= Z+ Za. This is not an endomorphism of the single elliptic lattice Λ_t. Section~[source reference] proves the exact rational fifth-torsion normalization and explains the covariance without omitting its s…
Exact mathematical passage
\label{eq:F-identities} \begin{aligned} F(z+1)&=F(z),&\frac{F(z+\tau)}{F(z)}&=q_s(z),& \frac{F(z+s)}{F(z)}&=q_t(z)^{-1},\\ q_s(z)&=q_t(\rho z),&F(0)&=\epsilon^2,&F(1/3)F(2/3)&=1. \end{aligned}
q_tUse in source · 32 passages
Search aliases: q_t
Use in source · H12_Cubic_Divisor_Traditional.tex, line 90
Put q_a(z)=θ_0(z;a)^5/θ_0(5z;5a). Direct product identities and the inherited modular normalization give F(z+1) · =F(z), · fracF(z+τ)F(z) · =q_s(z), · fracF(z+s)F(z) · =q_t(z)^-1, q_s(z) · =q_t(ρ z), · F(0) · =ϵ^2, · F(1/3)F(2/3) · =1. The two elliptic lattices are homothetic, ρΛ_s=Λ_t, with Λ_a= Z+ Za. This is not an endomorphism of the single elliptic lattice Λ_t. Section~[source reference] proves the exact rational fifth-torsion normalization and explains the covariance without omitting its s…
Exact mathematical passage
\label{eq:F-identities} \begin{aligned} F(z+1)&=F(z),&\frac{F(z+\tau)}{F(z)}&=q_s(z),& \frac{F(z+s)}{F(z)}&=q_t(z)^{-1},\\ q_s(z)&=q_t(\rho z),&F(0)&=\epsilon^2,&F(1/3)F(2/3)&=1. \end{aligned}
q_t'Use in source · 2 passages
Search aliases: q_t'
Use in source · norms.tex, line 56
The fifth-isogeny differential retains its constant. Let a=wp_t(1/5), b=wp_t(2/5) and L_t=(q_t'/q_t)'. The enlarged lattice is Λ_t^+=tfrac15 Z+ Zt, with wp_+(z)=25wp_5t(5z). The degree-five quotient satisfies L_t(z)=wp_+(z)-5wp_t(z)+2(a+b). At zero L_t has principal part -4/z^2; at the four nonzero kernel points it has +1/(z-P_j)^2, with no simple-pole terms. Thus L_t-wp_++5wp_t is a constant elliptic function. The canonical formula gives q_t(z)=(a-b)^2z^4[1+(a+b)z^2+O(z^4)], fixing that
Exact mathematical passage
L_t=(q_t'/q_t)'
RUse in source · 50 passages
Search aliases: R
Use in source · H12_Cubic_Divisor_Traditional.tex, line 99
We use F_0= Q(ω) for the imaginary-quadratic field, to separate it from the scalar gamma function F. The eta quotient is H(a)=η(5a)/η(a)^5; the integer step function H(n) used in the divisor formula is a different object. The symbol R_0 is the scalar anchor, while R_* is a counting cutoff radius. In the arithmetic block ϱ is a ray character and a radical lift, as explicitly indicated there; it is distinct from the contracting period multiplier ρ.
Exact mathematical passage
R_0
\mathcal RUse in source · 8 passages
Search aliases: R mathcal
Use in source · stark.tex, line 16
F_0= Q(ω), K= Q(β), ϵ=2-β, R=-log(2-7^1/3)>0.
Exact mathematical passage
F_0=\mathbb Q(\omega),\qquad K=\mathbb Q(\beta),\qquad \epsilon=2-\beta,\qquad \mathcal R=-\log(2-7^{1/3})>0.
rUse in source · 43 passages
Search aliases: r
Use in source · principal.tex, line 106
All transition multipliers and the scalar anchor. [Theta transitions] The following hold meromorphically: K(z):=fracR(z+1)R(z) · =∏_r=1^15q_t(β z-rs)^-1, fracR(z+s)R(z) · =fracq_s(β z-τ)∏_k=0^6q_t(z+kδ), fracR(z+τ)R(z) · = frac∏_k=0^6q_s(z+kδ)∏_r=0^3q_t(β z+rs) q_s(β z)q_s(β z+τ), fracR(z+δ)R(z) · = fracq_s(z)q_t(z)q_t(z+s)∏_r=1^8q_t(β z-rs). Use F(z+1)=F(z), F(z+τ)/F(z)=q_s(z) and F(z+s)/F(z)=q_t(z)^-1. For [source reference], P_7 is period one and β=-15s-2, producing fifteen negative s shifts …
Exact mathematical passage
K(z):=\frac{R(z+1)}{R(z)}&=\prod_{r=1}^{15}q_t(\beta z-rs)^{-1},\label{eq:K}\\ \frac{R(z+s)}{R(z)}&=\frac{q_s(\beta z-\tau)}{\prod_{k=0}^{6}q_t(z+k\delta)},\label{eq:Rs}\\ \frac{R(z+\tau)}{R(z)}&= \frac{\prod_{k=0}^{6}q_s(z+k\delta)\prod_{r=0}^{3}q_t(\beta z+rs)} {q_s(\beta z)q_s(\beta z+\tau)},\label{eq:Rtau}\\ \frac{R(z+\delta)}{R(z)}&= \frac{q_s(z)}{q_t(z)q_t(z+s)\prod_{r=1}^{8}q_t(\beta z-rs)}.\label{eq:Rdelta}
R'Use in source · 1 passage
Search aliases: R'
Use in source · reconstruction.tex, line 119
fracm_λ(z-λ)^4, locally normally away from D. Consequently R(z)=R_0exp(α_1z+α_2z^2+α_3z^3)W_D(z), α_j=frac(log R)^(j)(0)j!. The logarithmic derivatives are branch independent. The notation (log R)^(4) means the third derivative of R'/R, a single-valued meromorphic function. The sine partial fractions [source reference] give fracd^4dw^4log(1-e^2π iw) =-6∑_n∈ Z(w-n)^-4. The linear exponential factor relating 1-e^2π iw to sinπ w has zero fourth logarithmic derivative. Apply this identity to each fa…
Exact mathematical passage
R'/R
r'Use in source · 1 passage
Search aliases: r'
Use in source · reconstruction.tex, line 10
Z_i,j,k^(n)=λ(-i,j,5(n+k)), P_i,j,k,r^(n)=λ(-i,j,5(n+k)-r), 1≤ r≤4. Every such triple has a<0, b>0, 3c>2b. Hence p>0 and the bracket in [source reference] is one. The Z points have order four and the P points have order minus one. Rational independence makes the parameter map injective. The pole indices are distinct because 5(k-k')=r-r' forces both sides to vanish; the zero and pole families are disjoint by c modulo five. [Two-sided cubic growth] Let n_+(r) and n_-(r) count the zeros and poles o…
Exact mathematical passage
5(k-k')=r-r'
R^\#Use in source · 4 passages
Search aliases: R^#
Use in source · principal.tex, line 143
The conjugation divisor cannot be discarded. Put R^#(z)=overlineR(overline z). The cubic fields Q(β) and Q(overlineβ) are distinct: equality would put β/overlineβ=ω in a cubic field. Their intersection is Q. The support of divR lies in the first, and that of R^# in the second. Since τ is irrational (indeed β=2-1/(4+15τ)), ord_τ(R/R^#)=4. Thus this naive conjugation quotient is not a holomorphic unit. A constant polarization, root of unity or nowhere-zero
Exact mathematical passage
R^\#(z)=\overline{R(\overline z)}
r^{-4}Use in source · 1 passage
Search aliases: r^-4
Use in source · reconstruction.tex, line 87
The lower bound n_D(r)≥ c r^3 yields divergence of ∑_D|m_λ|/|λ|^3: partial summation contains 3∫ n_D(r)r^-4 dr, which diverges logarithmically. For a fixed z≠0, log E_2(z/λ)=-fracz^33λ^3+O(|λ|^-4). Its weighted absolute logarithm sum diverges, because the fourth-power remainder is summable. This proves failure of ungrouped absolute genus two, without a claim about cancellations in specially grouped signed products. [Explicit analytic tail] If n_D(r)=∑_|λ|≤ r|m_λ|≤ Ar^3 for all r≥ R_*, then for T…
Exact mathematical passage
3\int n_D(r)r^{-4}\,dr
r^{-5}Use in source · 1 passage
Search aliases: r^-5
Use in source · reconstruction.tex, line 101
n_D(r)=∑_|λ|≤ r|m_λ|≤ Ar^3 for all r≥ R_*, then for T≥ R_* and |z|≤ T/2, ∑_|λ|>Tfrac|m_λ||λ|^4≤frac4AT, |log W_D,>T(z)|≤frac2A|z|^4T. Partial summation gives the first upper bound by dropping its nonpositive boundary term and using 4∫_T^∞ Ar^3r^-5 dr=4A/T. For the tail factors the canonical branch satisfies |log E_3(w)|≤ |w|^4/[4(1-|w|)]≤ |w|^4/2, proving the second inequality. The condition |z|≤ T/2 keeps these tail factors away from zeros and fixes the logarithm through its power series. This …
Exact mathematical passage
4\int_T^\infty Ar^3r^{-5}\,dr=4A/T
r^2Use in source · 2 passages
Search aliases: r^2
Use in source · reconstruction.tex, line 54
parts ImΩ_ℓ, affine maps z↦ A_ℓ z+b_ℓ, and finite derivatives of their Riemann theta restrictions. Let P,Q be nonzero entire functions formed by finite polynomial expressions in these restrictions, derivatives and polynomials in z. Let u be a single global nowhere-zero entire function. Every meromorphic function u(z)fracP(z)Q(z) has zero and pole counts O(r^2). Thus R is not representable in this class. For Y=ImΩ>0, the defining theta series has terms bounded by exp{-π n^TYn+2π n Im(Az+b) }. Com…
Exact mathematical passage
O(r^2)
r^3Use in source · 7 passages
Search aliases: r^3
Use in source · reconstruction.tex, line 13
the P points have order minus one. Rational independence makes the parameter map injective. The pole indices are distinct because 5(k-k')=r-r' forces both sides to vanish; the zero and pole families are disjoint by c modulo five. [Two-sided cubic growth] Let n_+(r) and n_-(r) count the zeros and poles of R in |z|≤ r, with multiplicity. Then n_+(r)=Θ(r^3), n_-(r)=Θ(r^3), n_D(r):=n_+(r)+n_-(r)=Θ(r^3). In particular, for K_0=(|τ|+|s|+2)/|β|, n_+(K_0n)≥4n^3, n_-(K_0n)≥4n^3. These are unsigned counts…
Exact mathematical passage
n_+(r)=\Theta(r^3),\qquad n_-(r)=\Theta(r^3),\qquad n_D(r):=n_+(r)+n_-(r)=\Theta(r^3).
r^4Use in source · 1 passage
Search aliases: r^4
Use in source · theta.tex, line 96
r^4=r, r^9=r.
Exact mathematical passage
r^4=r,\qquad r^9=r.
r^8Use in source · 2 passages
Search aliases: r^8
Use in source · theta.tex, line 100
Thus r^3=r^8=1, and
Exact mathematical passage
r^3=r^8=1
r^9Use in source · 1 passage
Search aliases: r^9
Use in source · theta.tex, line 96
r^4=r, r^9=r.
Exact mathematical passage
r^4=r,\qquad r^9=r.
R_*Use in source · 5 passages
Search aliases: R_*
Use in source · H12_Cubic_Divisor_Traditional.tex, line 99
We use F_0= Q(ω) for the imaginary-quadratic field, to separate it from the scalar gamma function F. The eta quotient is H(a)=η(5a)/η(a)^5; the integer step function H(n) used in the divisor formula is a different object. The symbol R_0 is the scalar anchor, while R_* is a counting cutoff radius. In the arithmetic block ϱ is a ray character and a radical lift, as explicitly indicated there; it is distinct from the contracting period multiplier ρ.
Exact mathematical passage
R_*
R_+Use in source · 8 passages
Search aliases: R_+
Use in source · polarization.tex, line 236
R_+=|ϵ|^4|a_1|^6|a_2|^4.
Exact mathematical passage
R_+=|\epsilon|^4|a_1|^6|a_2|^4.
R_0Use in source · 6 passages
Search aliases: R_0
Use in source · H12_Cubic_Divisor_Traditional.tex, line 99
We use F_0= Q(ω) for the imaginary-quadratic field, to separate it from the scalar gamma function F. The eta quotient is H(a)=η(5a)/η(a)^5; the integer step function H(n) used in the divisor formula is a different object. The symbol R_0 is the scalar anchor, while R_* is a counting cutoff radius. In the arithmetic block ϱ is a ray character and a radical lift, as explicitly indicated there; it is distinct from the contracting period multiplier ρ.
Exact mathematical passage
R_0
R_0^2Use in source · 1 passage
Search aliases: R_0^2
Use in source · reconstruction.tex, line 147
P_7(z)P_7(-z). For 1≤ k≤6, F(z+kδ)F(-z+(7-k)δ)=q_t(z+kδ-s)^-1. The unpaired k=0 term retains q_t(z)/q_s(z). Dividing by the reflected denominator gives R(z)R(-z)=T(z):= fracq_t(z)q_s(β z)q_s(z)q_t(β z) ∏_k=1^6q_t(z+kδ-s)^-1. Its apparent singularities at zero are removable, and T(0)=R_0^2.
Exact mathematical passage
T(0)=R_0^2
R_0^3Use in source · 2 passages
Search aliases: R_0^3
Use in source · reconstruction.tex, line 187
[Branch-free special cube] The exact analytic reconstruction at z=1/3 is R(1/3)^3=R_0^3K(0)fracW_D(1/3)^3W_D(1) exp(-frac23α_2-frac89α_3). The right side consists of specified theta values and derivatives, algebraic period data and absolutely convergent genus-three products and sums. It contains no gamma value, arbitrary normalization scalar or new root choice. Cube [source reference] at 1/3 to get R_0^3e^α_1+α_2/3+α_3/9W_D(1/3)^3, then substitute [source reference]. The resulting exponents are …
Exact mathematical passage
\label{eq:cube} \boxed{R(1/3)^3=R_0^3K(0)\frac{W_D(1/3)^3}{W_D(1)} \exp\left(-\frac23\alpha_2-\frac89\alpha_3\right).}
\rceilUse in source · 2 passages
Search aliases: rceil
Use in source · principal.tex, line 70
0 · -5 · 4. Its quotient generator is (1,2,20)/7, representing δ. Thus β^-1Λ_5=Λ_5+ Zδ. Write a potential singular point as λ(a,b,c)=fracaτ+bs+c/5β, p=lceilfrac3c-2b7rceil, k=7p+2b-3c∈{0,…,6}. Exactly one argument λ+kδ lies in Λ_5. Its integral coordinates are (widehat a,widehat b,widehat c)=(p,2p-a,20p+5b-8c). Indeed N_5(widehat a,widehat b,widehat c)^T=(a,b-8k,c-5k)^T, which expresses β(λ+kδ)=βλ+k(-8s-1). The other six digits fail integrality. Also widehat c-2c=5(4p+b-2c), w(widehat c)=w(c). T…
Exact mathematical passage
\label{eq:lambda} \lambda(a,b,c)=\frac{a\tau+bs+c/5}{\beta},\qquad p=\left\lceil\frac{3c-2b}{7}\right\rceil,\qquad k=7p+2b-3c\in\{0,\ldots,6\}.
\ReUse in source · 2 passages
Search aliases: Re
Use in source · polarization.tex, line 311
3.6·10^6<Re(C U_Γ^3)<3.7·10^6.
Exact mathematical passage
3.6\cdot10^6<\Re(C U_\Gamma^3)<3.7\cdot10^6.
\rfloorUse in source · 1 passage
Search aliases: rfloor
Use in source · reconstruction.tex, line 30
A bounded complex radius bounds |a| and |b| by O(r); the real part then bounds |c| by O(r). There are O((1+r)^3) triples, and absolute weights are at most four. The seven translated numerator factors and the scaled denominator in [source reference] give a finite union of such cone supports and cannot increase this order. Combining the upper bound with the explicit lower families, choosing n=lfloor r/K_0rfloor, proves the assertion for all sufficiently large r.
Exact mathematical passage
n=\lfloor r/K_0\rfloor
SUse in source · 30 passages
Search aliases: S
Use in source · norms.tex, line 16
The point S=sbmodΛ_t has infinite order by rational independence. Put ψ(w)=fracq_t(w)^2q_t(2w)q_t(2w+s). [All finite norms of the return kernel] For every integer n≥1, N_nψ(w)= 1, · 5mid n,
Exact mathematical passage
S=s\bmod\Lambda_t
sUse in source · 105 passages
Search aliases: s
Use in source · H12_Cubic_Divisor_Traditional.tex, line 61
Fixed data, conventions, and the claim boundary. Fix throughout ω · =e^2π i/3, · β · =7^1/3ω^2, · ϵ · =2-β, · ρ · =ϵ^-1, τ · =(β^2+2β)/15, · t · =-τ, · s · =-(β+2)/15, · δ · =(τ+2s+4)/7. The real cube root 7^1/3 is positive; this fixes the lower complex embedding of β. We have Im t,Im s,Imδ>0, Imτ<0 and |ρ|<1. The last follows from N(ϵ)=1 and 0<2-7^1/3<1 at the real embedding. The numbers 1,t,s are rationally independent: a rational linear relation becomes a polynomial of degree at most two in β…
Exact mathematical passage
\label{eq:fixed-data} \begin{aligned} \omega&=e^{2\pi i/3},&\beta&=7^{1/3}\omega^2,&\epsilon&=2-\beta,&\rho&=\epsilon^{-1},\\ \tau&=(\beta^2+2\beta)/15,&t&=-\tau,&s&=-(\beta+2)/15,& \delta&=(\tau+2s+4)/7. \end{aligned}
s^2Use in source · 1 passage
Search aliases: s^2
Use in source · stark.tex, line 662
At zero its value vanishes. Its derivative equals 2π y B_2(a), using Γ(-1/2)=-2sqrtπ, 1/Γ(s)=s+O(s^2), and ζ(-1,a)=-B_2(a)/2.
Exact mathematical passage
1/\Gamma(s)=s+O(s^2)
s_0Use in source · 7 passages
Search aliases: s_0
Use in source · stark.tex, line 106
r:β↦ωβ, c:ω↦ω^2, s_0:θ↦θ^2-2,
Exact mathematical passage
r:\beta\mapsto\omega\beta,\qquad c:\omega\mapsto\omega^2,\qquad s_0:\theta\mapsto\theta^2-2,
s_0^kUse in source · 2 passages
Search aliases: s_0^k
Use in source · stark.tex, line 274
ζ_ f'(c^e s_0^k,0) =(-1)^e(frac R6-ℓ_k-1). tagA3.5
Exact mathematical passage
\boxed{\zeta_{\mathfrak f}'(c^e s_0^k,0) =(-1)^e\left(\frac{\mathcal R}{6}-\ell_{k-1}\right).} \tag{A3.5}
S_3Use in source · 5 passages
Search aliases: S_3
Use in source · stark.tex, line 95
The splitting field F_0(β)/ Q has group S_3, while Q(θ)/ Q is cyclic cubic. Their intersection is Galois over Q; it cannot have degree three because S_3 has no cyclic cubic quotient. Hence
Exact mathematical passage
S_3
S_5Use in source · 13 passages
Search aliases: S_5
Use in source · principal.tex, line 6
The seven-point product. Use the fixed periods of Section~[source reference]. Define S_5(z;a,b)=fracΓ_rm ell(z;a,b)^5 Γ_rm ell(5z;5a,5b), F(z)=S_5(z+t;t,s)^-1. All identities below are meromorphic identities; their quotients at apparent exceptional points mean meromorphic continuation. Let Λ_5= Zτ+ Zs+tfrac15 Z, δ=fracτ+2s+47. The period identities, obtained from β^3=7, are βτ · =2τ+4s+1, · β s · =-τ, · β · =-15s-2,
Exact mathematical passage
S_5(z;a,b)=\frac{\Gamma_{\rm ell}(z;a,b)^5} {\Gamma_{\rm ell}(5z;5a,5b)},\qquad F(z)=S_5(z+t;t,s)^{-1}.
S_5^2Use in source · 1 passage
Search aliases: S_5^2
Use in source · theta.tex, line 258
q_t(z)=fracS_5^2-4P_5x(z)^2-S_5x(z)+P_5. tagN5.2
Exact mathematical passage
q_t(z)=\frac{S_5^2-4P_5}{x(z)^2-S_5x(z)+P_5}. \tag{N5.2}
S_NUse in source · 3 passages
Search aliases: S_N
Use in source · rigidity.tex, line 24
The convergent transfer proof. For functions on T^3, let S_Nh(v)=∑_κ∈ker Nh(v+κ), C_Nh(v)=h(Nv), L_Nh(v)=∑_Nw=vh(w), δ_Mh=h-h∘ M. The transfer L_N is unnormalized. For h=∑ c_ke_k, S_Nh=∑_k∈ B Z^37c_ke_k, L_Nh=∑_k7c_Bke_k, S_Nh=(L_Nh)∘ N, L_NC_Nh=7h. Since M permutes ker N and commutes with N, the required operators commute with composition by M.
Exact mathematical passage
S_Nh(v)=\sum_{\kappa\in\ker N}h(v+\kappa),\quad C_Nh(v)=h(Nv),\quad L_Nh(v)=\sum_{Nw=v}h(w),\quad \delta_Mh=h-h\circ M.
s_{\rm per}Use in source · 1 passage
Search aliases: s_rm per rm
Use in source · stark.tex, line 576
U_Γ=fracΓ_rm ell(1/3;τ,s_rm per)^5 Γ_rm ell(5/3;5τ,5s_rm per), τ=(β^2+2β)/15, s_rm per=-(β+2)/15.
Exact mathematical passage
U_\Gamma=\frac{\Gamma_{\rm ell}(1/3;\tau,s_{\rm per})^5} {\Gamma_{\rm ell}(5/3;5\tau,5s_{\rm per})}, \quad \tau=(\beta^2+2\beta)/15,\quad s_{\rm per}=-(\beta+2)/15.
S_sUse in source · 1 passage
Search aliases: S_s
Use in source · reconstruction.tex, line 150
Let L_t=(q_t'/q_t)' and S_t=wp_t(1/5)+wp_t(2/5). The normalized theta formula gives L_t(z)=-4z^-2+2S_t+O(z^2), S_s=ρ^2S_t. Since 7δ-2s=-t+4, evenness and t-periodicity pair the six translated L_t terms. Using s=u_0+3δ-2, their sum is 2∑_j=0^2L_t(u_j). The second logarithmic derivative of [source reference] cancels all z^-2 terms, leaving (log T)''(0)=2(1-β^2)(1-ρ^2)S_t-2∑_j=0^2L_t(u_j). The left side is 4α_2. Therefore α_2=frac12[(1-β^2)(1-ρ^2)S_t- ∑_j=0^2L_t(u_j)]. The factor 1/2 is fixed by re…
Exact mathematical passage
L_t(z)=-4z^{-2}+2S_t+O(z^2),\qquad S_s=\rho^2S_t.
S_tUse in source · 4 passages
Search aliases: S_t
Use in source · reconstruction.tex, line 149
Let L_t=(q_t'/q_t)' and S_t=wp_t(1/5)+wp_t(2/5). The normalized theta formula gives L_t(z)=-4z^-2+2S_t+O(z^2), S_s=ρ^2S_t. Since 7δ-2s=-t+4, evenness and t-periodicity pair the six translated L_t terms. Using s=u_0+3δ-2, their sum is 2∑_j=0^2L_t(u_j). The second logarithmic derivative of [source reference] cancels all z^-2 terms, leaving (log T)''(0)=2(1-β^2)(1-ρ^2)S_t-2∑_j=0^2L_t(u_j). The left side is 4α_2. Therefore α_2=frac12[(1-β^2)(1-ρ^2)S_t- ∑_j=0^2L_t(u_j)]. The factor 1/2 is fixed by re…
Exact mathematical passage
S_t=\wp_t(1/5)+\wp_t(2/5)
\operatorname{sgn}Use in source · 1 passage
Search aliases: sgn operatorname
Use in source · stark.tex, line 132
Ind_C_2^S_3(sgn) =sgn⊕Std, Std=Ind_C_3^S_3(ϱ)
Exact mathematical passage
\operatorname{Ind}_{C_2}^{S_3}(\operatorname{sgn}) =\operatorname{sgn}\oplus\operatorname{Std}, \qquad \operatorname{Std}=\operatorname{Ind}_{C_3}^{S_3}(\varrho)
\sinUse in source · 1 passage
Search aliases: sin
Use in source · reconstruction.tex, line 124
j)(0)j!. The logarithmic derivatives are branch independent. The notation (log R)^(4) means the third derivative of R'/R, a single-valued meromorphic function. The sine partial fractions [source reference] give fracd^4dw^4log(1-e^2π iw) =-6∑_n∈ Z(w-n)^-4. The linear exponential factor relating 1-e^2π iw to sinπ w has zero fourth logarithmic derivative. Apply this identity to each factor of the defining double gamma products. Their three-index cone supports have O(r^3) counting. Initially work of…
Exact mathematical passage
\sin\pi w
\operatorname{Std}Use in source · 1 passage
Search aliases: Std operatorname
Use in source · stark.tex, line 132
Ind_C_2^S_3(sgn) =sgn⊕Std, Std=Ind_C_3^S_3(ϱ)
Exact mathematical passage
\operatorname{Ind}_{C_2}^{S_3}(\operatorname{sgn}) =\operatorname{sgn}\oplus\operatorname{Std}, \qquad \operatorname{Std}=\operatorname{Ind}_{C_3}^{S_3}(\varrho)
\mathbb TUse in source · 4 passages
Search aliases: T mathbb
Use in source · reconstruction.tex, line 206
# =fracR_0overlineR_0 exp(∑_j=1^3(α_j-overlineα_j)z^j). This is entire and nowhere zero, with modulus one on the real axis. It cancels all unmatched divisor orders by a canonical meromorphic factor. That factor is not a holomorphic-unit gauge or a finite theta section. Regularity on the scalar plane does not establish periodic descent to T^3, identify a based multiplier with U_Γ/U_rm alg, or supply the principal-norm coboundary of Section~[source reference]. Those remain distinct obligations.
Exact mathematical passage
\mathbb T^3
tUse in source · 122 passages
Search aliases: t
Use in source · H12_Cubic_Divisor_Traditional.tex, line 61
Fixed data, conventions, and the claim boundary. Fix throughout ω · =e^2π i/3, · β · =7^1/3ω^2, · ϵ · =2-β, · ρ · =ϵ^-1, τ · =(β^2+2β)/15, · t · =-τ, · s · =-(β+2)/15, · δ · =(τ+2s+4)/7. The real cube root 7^1/3 is positive; this fixes the lower complex embedding of β. We have Im t,Im s,Imδ>0, Imτ<0 and |ρ|<1. The last follows from N(ϵ)=1 and 0<2-7^1/3<1 at the real embedding. The numbers 1,t,s are rationally independent: a rational linear relation becomes a polynomial of degree at most two in β…
Exact mathematical passage
\label{eq:fixed-data} \begin{aligned} \omega&=e^{2\pi i/3},&\beta&=7^{1/3}\omega^2,&\epsilon&=2-\beta,&\rho&=\epsilon^{-1},\\ \tau&=(\beta^2+2\beta)/15,&t&=-\tau,&s&=-(\beta+2)/15,& \delta&=(\tau+2s+4)/7. \end{aligned}
T^2Use in source · 2 passages
Search aliases: T^2
Use in source · theta.tex, line 397
F(ρ w_v)=F(w_Tv)B_v, B_v=∏_j=0^A_v-1q_t(w_T^2v+js), A_v=5+4b-2c+7a.
Exact mathematical passage
F(\rho w_v)=F(w_{Tv})B_v,\qquad B_v=\prod_{j=0}^{A_v-1}q_t(w_{T^2v}+js), \quad A_v=5+4b-2c+7a.
\mathbb T^3Use in source · 4 passages
Search aliases: T^3 mathbb
Use in source · reconstruction.tex, line 206
# =fracR_0overlineR_0 exp(∑_j=1^3(α_j-overlineα_j)z^j). This is entire and nowhere zero, with modulus one on the real axis. It cancels all unmatched divisor orders by a canonical meromorphic factor. That factor is not a holomorphic-unit gauge or a finite theta section. Regularity on the scalar plane does not establish periodic descent to T^3, identify a based multiplier with U_Γ/U_rm alg, or supply the principal-norm coboundary of Section~[source reference]. Those remain distinct obligations.
Exact mathematical passage
\mathbb T^3
t^5Use in source · 1 passage
Search aliases: t^5
Use in source · stark.tex, line 560
For t^6=W, the lift ϱ(t)=t^5/U_rm alg commutes with t↦ c_6t, and satisfies ϱ^6(t)=c_6^4t. Irreducibility is checked at the residue assignment (β,v,ω)=(32,99,48) over F_181, where W=23 and T^6-23 is irreducible. The earlier local-integrality argument validates this reduction. The resulting abelian group is C_2× C_18.
Exact mathematical passage
\varrho(t)=t^5/U_{\rm alg}
T^6Use in source · 1 passage
Search aliases: T^6
Use in source · stark.tex, line 563
For t^6=W, the lift ϱ(t)=t^5/U_rm alg commutes with t↦ c_6t, and satisfies ϱ^6(t)=c_6^4t. Irreducibility is checked at the residue assignment (β,v,ω)=(32,99,48) over F_181, where W=23 and T^6-23 is irreducible. The earlier local-integrality argument validates this reduction. The resulting abelian group is C_2× C_18.
Exact mathematical passage
T^6-23
t^6Use in source · 1 passage
Search aliases: t^6
Use in source · stark.tex, line 560
For t^6=W, the lift ϱ(t)=t^5/U_rm alg commutes with t↦ c_6t, and satisfies ϱ^6(t)=c_6^4t. Irreducibility is checked at the residue assignment (β,v,ω)=(32,99,48) over F_181, where W=23 and T^6-23 is irreducible. The earlier local-integrality argument validates this reduction. The resulting abelian group is C_2× C_18.
Exact mathematical passage
t^6=W
uUse in source · 30 passages
Search aliases: u
Use in source · principal.tex, line 128
[Three-theta anchor] R is regular and nonzero at zero, and R_0:=R(0)=∏_j=0^2q_t(u_j)^-1, u_j=fract+δ2+jδ. Let H(a)=η(5a)/η(a)^5. Direct product cancellation gives S_5(z;a,b)∼frac5(2π)^4 H(a) H(b)z^-4. The ratio in [source reference] therefore has value P_7(0)=[ H(s)/ H(t)]/S_5(s;t,δ). Since F(0)= H(s)/ H(t)=ϵ^2, the eta factors cancel and R_0=S_5(s;t,δ)^-1. The period relation is s=(t+δ)/2+3δ-2. At the central point, S_5((t+δ)/2;t,δ)=1 by termwise coincidence of the gamma numerator and denominat…
Exact mathematical passage
\label{eq:R0} R_0:=R(0)=\prod_{j=0}^{2}q_t(u_j)^{-1},\qquad u_j=\frac{t+\delta}{2}+j\delta.
u^2Use in source · 7 passages
Search aliases: u^2
Use in source · theta.tex, line 311
with P=(0,0), 2P=(u,u^2), and
Exact mathematical passage
2P=(u,u^2)
u_0Use in source · 1 passage
Search aliases: u_0
Use in source · reconstruction.tex, line 153
Let L_t=(q_t'/q_t)' and S_t=wp_t(1/5)+wp_t(2/5). The normalized theta formula gives L_t(z)=-4z^-2+2S_t+O(z^2), S_s=ρ^2S_t. Since 7δ-2s=-t+4, evenness and t-periodicity pair the six translated L_t terms. Using s=u_0+3δ-2, their sum is 2∑_j=0^2L_t(u_j). The second logarithmic derivative of [source reference] cancels all z^-2 terms, leaving (log T)''(0)=2(1-β^2)(1-ρ^2)S_t-2∑_j=0^2L_t(u_j). The left side is 4α_2. Therefore α_2=frac12[(1-β^2)(1-ρ^2)S_t- ∑_j=0^2L_t(u_j)]. The factor 1/2 is fixed by re…
Exact mathematical passage
s=u_0+3\delta-2
U_{21}Use in source · 2 passages
Search aliases: U_21
Use in source · polarization.tex, line 168
The normalized polynomial U_21(β,ξ,ω)=21U is explicitly defined in H12CMNormCertificate.lean. It is the normal form of the inherited candidate, as checked in both rational implementations. Also
Exact mathematical passage
U_{21}(\beta,\xi,\omega)=21U
U_{21}^3Use in source · 1 passage
Search aliases: U_21^3
Use in source · polarization.tex, line 185
U_21^3(2-ωβ)(7ι A) =64827 ω[2+(1+ω)β] A,
Exact mathematical passage
U_{21}^3(2-\omega\beta)(7\iota\mathcal A) =64827\,\omega[2+(1+\omega)\beta]\mathcal A,
u_jUse in source · 4 passages
Search aliases: u_j
Use in source · principal.tex, line 128
[Three-theta anchor] R is regular and nonzero at zero, and R_0:=R(0)=∏_j=0^2q_t(u_j)^-1, u_j=fract+δ2+jδ. Let H(a)=η(5a)/η(a)^5. Direct product cancellation gives S_5(z;a,b)∼frac5(2π)^4 H(a) H(b)z^-4. The ratio in [source reference] therefore has value P_7(0)=[ H(s)/ H(t)]/S_5(s;t,δ). Since F(0)= H(s)/ H(t)=ϵ^2, the eta factors cancel and R_0=S_5(s;t,δ)^-1. The period relation is s=(t+δ)/2+3δ-2. At the central point, S_5((t+δ)/2;t,δ)=1 by termwise coincidence of the gamma numerator and denominat…
Exact mathematical passage
\label{eq:R0} R_0:=R(0)=\prod_{j=0}^{2}q_t(u_j)^{-1},\qquad u_j=\frac{t+\delta}{2}+j\delta.
U_{\rm alg}^{25}Use in source · 1 passage
Search aliases: U_rm alg^25 rm
Use in source · stark.tex, line 556
W^21=c_6U_rm alg^25σ(U_rm alg)^5σ^2(U_rm alg).
Exact mathematical passage
W^{21}=c_6U_{\rm alg}^{25}\sigma(U_{\rm alg})^5\sigma^2(U_{\rm alg}).
U_{\rm alg}^3Use in source · 7 passages
Search aliases: U_rm alg^3 rm
Use in source · polarization.tex, line 139
U_rm alg^3=ωfracϵ_2ϵ_1 frac Aι( A). tagQ3
Exact mathematical passage
\boxed{ U_{\rm alg}^3=\omega\frac{\epsilon_2}{\epsilon_1} \frac{\mathcal A}{\iota(\mathcal A)}. } \tag{Q3}
U_{\rm alg}^6Use in source · 2 passages
Search aliases: U_rm alg^6 rm
Use in source · stark.tex, line 550
W^5=σ(W)U_rm alg^6,
Exact mathematical passage
W^5=\sigma(W)U_{\rm alg}^6,
U_\Gamma^3Use in source · 10 passages
Search aliases: U_Γ^3 Gamma
Use in source · polarization.tex, line 305
|C U_Γ^3|=R_+.
Exact mathematical passage
|C U_\Gamma^3|=R_+.
VUse in source · 5 passages
Search aliases: V
Use in source · stark.tex, line 342
V_0=H(1/(3J))H(13/(3J)), V_1=H(4/(3J))H(10/(3J)), V_2=H(16/(3J))H(19/(3J)).
Exact mathematical passage
V_0=H(1/(3J))H(13/(3J)),
v^2Use in source · 3 passages
Search aliases: v^2
Use in source · H12_Cubic_Divisor_Traditional.tex, line 108
The arithmetic candidate is fixed independently. Let J=2+3ω and choose the Gauss branch X=β^-1∑_a=1^6χ_7(a)e^6π ia/7, (χ_7(1),…,χ_7(6))=(1,ω,ω^2,ω^2,ω,1), X^3=J. Then E= Q(ω,β,X) has degree eighteen. The retained arithmetic construction defines ν · =(X+β/X-β^2)/3, · v · =-β^2ν-2ν^2, Y · =(4β^2+β-14)v^2+(11β-23)v-β^2+3β-3, Z · =frac(-2β^2+7β-8)v^2+(-10β^2+11β+17)v3 +7-β-β^2, U_rm alg · =fracY-(1+2ω)Z2. These formulas choose no coefficient or branch by fitting U_Γ. The original desired identity is…
Exact mathematical passage
\nu&=(X+\beta/X-\beta^2)/3,&v&=-\beta^2\nu-2\nu^2,\\ Y&=(4\beta^2+\beta-14)v^2+(11\beta-23)v-\beta^2+3\beta-3,\\ Z&=\frac{(-2\beta^2+7\beta-8)v^2+(-10\beta^2+11\beta+17)v}{3} +7-\beta-\beta^2,\\ U_{\rm alg}&=\frac{Y-(1+2\omega)Z}{2}.
v^3Use in source · 1 passage
Search aliases: v^3
Use in source · stark.tex, line 478
ν=(X+β/X-β^2)/3, v=-β^2ν-2ν^2, v^3-β v^2+1=0.
Exact mathematical passage
\nu=(X+\beta/X-\beta^2)/3,\quad v=-\beta^2\nu-2\nu^2, \qquad v^3-\beta v^2+1=0.
V_0Use in source · 3 passages
Search aliases: V_0
Use in source · stark.tex, line 342
V_0=H(1/(3J))H(13/(3J)), V_1=H(4/(3J))H(10/(3J)), V_2=H(16/(3J))H(19/(3J)).
Exact mathematical passage
V_0=H(1/(3J))H(13/(3J)),
V_1Use in source · 3 passages
Search aliases: V_1
Use in source · stark.tex, line 344
V_0=H(1/(3J))H(13/(3J)), V_1=H(4/(3J))H(10/(3J)), V_2=H(16/(3J))H(19/(3J)).
Exact mathematical passage
V_1=H(4/(3J))H(10/(3J)),\qquad V_2=H(16/(3J))H(19/(3J)).
V_2Use in source · 3 passages
Search aliases: V_2
Use in source · stark.tex, line 344
V_0=H(1/(3J))H(13/(3J)), V_1=H(4/(3J))H(10/(3J)), V_2=H(16/(3J))H(19/(3J)).
Exact mathematical passage
V_1=H(4/(3J))H(10/(3J)),\qquad V_2=H(16/(3J))H(19/(3J)).
V_mUse in source · 1 passage
Search aliases: V_m
Use in source · stark.tex, line 351
ℓ_m=-2log V_m, V_0V_1V_2=1. tagA4.3
Exact mathematical passage
\ell_m=-2\log V_m, \qquad V_0V_1V_2=1. \tag{A4.3}
V_m^3Use in source · 1 passage
Search aliases: V_m^3
Use in source · stark.tex, line 425
V_m^3=|a_m|, tagA5.6
Exact mathematical passage
\boxed{V_m^3=|a_m|,} \tag{A5.6}
WUse in source · 33 passages
Search aliases: W
Use in source · polarization.tex, line 146
The same calculation for the inherited pre-smoothing unit W gives line norms
Exact mathematical passage
W
wUse in source · 30 passages
Search aliases: w
Use in source · norms.tex, line 5
Exactly normalized elliptic norms. On E_t= C/Λ_t define N_nf(w)=∏_nz=wf(z) for integers n≥1, with all n^2 preimages counted. The product is independent of the representative lift. Root-of-unity averaging and vertical residue splitting in [source reference] give N_nq_t(w)=q_t(w), · 5nmid n,
Exact mathematical passage
N_nf(w)=\prod_{nz=w}f(z)
w^2Use in source · 1 passage
Search aliases: w^2
Use in source · reconstruction.tex, line 67
The genus-three divisor product. Let D={λ:m_λ≠0}; zero is not in D. Define E_3(w)=(1-w)e^w+w^2/2+w^3/3, W_D(z)=∏_λ∈ DE_3(z/λ)^m_λ. Negative exponents mean the quotient of the separate positive and negative entire products. [Canonical regularizer and absolute genus] The products defining W_D converge normally on compact sets away from their poles, and divW_D=divR, W_D(0)=1, (log W_D)^(j)(0)=0 (1≤ j≤3). Genus three is sufficient and minimal for this ungrouped absolute construction. Special signed …
Exact mathematical passage
\label{eq:WD} E_3(w)=(1-w)e^{w+w^2/2+w^3/3},\qquad W_D(z)=\prod_{\lambda\in D}E_3(z/\lambda)^{m_\lambda}.
W^{21}Use in source · 1 passage
Search aliases: W^21
Use in source · stark.tex, line 556
W^21=c_6U_rm alg^25σ(U_rm alg)^5σ^2(U_rm alg).
Exact mathematical passage
W^{21}=c_6U_{\rm alg}^{25}\sigma(U_{\rm alg})^5\sigma^2(U_{\rm alg}).
W^3Use in source · 1 passage
Search aliases: W^3
Use in source · polarization.tex, line 156
W^3=-fracϵ_2ϵ_1 frac Bι( B). tagQ4
Exact mathematical passage
\boxed{W^3=-\frac{\epsilon_2}{\epsilon_1} \frac{\mathcal B}{\iota(\mathcal B)}.} \tag{Q4}
w^3Use in source · 1 passage
Search aliases: w^3
Use in source · reconstruction.tex, line 67
The genus-three divisor product. Let D={λ:m_λ≠0}; zero is not in D. Define E_3(w)=(1-w)e^w+w^2/2+w^3/3, W_D(z)=∏_λ∈ DE_3(z/λ)^m_λ. Negative exponents mean the quotient of the separate positive and negative entire products. [Canonical regularizer and absolute genus] The products defining W_D converge normally on compact sets away from their poles, and divW_D=divR, W_D(0)=1, (log W_D)^(j)(0)=0 (1≤ j≤3). Genus three is sufficient and minimal for this ungrouped absolute construction. Special signed …
Exact mathematical passage
\label{eq:WD} E_3(w)=(1-w)e^{w+w^2/2+w^3/3},\qquad W_D(z)=\prod_{\lambda\in D}E_3(z/\lambda)^{m_\lambda}.
w^4Use in source · 1 passage
Search aliases: w^4
Use in source · reconstruction.tex, line 120
beginequation R(z)=R_0exp(α_1z+α_2z^2+α_3z^3)W_D(z), α_j=frac(log R)^(j)(0)j!. The logarithmic derivatives are branch independent. The notation (log R)^(4) means the third derivative of R'/R, a single-valued meromorphic function. The sine partial fractions [source reference] give fracd^4dw^4log(1-e^2π iw) =-6∑_n∈ Z(w-n)^-4. The linear exponential factor relating 1-e^2π iw to sinπ w has zero fourth logarithmic derivative. Apply this identity to each factor of the defining double gamma products. T…
Exact mathematical passage
\frac{d^4}{dw^4}\log(1-e^{2\pi iw}) =-6\sum_{n\in\mathbb Z}(w-n)^{-4}.
W^5Use in source · 2 passages
Search aliases: W^5
Use in source · stark.tex, line 550
W^5=σ(W)U_rm alg^6,
Exact mathematical passage
W^5=\sigma(W)U_{\rm alg}^6,
w^kUse in source · 1 passage
Search aliases: w^k
Use in source · reconstruction.tex, line 82
ir poles, and divW_D=divR, W_D(0)=1, (log W_D)^(j)(0)=0 (1≤ j≤3). Genus three is sufficient and minimal for this ungrouped absolute construction. Special signed groupings are not excluded. The cubic upper bound gives ∑_D|m_λ|/|λ|^4<∞, by partial summation or dyadic shells. For |w|<1, log E_3(w)=-∑_k≥4fracw^kk. On each fixed compact set the tail logarithms are thus absolutely and uniformly summable. The finitely many remaining factors supply exactly the stated orders, and each factor has its firs…
Exact mathematical passage
\log E_3(w)=-\sum_{k\ge4}\frac{w^k}{k}.
W_DUse in source · 13 passages
Search aliases: W_D
Use in source · reconstruction.tex, line 67
The genus-three divisor product. Let D={λ:m_λ≠0}; zero is not in D. Define E_3(w)=(1-w)e^w+w^2/2+w^3/3, W_D(z)=∏_λ∈ DE_3(z/λ)^m_λ. Negative exponents mean the quotient of the separate positive and negative entire products. [Canonical regularizer and absolute genus] The products defining W_D converge normally on compact sets away from their poles, and divW_D=divR, W_D(0)=1, (log W_D)^(j)(0)=0 (1≤ j≤3). Genus three is sufficient and minimal for this ungrouped absolute construction. Special signed …
Exact mathematical passage
\label{eq:WD} E_3(w)=(1-w)e^{w+w^2/2+w^3/3},\qquad W_D(z)=\prod_{\lambda\in D}E_3(z/\lambda)^{m_\lambda}.
W_{D,>T}Use in source · 1 passage
Search aliases: W_D,>T
Use in source · reconstruction.tex, line 95
logarithm sum diverges, because the fourth-power remainder is summable. This proves failure of ungrouped absolute genus two, without a claim about cancellations in specially grouped signed products. [Explicit analytic tail] If n_D(r)=∑_|λ|≤ r|m_λ|≤ Ar^3 for all r≥ R_*, then for T≥ R_* and |z|≤ T/2, ∑_|λ|>Tfrac|m_λ||λ|^4≤frac4AT, |log W_D,>T(z)|≤frac2A|z|^4T. Partial summation gives the first upper bound by dropping its nonpositive boundary term and using 4∫_T^∞ Ar^3r^-5 dr=4A/T. For the tail fac…
Exact mathematical passage
\label{eq:tail} \sum_{|\lambda|>T}\frac{|m_\lambda|}{|\lambda|^4}\le\frac{4A}{T}, \qquad |\log W_{D,>T}(z)|\le\frac{2A|z|^4}{T}.
W_D^\#Use in source · 2 passages
Search aliases: W_D^#
Use in source · reconstruction.tex, line 200
The repaired conjugation quotient and its boundary. For f^#(z)=overlinef(overline z), the canonical product W_D^# has the conjugate divisor. Hence fracR/R^#W_D/W_D^# =fracR_0overlineR_0 exp(∑_j=1^3(α_j-overlineα_j)z^j). This is entire and nowhere zero, with modulus one on the real axis. It cancels all unmatched divisor orders by a canonical meromorphic factor. That factor is not a holomorphic-unit gauge or a finite theta section. Regularity on the scalar plane does not establish periodic des
Exact mathematical passage
W_D^\#
W_{\rm St}Use in source · 3 passages
Search aliases: W_rm St rm
Use in source · stark.tex, line 525
Consequently W_rm St=σ(W) satisfies
Exact mathematical passage
W_{\rm St}=\sigma(W)
w_{T^2v}Use in source · 2 passages
Search aliases: w_T^2v
Use in source · theta.tex, line 397
F(ρ w_v)=F(w_Tv)B_v, B_v=∏_j=0^A_v-1q_t(w_T^2v+js), A_v=5+4b-2c+7a.
Exact mathematical passage
F(\rho w_v)=F(w_{Tv})B_v,\qquad B_v=\prod_{j=0}^{A_v-1}q_t(w_{T^2v}+js), \quad A_v=5+4b-2c+7a.
w_{Tv}Use in source · 1 passage
Search aliases: w_Tv
Use in source · theta.tex, line 397
F(ρ w_v)=F(w_Tv)B_v, B_v=∏_j=0^A_v-1q_t(w_T^2v+js), A_v=5+4b-2c+7a.
Exact mathematical passage
F(\rho w_v)=F(w_{Tv})B_v,\qquad B_v=\prod_{j=0}^{A_v-1}q_t(w_{T^2v}+js), \quad A_v=5+4b-2c+7a.
w_vUse in source · 2 passages
Search aliases: w_v
Use in source · theta.tex, line 390
w_v=tfrac13+(bτ+cs+a)/2, v=(b,c,a)∈{0,1}^3, T(b,c,a)=(a,b,c), τ=-t.
Exact mathematical passage
w_v=\tfrac13+(b\tau+cs+a)/2,\quad v=(b,c,a)\in\{0,1\}^3, \quad T(b,c,a)=(a,b,c),\quad \tau=-t.
\widehat aUse in source · 3 passages
Search aliases: widehat a widehat
Use in source · principal.tex, line 76
epresenting δ. Thus β^-1Λ_5=Λ_5+ Zδ. Write a potential singular point as λ(a,b,c)=fracaτ+bs+c/5β, p=lceilfrac3c-2b7rceil, k=7p+2b-3c∈{0,…,6}. Exactly one argument λ+kδ lies in Λ_5. Its integral coordinates are (widehat a,widehat b,widehat c)=(p,2p-a,20p+5b-8c). Indeed N_5(widehat a,widehat b,widehat c)^T=(a,b-8k,c-5k)^T, which expresses β(λ+kδ)=βλ+k(-8s-1). The other six digits fail integrality. Also widehat c-2c=5(4p+b-2c), w(widehat c)=w(c). This transports the fifth-smoothing weight exactly.
Exact mathematical passage
\label{eq:lift} (\widehat a,\widehat b,\widehat c)=(p,2p-a,20p+5b-8c).
\widehat bUse in source · 3 passages
Search aliases: widehat b widehat
Use in source · principal.tex, line 76
epresenting δ. Thus β^-1Λ_5=Λ_5+ Zδ. Write a potential singular point as λ(a,b,c)=fracaτ+bs+c/5β, p=lceilfrac3c-2b7rceil, k=7p+2b-3c∈{0,…,6}. Exactly one argument λ+kδ lies in Λ_5. Its integral coordinates are (widehat a,widehat b,widehat c)=(p,2p-a,20p+5b-8c). Indeed N_5(widehat a,widehat b,widehat c)^T=(a,b-8k,c-5k)^T, which expresses β(λ+kδ)=βλ+k(-8s-1). The other six digits fail integrality. Also widehat c-2c=5(4p+b-2c), w(widehat c)=w(c). This transports the fifth-smoothing weight exactly.
Exact mathematical passage
\label{eq:lift} (\widehat a,\widehat b,\widehat c)=(p,2p-a,20p+5b-8c).
\widehat cUse in source · 4 passages
Search aliases: widehat c widehat
Use in source · principal.tex, line 76
epresenting δ. Thus β^-1Λ_5=Λ_5+ Zδ. Write a potential singular point as λ(a,b,c)=fracaτ+bs+c/5β, p=lceilfrac3c-2b7rceil, k=7p+2b-3c∈{0,…,6}. Exactly one argument λ+kδ lies in Λ_5. Its integral coordinates are (widehat a,widehat b,widehat c)=(p,2p-a,20p+5b-8c). Indeed N_5(widehat a,widehat b,widehat c)^T=(a,b-8k,c-5k)^T, which expresses β(λ+kδ)=βλ+k(-8s-1). The other six digits fail integrality. Also widehat c-2c=5(4p+b-2c), w(widehat c)=w(c). This transports the fifth-smoothing weight exactly.
Exact mathematical passage
\label{eq:lift} (\widehat a,\widehat b,\widehat c)=(p,2p-a,20p+5b-8c).
\wpUse in source · 14 passages
Search aliases: wp
Use in source · norms.tex, line 56
The fifth-isogeny differential retains its constant. Let a=wp_t(1/5), b=wp_t(2/5) and L_t=(q_t'/q_t)'. The enlarged lattice is Λ_t^+=tfrac15 Z+ Zt, with wp_+(z)=25wp_5t(5z). The degree-five quotient satisfies L_t(z)=wp_+(z)-5wp_t(z)+2(a+b). At zero L_t has principal part -4/z^2; at the four nonzero kernel points it has +1/(z-P_j)^2, with no simple-pole terms. Thus L_t-wp_++5wp_t is a constant elliptic function. The canonical formula gives q_t(z)=(a-b)^
Exact mathematical passage
a=\wp_t(1/5)
\wp'^2Use in source · 1 passage
Search aliases: wp'^2 wp
Use in source · stark.tex, line 393
For completeness, T is nonzero three-torsion and its normalized x-coordinate is a root of 3x(x^3+1). The lattice is invariant under ordinary conjugation, so wp(1/3) and its derivative are real. Since g_3(Λ_0)>0, the real differential equation wp'^2=4wp^3-g_3 implies wp(1/3)>0. The argument of Ω^2 is π/3, selecting -ω among the three roots of x^3=-1.
Exact mathematical passage
\wp'^2=4\wp^3-g_3
\wp^3Use in source · 1 passage
Search aliases: wp^3 wp
Use in source · stark.tex, line 393
For completeness, T is nonzero three-torsion and its normalized x-coordinate is a root of 3x(x^3+1). The lattice is invariant under ordinary conjugation, so wp(1/3) and its derivative are real. Since g_3(Λ_0)>0, the real differential equation wp'^2=4wp^3-g_3 implies wp(1/3)>0. The argument of Ω^2 is π/3, selecting -ω among the three roots of x^3=-1.
Exact mathematical passage
\wp'^2=4\wp^3-g_3
\wp_+Use in source · 4 passages
Search aliases: wp_+ wp
Use in source · norms.tex, line 56
The fifth-isogeny differential retains its constant. Let a=wp_t(1/5), b=wp_t(2/5) and L_t=(q_t'/q_t)'. The enlarged lattice is Λ_t^+=tfrac15 Z+ Zt, with wp_+(z)=25wp_5t(5z). The degree-five quotient satisfies L_t(z)=wp_+(z)-5wp_t(z)+2(a+b). At zero L_t has principal part -4/z^2; at the four nonzero kernel points it has +1/(z-P_j)^2, with no simple-pole terms. Thus L_t-wp_++5wp_t is a constant elliptic function. The canonical formula gives q_t(z)=(a-b)^2z^4[1+(a+b)z^2+O(z^4)], fixing that constan…
Exact mathematical passage
\wp_+(z)=25\wp_{5t}(5z)
\wp_{5t}Use in source · 1 passage
Search aliases: wp_5t wp
Use in source · norms.tex, line 56
The fifth-isogeny differential retains its constant. Let a=wp_t(1/5), b=wp_t(2/5) and L_t=(q_t'/q_t)'. The enlarged lattice is Λ_t^+=tfrac15 Z+ Zt, with wp_+(z)=25wp_5t(5z). The degree-five quotient satisfies L_t(z)=wp_+(z)-5wp_t(z)+2(a+b). At zero L_t has principal part -4/z^2; at the four nonzero kernel points it has +1/(z-P_j)^2, with no simple-pole terms. Thus L_t-wp_++5wp_t is a constant elliptic function. The canonical formula gives q_t(z)=(a-b)^2z^4[1+(a+b)z^2+O(z^4)], fixing that constan…
Exact mathematical passage
\wp_+(z)=25\wp_{5t}(5z)
\wp_tUse in source · 7 passages
Search aliases: wp_t wp
Use in source · norms.tex, line 56
The fifth-isogeny differential retains its constant. Let a=wp_t(1/5), b=wp_t(2/5) and L_t=(q_t'/q_t)'. The enlarged lattice is Λ_t^+=tfrac15 Z+ Zt, with wp_+(z)=25wp_5t(5z). The degree-five quotient satisfies L_t(z)=wp_+(z)-5wp_t(z)+2(a+b). At zero L_t has principal part -4/z^2; at the four nonzero kernel points it has +1/(z-P_j)^2, with no simple-pole terms. Thus L_t-wp_++5wp_t is a constant elliptic function. The canonical formula gives q_t(z)=(a-b)^
Exact mathematical passage
a=\wp_t(1/5)
\wp_t'Use in source · 1 passage
Search aliases: wp_t' wp
Use in source · theta.tex, line 55
x(z)=wp_t(z), y(z)=wp_t'(z), f(X)=4X^3-g_2X-g_3.
Exact mathematical passage
x(z)=\wp_t(z),\quad y(z)=\wp_t'(z),\quad f(X)=4X^3-g_2X-g_3.
\wp_\LambdaUse in source · 1 passage
Search aliases: wp_Λ wp Lambda
Use in source · stark.tex, line 399
fracσ_Λ(T+z)σ_Λ(T-z) σ_Λ(T)^2σ_Λ(z)^2 =wp_Λ(z)-wp_Λ(T). tagA5.4
Exact mathematical passage
\frac{\sigma_\Lambda(T+z)\sigma_\Lambda(T-z)}{ \sigma_\Lambda(T)^2\sigma_\Lambda(z)^2} =\wp_\Lambda(z)-\wp_\Lambda(T). \tag{A5.4}
\wp_{\Lambda_0}Use in source · 2 passages
Search aliases: wp_Λ_0 wp Lambda
Use in source · stark.tex, line 60
X=fracΩ^2wp_Λ_0(1/J), Λ_0= Z+ Zω, Ω=e^π i/6fracΓ_rm Euler(1/3)^32π. tagA1.3
Exact mathematical passage
X=\frac{\Omega^2}{\wp_{\Lambda_0}(1/J)},\qquad \Lambda_0=\mathbb Z+\mathbb Z\omega,\qquad \Omega=e^{\pi i/6}\frac{\Gamma_{\rm Euler}(1/3)^3}{2\pi}. \tag{A1.3}
XUse in source · 34 passages
Search aliases: X
Use in source · H12_Cubic_Divisor_Traditional.tex, line 103
The arithmetic candidate is fixed independently. Let J=2+3ω and choose the Gauss branch X=β^-1∑_a=1^6χ_7(a)e^6π ia/7, (χ_7(1),…,χ_7(6))=(1,ω,ω^2,ω^2,ω,1), X^3=J. Then E= Q(ω,β,X) has degree eighteen. The retained arithmetic construction defines ν · =(X+β/X-β^2)/3, · v · =-β^2ν-2ν^2,
Exact mathematical passage
X=\beta^{-1}\sum_{a=1}^{6}\chi_7(a)e^{6\pi ia/7},\qquad (\chi_7(1),\ldots,\chi_7(6))=(1,\omega,\omega^2,\omega^2,\omega,1),\quad X^3=J.
\mathsf XUse in source · 6 passages
Search aliases: X mathsf
Use in source · theta.tex, line 268
X_t(z)=fracx(z)-aa-b.
Exact mathematical passage
\mathsf X_t(z)=\frac{x(z)-a}{a-b}.
xUse in source · 21 passages
Search aliases: x
Use in source · rigidity.tex, line 15
The cubic characteristic polynomial (x-2)^3+7 gives a stable real eigenvalue a=2-7^1/3∈(0,1) and a conjugate unstable pair of modulus a^-1/2>1. Every nonzero integer A-orbit escapes in both directions: otherwise an integer vector would tend to zero along positive or negative iterates. Fourier orbit sums are therefore absolutely convergent for smooth coefficients. The dual degree-seven image is exactly k∈ B Z^3 Longleftrightarrow ℓ(k):=k_0+2k_1+4k_2≡0pmod7. Indeed k-ℓ(k)e_0=B(-k_1,k_2,-8k_2)^T an…
Exact mathematical passage
(x-2)^3+7
X^{-1}Use in source · 1 passage
Search aliases: X^-1
Use in source · stark.tex, line 379
x(u)=X^-1, x(2u)=ω^2X^-1, x(3u)=ω X^-1. tagA5.2
Exact mathematical passage
x(u)=X^{-1},\quad x(2u)=\omega^2X^{-1},\quad x(3u)=\omega X^{-1}. \tag{A5.2}
X^{-3}Use in source · 1 passage
Search aliases: X^-3
Use in source · stark.tex, line 422
Multiplication and (A4.3) give C^3|X^-3+1|=1. Cubing each equation in (A5.5) now removes the unknown constant exactly:
Exact mathematical passage
C^3|X^{-3}+1|=1
X^2Use in source · 4 passages
Search aliases: X^2
Use in source · stark.tex, line 431
a_0 · =frac(ω^2+ω X)^31+J =(1+ω)X^2-X+1, a_1 · =frac(1+ω X)^31+J =-X^2+(1+ω)X+1, a_2 · =frac(ω+ω X)^31+J =-ω X^2-ω X+1. tagA5.7
Exact mathematical passage
\begin{aligned} a_0&=\frac{(\omega^2+\omega X)^3}{1+J} =(1+\omega)X^2-X+1,\\ a_1&=\frac{(1+\omega X)^3}{1+J} =-X^2+(1+\omega)X+1,\\ a_2&=\frac{(\omega+\omega X)^3}{1+J} =-\omega X^2-\omega X+1. \end{aligned} \tag{A5.7}
X^3Use in source · 5 passages
Search aliases: X^3
Use in source · H12_Cubic_Divisor_Traditional.tex, line 103
The arithmetic candidate is fixed independently. Let J=2+3ω and choose the Gauss branch X=β^-1∑_a=1^6χ_7(a)e^6π ia/7, (χ_7(1),…,χ_7(6))=(1,ω,ω^2,ω^2,ω,1), X^3=J. Then E= Q(ω,β,X) has degree eighteen. The retained arithmetic construction defines ν · =(X+β/X-β^2)/3, · v · =-β^2ν-2ν^2,
Exact mathematical passage
X=\beta^{-1}\sum_{a=1}^{6}\chi_7(a)e^{6\pi ia/7},\qquad (\chi_7(1),\ldots,\chi_7(6))=(1,\omega,\omega^2,\omega^2,\omega,1),\quad X^3=J.
x^3Use in source · 3 passages
Search aliases: x^3
Use in source · stark.tex, line 371
On the normalized curve E_0:y^2+y=x^3, set
Exact mathematical passage
E_0:y^2+y=x^3
X^4Use in source · 2 passages
Search aliases: X^4
Use in source · theta.tex, line 155
ψ_3(X)=3X^4-tfrac32g_2X^2-3g_3X-tfrac116g_2^2,
Exact mathematical passage
\psi_3(X)=3X^4-\tfrac32g_2X^2-3g_3X-\tfrac1{16}g_2^2,
X^jUse in source · 1 passage
Search aliases: X^j
Use in source · stark.tex, line 512
A basis for the rational calculation is β^iX^jω^k, 0≤ i,j<3, 0≤ k<2. For odd h, multiply the two norms instead of dividing: the residual is their product minus one. Both supplied implementations obtain the exact zero polynomial in all six cases.
Exact mathematical passage
\beta^iX^j\omega^k
\mathsf X_sUse in source · 1 passage
Search aliases: X_s mathsf
Use in source · theta.tex, line 374
X_t(ρ z)= X_s(z), q_t(ρ z)=q_s(z). tagN7.1
Exact mathematical passage
\mathsf X_t(\rho z)=\mathsf X_s(z),\qquad \boxed{q_t(\rho z)=q_s(z).} \tag{N7.1}
\mathsf X_tUse in source · 4 passages
Search aliases: X_t mathsf
Use in source · theta.tex, line 268
X_t(z)=fracx(z)-aa-b.
Exact mathematical passage
\mathsf X_t(z)=\frac{x(z)-a}{a-b}.
YUse in source · 5 passages
Search aliases: Y
Use in source · H12_Cubic_Divisor_Traditional.tex, line 108
The arithmetic candidate is fixed independently. Let J=2+3ω and choose the Gauss branch X=β^-1∑_a=1^6χ_7(a)e^6π ia/7, (χ_7(1),…,χ_7(6))=(1,ω,ω^2,ω^2,ω,1), X^3=J. Then E= Q(ω,β,X) has degree eighteen. The retained arithmetic construction defines ν · =(X+β/X-β^2)/3, · v · =-β^2ν-2ν^2, Y · =(4β^2+β-14)v^2+(11β-23)v-β^2+3β-3, Z · =frac(-2β^2+7β-8)v^2+(-10β^2+11β+17)v3 +7-β-β^2, U_rm alg · =fracY-(1+2ω)Z2. These formulas choose no coefficient or branch by fitting U_Γ. The original desired identity is…
Exact mathematical passage
\nu&=(X+\beta/X-\beta^2)/3,&v&=-\beta^2\nu-2\nu^2,\\ Y&=(4\beta^2+\beta-14)v^2+(11\beta-23)v-\beta^2+3\beta-3,\\ Z&=\frac{(-2\beta^2+7\beta-8)v^2+(-10\beta^2+11\beta+17)v}{3} +7-\beta-\beta^2,\\ U_{\rm alg}&=\frac{Y-(1+2\omega)Z}{2}.
yUse in source · 18 passages
Search aliases: y
Use in source · stark.tex, line 371
On the normalized curve E_0:y^2+y=x^3, set
Exact mathematical passage
E_0:y^2+y=x^3
y^{1-2s}Use in source · 1 passage
Search aliases: y^1-2s
Use in source · stark.tex, line 656
sqrtπ fracΓ(s-1/2)Γ(s) y^1-2s[ζ(2s-1,a)+ζ(2s-1,1-a)].
Exact mathematical passage
\sqrt\pi\,\frac{\Gamma(s-1/2)}{\Gamma(s)}\, y^{1-2s}\bigl[\zeta(2s-1,a)+\zeta(2s-1,1-a)\bigr].
Y^2Use in source · 1 passage
Search aliases: Y^2
Use in source · theta.tex, line 307
Y^2+(1-u)XY-uY=X^3-uX^2, tagN6.1
Exact mathematical passage
\boxed{Y^2+(1-u)XY-uY=X^3-uX^2,} \tag{N6.1}
y^2Use in source · 2 passages
Search aliases: y^2
Use in source · stark.tex, line 371
On the normalized curve E_0:y^2+y=x^3, set
Exact mathematical passage
E_0:y^2+y=x^3
y_aUse in source · 8 passages
Search aliases: y_a
Use in source · theta.tex, line 67
a=x(P), b=x(2P), d=a-b, y_a=y(P), y_b=y(2P),
Exact mathematical passage
a=x(P),\quad b=x(2P),\quad d=a-b,\quad y_a=y(P),\quad y_b=y(2P),
y_bUse in source · 7 passages
Search aliases: y_b
Use in source · theta.tex, line 67
a=x(P), b=x(2P), d=a-b, y_a=y(P), y_b=y(2P),
Exact mathematical passage
a=x(P),\quad b=x(2P),\quad d=a-b,\quad y_a=y(P),\quad y_b=y(2P),
ZUse in source · 7 passages
Search aliases: Z
Use in source · H12_Cubic_Divisor_Traditional.tex, line 108
The arithmetic candidate is fixed independently. Let J=2+3ω and choose the Gauss branch X=β^-1∑_a=1^6χ_7(a)e^6π ia/7, (χ_7(1),…,χ_7(6))=(1,ω,ω^2,ω^2,ω,1), X^3=J. Then E= Q(ω,β,X) has degree eighteen. The retained arithmetic construction defines ν · =(X+β/X-β^2)/3, · v · =-β^2ν-2ν^2, Y · =(4β^2+β-14)v^2+(11β-23)v-β^2+3β-3, Z · =frac(-2β^2+7β-8)v^2+(-10β^2+11β+17)v3 +7-β-β^2, U_rm alg · =fracY-(1+2ω)Z2. These formulas choose no coefficient or branch by fitting U_Γ. The original desired identity is…
Exact mathematical passage
\nu&=(X+\beta/X-\beta^2)/3,&v&=-\beta^2\nu-2\nu^2,\\ Y&=(4\beta^2+\beta-14)v^2+(11\beta-23)v-\beta^2+3\beta-3,\\ Z&=\frac{(-2\beta^2+7\beta-8)v^2+(-10\beta^2+11\beta+17)v}{3} +7-\beta-\beta^2,\\ U_{\rm alg}&=\frac{Y-(1+2\omega)Z}{2}.
zUse in source · 103 passages
Search aliases: z
Use in source · H12_Cubic_Divisor_Traditional.tex, line 72
For positive imaginary periods use the additive product convention θ_0(z;a) · =∏_n≥0(1-e^2π i(z+na))(1-e^2π i((n+1)a-z)), Γ_rm ell(z;a,b) · =∏_j,k≥0 frac1-e^2π i((j+1)a+(k+1)b-z)1-e^2π i(ja+kb+z). The negative-first-period continuation is Γ_rm ell(z;-t,s)=Γ_rm ell(z+t;t,s)^-1. Consequently the main scalar function is F(z)=fracΓ_rm ell(z;τ,s)^5Γ_rm ell(5z;5τ,5s) =fracΓ_rm ell(5(z+t);5t,5s)Γ_rm ell(z+t;t,s)^5, U_Γ=F(1/3). This convention agrees with the checked primary authors' copy of Bergeron--C…
Exact mathematical passage
\theta_0(z;a)&=\prod_{n\ge0}(1-e^{2\pi i(z+na)})(1-e^{2\pi i((n+1)a-z)}),\label{eq:theta-product}\\ \Gamma_{\rm ell}(z;a,b)&=\prod_{j,k\ge0} \frac{1-e^{2\pi i((j+1)a+(k+1)b-z)}}{1-e^{2\pi i(ja+kb+z)}}.\label{eq:gamma-product}
z^{-2}Use in source · 2 passages
Search aliases: z^-2
Use in source · reconstruction.tex, line 150
Let L_t=(q_t'/q_t)' and S_t=wp_t(1/5)+wp_t(2/5). The normalized theta formula gives L_t(z)=-4z^-2+2S_t+O(z^2), S_s=ρ^2S_t. Since 7δ-2s=-t+4, evenness and t-periodicity pair the six translated L_t terms. Using s=u_0+3δ-2, their sum is 2∑_j=0^2L_t(u_j). The second logarithmic derivative of [source reference] cancels all z^-2 terms, leaving (log T)''(0)=2(1-β^2)(1-ρ^2)S_t-2∑_j=0^2L_t(u_j). The left side is 4α_2. Therefore α_2=frac12[(1-β^2)(1-ρ^2)S_t- ∑_j=0^2L_t(u_j)]. The factor 1/2 is fixed by re…
Exact mathematical passage
L_t(z)=-4z^{-2}+2S_t+O(z^2),\qquad S_s=\rho^2S_t.
z^{-4}Use in source · 1 passage
Search aliases: z^-4
Use in source · principal.tex, line 135
[Three-theta anchor] R is regular and nonzero at zero, and R_0:=R(0)=∏_j=0^2q_t(u_j)^-1, u_j=fract+δ2+jδ. Let H(a)=η(5a)/η(a)^5. Direct product cancellation gives S_5(z;a,b)∼frac5(2π)^4 H(a) H(b)z^-4. The ratio in [source reference] therefore has value P_7(0)=[ H(s)/ H(t)]/S_5(s;t,δ). Since F(0)= H(s)/ H(t)=ϵ^2, the eta factors cancel and R_0=S_5(s;t,δ)^-1. The period relation is s=(t+δ)/2+3δ-2. At the central point, S_5((t+δ)/2;t,δ)=1 by termwise coincidence of the gamma numerator and denominat…
Exact mathematical passage
S_5(z;a,b)\sim\frac5{(2\pi)^4}\mathcal H(a)\mathcal H(b)z^{-4}.
Z^{-n}Use in source · 1 passage
Search aliases: Z^-n
Use in source · product_proof.tex, line 3
Full seven-product series proof. For positive imaginary periods t,s,d with t+7d=2s+4, put P=e^2π it, Q=e^2π is, D=e^2π id and Z=e^2π iz. Then PD^7=Q^2. In a common convergence region the fifth-smoothed logarithms are log S_5(z;a,b) · =5∑_substackn≥1 5nmid n fracZ^n-(A_0B_0/Z)^nn(1-A_0^n)(1-B_0^n), log F_t,s(z) · =-5∑_substackn≥1 5nmid n fracP^nZ^n-Q^nZ^-nn(1-P^n)(1-Q^n), where A_0=e^2π ia,B_0=e^2π ib in the first line. At one surviving frequency suppress the index n. Summing seven shifts multipl…
Exact mathematical passage
\log S_5(z;a,b)&=5\sum_{\substack{n\ge1\\5\nmid n}} \frac{Z^n-(A_0B_0/Z)^n}{n(1-A_0^n)(1-B_0^n)},\\ \log F_{t,s}(z)&=-5\sum_{\substack{n\ge1\\5\nmid n}} \frac{P^nZ^n-Q^nZ^{-n}}{n(1-P^n)(1-Q^n)},
\mathbb Z^2Use in source · 1 passage
Search aliases: Z^2 mathbb
Use in source · norms.tex, line 45
The exact information lost by doubling. Quotient nonzero meromorphic elliptic functions by constants and translation coboundaries A(w+s)/A(w). Set e=[q_t], p=[ψ] in additive notation. These generate a free Z^2: the divisor charges on the disjoint infinite translation orbits of 0 and -S/2 are respectively (4,4) and (0,-4). A finite divisor coboundary has charge zero on every orbit by telescoping, so a relation ae+bp=0 forces b=0 then a=0.
Exact mathematical passage
\mathbb Z^2
z^2Use in source · 5 passages
Search aliases: z^2
Use in source · norms.tex, line 60
The fifth-isogeny differential retains its constant. Let a=wp_t(1/5), b=wp_t(2/5) and L_t=(q_t'/q_t)'. The enlarged lattice is Λ_t^+=tfrac15 Z+ Zt, with wp_+(z)=25wp_5t(5z). The degree-five quotient satisfies L_t(z)=wp_+(z)-5wp_t(z)+2(a+b). At zero L_t has principal part -4/z^2; at the four nonzero kernel points it has +1/(z-P_j)^2, with no simple-pole terms. Thus L_t-wp_++5wp_t is a constant elliptic function. The canonical formula gives q_t(z)=(a-b)^2z^4[1+(a+b)z^2+O(z^4)], fixing that constan…
Exact mathematical passage
-4/z^2
z^3Use in source · 5 passages
Search aliases: z^3
Use in source · polarization.tex, line 103
∏_H a lineN_E/E^H(z) =z^3N_E/F_0(z). tagQ2
Exact mathematical passage
\boxed{ \prod_{H\text{ a line}}N_{E/E^H}(z) =z^3N_{E/F_0}(z). } \tag{Q2}
z^4Use in source · 2 passages
Search aliases: z^4
Use in source · norms.tex, line 60
mathbb Z+ Zt, with wp_+(z)=25wp_5t(5z). The degree-five quotient satisfies L_t(z)=wp_+(z)-5wp_t(z)+2(a+b). At zero L_t has principal part -4/z^2; at the four nonzero kernel points it has +1/(z-P_j)^2, with no simple-pole terms. Thus L_t-wp_++5wp_t is a constant elliptic function. The canonical formula gives q_t(z)=(a-b)^2z^4[1+(a+b)z^2+O(z^4)], fixing that constant as 2(a+b). The degree-five quotient is equivalently the paired sum wp_+(z)=wp_t(z)+∑_j=1^4[wp_t(z+j/5)-wp_t(j/5)]. The analytic iden…
Exact mathematical passage
q_t(z)=(a-b)^2z^4[1+(a+b)z^2+O(z^4)]
z^jUse in source · 1 passage
Search aliases: z^j
Use in source · reconstruction.tex, line 201
The repaired conjugation quotient and its boundary. For f^#(z)=overlinef(overline z), the canonical product W_D^# has the conjugate divisor. Hence fracR/R^#W_D/W_D^# =fracR_0overlineR_0 exp(∑_j=1^3(α_j-overlineα_j)z^j). This is entire and nowhere zero, with modulus one on the real axis. It cancels all unmatched divisor orders by a canonical meromorphic factor. That factor is not a holomorphic-unit gauge or a finite theta section. Regularity on the scalar plane does not establish periodic descent…
Exact mathematical passage
\label{eq:regularized-conjugation} \frac{R/R^\#}{W_D/W_D^\#} =\frac{R_0}{\overline{R_0}} \exp\left(\sum_{j=1}^{3}(\alpha_j-\overline{\alpha_j})z^j\right).
Z^nUse in source · 1 passage
Search aliases: Z^n
Use in source · product_proof.tex, line 3
Full seven-product series proof. For positive imaginary periods t,s,d with t+7d=2s+4, put P=e^2π it, Q=e^2π is, D=e^2π id and Z=e^2π iz. Then PD^7=Q^2. In a common convergence region the fifth-smoothed logarithms are log S_5(z;a,b) · =5∑_substackn≥1 5nmid n fracZ^n-(A_0B_0/Z)^nn(1-A_0^n)(1-B_0^n), log F_t,s(z) · =-5∑_substackn≥1 5nmid n fracP^nZ^n-Q^nZ^-nn(1-P^n)(1-Q^n), where A_0=e^2π ia,B_0=e^2π ib in the first line. At one surviving frequency suppress the index n. Summing seven shifts multipl…
Exact mathematical passage
\log S_5(z;a,b)&=5\sum_{\substack{n\ge1\\5\nmid n}} \frac{Z^n-(A_0B_0/Z)^n}{n(1-A_0^n)(1-B_0^n)},\\ \log F_{t,s}(z)&=-5\sum_{\substack{n\ge1\\5\nmid n}} \frac{P^nZ^n-Q^nZ^{-n}}{n(1-P^n)(1-Q^n)},
Z_{i,j,k}^{(n)}Use in source · 1 passage
Search aliases: Z_i,j,k^(n)
Use in source · reconstruction.tex, line 6
Explicit three-dimensional families. For integers n≥1, 1≤ i,j,k≤ n, define Z_i,j,k^(n)=λ(-i,j,5(n+k)), P_i,j,k,r^(n)=λ(-i,j,5(n+k)-r), 1≤ r≤4. Every such triple has a<0, b>0, 3c>2b. Hence p>0 and the bracket in [source reference] is one. The Z points have order four and the P points have order minus one. Rational independence makes the parameter map injective. The pole indices are distinct because 5(k-k')=r-r' forces both sides to vanish; the zero and pole families are disjoint by c modulo five.…
Exact mathematical passage
\label{eq:families} Z_{i,j,k}^{(n)}=\lambda(-i,j,5(n+k)),\qquad P_{i,j,k,r}^{(n)}=\lambda(-i,j,5(n+k)-r),\quad 1\le r\le4.
z_jUse in source · 2 passages
Search aliases: z_j
Use in source · history.tex, line 25
The scalar missing from a closed finite orbit. The earlier index-23 and index-10 orbit decompositions are 529=1+528 and 100=1+3+8·12. They are exact finite permutation data. For values z_j on a cycle of length d, replacing every value by cz_j leaves all adjacent ratios unchanged, including their product one, but multiplies the cycle product by c^d. An anchor on another fixed-point orbit does not remove this freedom. Thus orbit closure alone does not evaluate a primitive analytic sector or fix it…
Exact mathematical passage
z_j
Z_mUse in source · 1 passage
Search aliases: Z_m
Use in source · stark.tex, line 243
Let Z_m(s) be the sum of the two ray partial zeta functions in row m, and write ℓ_m=Z_m'(0). Then
Exact mathematical passage
Z_m(s)
Z_m'Use in source · 1 passage
Search aliases: Z_m'
Use in source · stark.tex, line 244
Let Z_m(s) be the sum of the two ray partial zeta functions in row m, and write ℓ_m=Z_m'(0). Then
Exact mathematical passage
\ell_m=Z_m'(0)
|Use in source · 51 passages
Search aliases: |
Use in source · H12_Cubic_Divisor_Traditional.tex, line 68
τ · =(β^2+2β)/15, · t · =-τ, · s · =-(β+2)/15, · δ · =(τ+2s+4)/7. The real cube root 7^1/3 is positive; this fixes the lower complex embedding of β. We have Im t,Im s,Imδ>0, Imτ<0 and |ρ|<1. The last follows from N(ϵ)=1 and 0<2-7^1/3<1 at the real embedding. The numbers 1,t,s are rationally independent: a rational linear relation becomes a polynomial of degree at most two in β, contradicting its irreducible cubic minimal polynomial. [htbp]centeringincludegraphics[width=.96linewidth]figures/01_pe…
Exact mathematical passage
|\rho|<1
|^{-2s}Use in source · 1 passage
Search aliases: |^-2s
Use in source · stark.tex, line 317
E_z(s)=∑_m,n∈ Z|m+nω+z|^-2s,
Exact mathematical passage
\mathcal E_z(s)=\sum_{m,n\in\mathbb Z}|m+n\omega+z|^{-2s},
|^{-4}Use in source · 2 passages
Search aliases: |^-4
Use in source · reconstruction.tex, line 88
The lower bound n_D(r)≥ c r^3 yields divergence of ∑_D|m_λ|/|λ|^3: partial summation contains 3∫ n_D(r)r^-4 dr, which diverges logarithmically. For a fixed z≠0, log E_2(z/λ)=-fracz^33λ^3+O(|λ|^-4). Its weighted absolute logarithm sum diverges, because the fourth-power remainder is summable. This proves failure of ungrouped absolute genus two, without a claim about cancellations in specially grouped signed products. [Explicit analytic tail] If n_D(r)=∑_|λ|≤ r|m_λ|≤ Ar^3 for all r≥ R_*, then for T…
Exact mathematical passage
\log E_2(z/\lambda)=-\frac{z^3}{3\lambda^3}+O(|\lambda|^{-4}).
|^2Use in source · 5 passages
Search aliases: |^2
Use in source · reconstruction.tex, line 61
on. Every meromorphic function u(z)fracP(z)Q(z) has zero and pole counts O(r^2). Thus R is not representable in this class. For Y=ImΩ>0, the defining theta series has terms bounded by exp{-π n^TYn+2π n Im(Az+b) }. Completing the square, and absorbing half of the positive quadratic form into a convergent Gaussian sum, gives Cexp(C'|z|^2) for a fixed affine restriction. Cauchy estimates on fixed-radius circles give the same form for each fixed derivative. Finite sums, products and polynomial facto…
Exact mathematical passage
C\exp(C'|z|^2)
|^3Use in source · 3 passages
Search aliases: |^3
Use in source · polarization.tex, line 269
|U_Γ|^3=|ϵ^3a_2a_1^5|.
Exact mathematical passage
|U_\Gamma|^3=|\epsilon^3a_2a_1^5|.
|^4Use in source · 5 passages
Search aliases: |^4
Use in source · polarization.tex, line 236
R_+=|ϵ|^4|a_1|^6|a_2|^4.
Exact mathematical passage
R_+=|\epsilon|^4|a_1|^6|a_2|^4.
|^6Use in source · 3 passages
Search aliases: |^6
Use in source · polarization.tex, line 236
R_+=|ϵ|^4|a_1|^6|a_2|^4.
Exact mathematical passage
R_+=|\epsilon|^4|a_1|^6|a_2|^4.
\pmUse in source · 1 passage
Search aliases: ± pm
Use in source · stark.tex, line 363
{T±2u}, {T± u}, {T±3u}. tagA5.1
Exact mathematical passage
\{T\pm2u\},\quad\{T\pm u\},\quad\{T\pm3u\}. \tag{A5.1}
\cdotUse in source · 3 passages
Search aliases: · cdot
Use in source · history.tex, line 25
The scalar missing from a closed finite orbit. The earlier index-23 and index-10 orbit decompositions are 529=1+528 and 100=1+3+8·12. They are exact finite permutation data. For values z_j on a cycle of length d, replacing every value by cz_j leaves all adjacent ratios unchanged, including their product one, but multiplies the cycle product by c^d. An anchor on another fixed-point orbit does not remove this freedom. Thus orbit closure alone does not evaluate a primitive analytic sector or fix it…
Exact mathematical passage
100=1+3+8\cdot12
\timesUse in source · 8 passages
Search aliases: × times
Use in source · stark.tex, line 100
Gal(E/ Q)=S_3× C_3. tagA2.1
Exact mathematical passage
\operatorname{Gal}(E/\mathbb Q)=S_3\times C_3. \tag{A2.1}
\alpha^6Use in source · 1 passage
Search aliases: α^6 alpha
Use in source · rigidity.tex, line 75
, leaving fraca_Δ(p_3)a_Δ(0)=e^2π im_2/3e^real number. After the based identification the exact modulus equality forces that real exponent to vanish. The ratio then has cube one, and an independent branch enclosure can select one. The constant α cancels in this based ratio; it cannot simply be discarded in the unbased principal norm, where it contributes α^6.
Exact mathematical passage
\alpha^6
\alpha_1Use in source · 4 passages
Search aliases: α_1 alpha
Use in source · reconstruction.tex, line 112
The fourth logarithmic derivative fixes the remaining degree. [Cubic-exponential reconstruction] For the actual correction R, (log R)^(4)(z)=-6∑_λ∈ Dfracm_λ(z-λ)^4, locally normally away from D. Consequently R(z)=R_0exp(α_1z+α_2z^2+α_3z^3)W_D(z), α_j=frac(log R)^(j)(0)j!. The logarithmic derivatives are branch independent. The notation (log R)^(4) means the third derivative of R'/R, a single-valued meromorphic function. The sine partial fractions [source reference] give fracd^4dw^4log(1-e^2π iw)…
Exact mathematical passage
\label{eq:cubic-reconstruction} R(z)=R_0\exp(\alpha_1z+\alpha_2z^2+\alpha_3z^3)W_D(z),\qquad \alpha_j=\frac{(\log R)^{(j)}(0)}{j!}.
\alpha_2Use in source · 8 passages
Search aliases: α_2 alpha
Use in source · reconstruction.tex, line 112
The fourth logarithmic derivative fixes the remaining degree. [Cubic-exponential reconstruction] For the actual correction R, (log R)^(4)(z)=-6∑_λ∈ Dfracm_λ(z-λ)^4, locally normally away from D. Consequently R(z)=R_0exp(α_1z+α_2z^2+α_3z^3)W_D(z), α_j=frac(log R)^(j)(0)j!. The logarithmic derivatives are branch independent. The notation (log R)^(4) means the third derivative of R'/R, a single-valued meromorphic function. The sine partial fractions [source reference] give fracd^4dw^4log(1-e^2π iw)…
Exact mathematical passage
\label{eq:cubic-reconstruction} R(z)=R_0\exp(\alpha_1z+\alpha_2z^2+\alpha_3z^3)W_D(z),\qquad \alpha_j=\frac{(\log R)^{(j)}(0)}{j!}.
\alpha_3Use in source · 8 passages
Search aliases: α_3 alpha
Use in source · reconstruction.tex, line 112
The fourth logarithmic derivative fixes the remaining degree. [Cubic-exponential reconstruction] For the actual correction R, (log R)^(4)(z)=-6∑_λ∈ Dfracm_λ(z-λ)^4, locally normally away from D. Consequently R(z)=R_0exp(α_1z+α_2z^2+α_3z^3)W_D(z), α_j=frac(log R)^(j)(0)j!. The logarithmic derivatives are branch independent. The notation (log R)^(4) means the third derivative of R'/R, a single-valued meromorphic function. The sine partial fractions [source reference] give fracd^4dw^4log(1-e^2π iw)…
Exact mathematical passage
\label{eq:cubic-reconstruction} R(z)=R_0\exp(\alpha_1z+\alpha_2z^2+\alpha_3z^3)W_D(z),\qquad \alpha_j=\frac{(\log R)^{(j)}(0)}{j!}.
\alpha_jUse in source · 2 passages
Search aliases: α_j alpha
Use in source · reconstruction.tex, line 112
The fourth logarithmic derivative fixes the remaining degree. [Cubic-exponential reconstruction] For the actual correction R, (log R)^(4)(z)=-6∑_λ∈ Dfracm_λ(z-λ)^4, locally normally away from D. Consequently R(z)=R_0exp(α_1z+α_2z^2+α_3z^3)W_D(z), α_j=frac(log R)^(j)(0)j!. The logarithmic derivatives are branch independent. The notation (log R)^(4) means the third derivative of R'/R, a single-valued meromorphic function. The sine partial fractions [source reference] give fracd^4dw^4log(1-e^2π iw)…
Exact mathematical passage
\label{eq:cubic-reconstruction} R(z)=R_0\exp(\alpha_1z+\alpha_2z^2+\alpha_3z^3)W_D(z),\qquad \alpha_j=\frac{(\log R)^{(j)}(0)}{j!}.
\betaUse in source · 67 passages
Search aliases: β beta
Use in source · H12_Cubic_Divisor_Traditional.tex, line 61
Fixed data, conventions, and the claim boundary. Fix throughout ω · =e^2π i/3, · β · =7^1/3ω^2, · ϵ · =2-β, · ρ · =ϵ^-1, τ · =(β^2+2β)/15, · t · =-τ, · s · =-(β+2)/15, · δ · =(τ+2s+4)/7. The real cube root 7^1/3 is positive; this fixes the lower complex embedding of β. We have Im t,Im s,Imδ>0, Imτ<0 and |ρ|<1. The last follows from N(ϵ)=1 and 0<2-7^1/3<1 at the real embedding. The numbers 1,t,s are rationally independent: a rational linear relation becomes a polynomial of degree at most two in β…
Exact mathematical passage
\label{eq:fixed-data} \begin{aligned} \omega&=e^{2\pi i/3},&\beta&=7^{1/3}\omega^2,&\epsilon&=2-\beta,&\rho&=\epsilon^{-1},\\ \tau&=(\beta^2+2\beta)/15,&t&=-\tau,&s&=-(\beta+2)/15,& \delta&=(\tau+2s+4)/7. \end{aligned}
\beta^{-1}Use in source · 4 passages
Search aliases: β^-1 beta
Use in source · H12_Cubic_Divisor_Traditional.tex, line 103
The arithmetic candidate is fixed independently. Let J=2+3ω and choose the Gauss branch X=β^-1∑_a=1^6χ_7(a)e^6π ia/7, (χ_7(1),…,χ_7(6))=(1,ω,ω^2,ω^2,ω,1), X^3=J. Then E= Q(ω,β,X) has degree eighteen. The retained arithmetic construction defines ν · =(X+β/X-β^2)/3, · v · =-β^2ν-2ν^2,
Exact mathematical passage
X=\beta^{-1}\sum_{a=1}^{6}\chi_7(a)e^{6\pi ia/7},\qquad (\chi_7(1),\ldots,\chi_7(6))=(1,\omega,\omega^2,\omega^2,\omega,1),\quad X^3=J.
\beta^2Use in source · 10 passages
Search aliases: β^2 beta
Use in source · H12_Cubic_Divisor_Traditional.tex, line 61
Fixed data, conventions, and the claim boundary. Fix throughout ω · =e^2π i/3, · β · =7^1/3ω^2, · ϵ · =2-β, · ρ · =ϵ^-1, τ · =(β^2+2β)/15, · t · =-τ, · s · =-(β+2)/15, · δ · =(τ+2s+4)/7. The real cube root 7^1/3 is positive; this fixes the lower complex embedding of β. We have Im t,Im s,Imδ>0, Imτ<0 and |ρ|<1. The last follows from N(ϵ)=1 and 0<2-7^1/3<1 at the real embedding. The numbers 1,t,s are rationally independent: a rational linear relation becomes a polynomial of degree at most two in β…
Exact mathematical passage
\label{eq:fixed-data} \begin{aligned} \omega&=e^{2\pi i/3},&\beta&=7^{1/3}\omega^2,&\epsilon&=2-\beta,&\rho&=\epsilon^{-1},\\ \tau&=(\beta^2+2\beta)/15,&t&=-\tau,&s&=-(\beta+2)/15,& \delta&=(\tau+2s+4)/7. \end{aligned}
\beta^3Use in source · 5 passages
Search aliases: β^3 beta
Use in source · polarization.tex, line 10
ω^2+ω+1=0, J=2+3ω, β^3=7, ξ^3=J.
Exact mathematical passage
\omega^2+\omega+1=0,\quad J=2+3\omega,\quad \beta^3=7,\quad \xi^3=J.
\beta^iUse in source · 1 passage
Search aliases: β^i beta
Use in source · stark.tex, line 512
A basis for the rational calculation is β^iX^jω^k, 0≤ i,j<3, 0≤ k<2. For odd h, multiply the two norms instead of dividing: the residual is their product minus one. Both supplied implementations obtain the exact zero polynomial in all six cases.
Exact mathematical passage
\beta^iX^j\omega^k
\gammaUse in source · 1 passage
Search aliases: γ gamma
Use in source · polarization.tex, line 87
Let γ_ij fix ω and send
Exact mathematical passage
\gamma_{ij}
\mathbf\GammaUse in source · 1 passage
Search aliases: Γ mathbf Gamma
Use in source · stark.tex, line 632
|Γ_ a,h( f b^-1)|^6 =|(σ_ bW_rm St)^N( a)-σ_ a|. tagA7.4
Exact mathematical passage
|\mathbf\Gamma_{\mathfrak a,h}(\mathfrak f\mathfrak b^{-1})|^6 =\left|(\sigma_{\mathfrak b}W_{\rm St})^{N(\mathfrak a)-\sigma_{\mathfrak a}}\right|. \tag{A7.4}
\mathbf\Gamma_{\mathfrak a,h}Use in source · 1 passage
Search aliases: Γ_ a,h mathbf Gamma mathfrak
Use in source · stark.tex, line 632
|Γ_ a,h( f b^-1)|^6 =|(σ_ bW_rm St)^N( a)-σ_ a|. tagA7.4
Exact mathematical passage
|\mathbf\Gamma_{\mathfrak a,h}(\mathfrak f\mathfrak b^{-1})|^6 =\left|(\sigma_{\mathfrak b}W_{\rm St})^{N(\mathfrak a)-\sigma_{\mathfrak a}}\right|. \tag{A7.4}
\gamma_{ij}Use in source · 1 passage
Search aliases: γ_ij gamma
Use in source · polarization.tex, line 87
Let γ_ij fix ω and send
Exact mathematical passage
\gamma_{ij}
\Gamma_{\rm ell}Use in source · 6 passages
Search aliases: Γ_rm ell Gamma rm
Use in source · H12_Cubic_Divisor_Traditional.tex, line 72
For positive imaginary periods use the additive product convention θ_0(z;a) · =∏_n≥0(1-e^2π i(z+na))(1-e^2π i((n+1)a-z)), Γ_rm ell(z;a,b) · =∏_j,k≥0 frac1-e^2π i((j+1)a+(k+1)b-z)1-e^2π i(ja+kb+z). The negative-first-period continuation is Γ_rm ell(z;-t,s)=Γ_rm ell(z+t;t,s)^-1. Consequently the main scalar function is F(z)=fracΓ_rm ell(z;τ,s)^5Γ_rm ell(5z;5τ,5s) =fracΓ_rm ell(5(z+t);5t,5s)Γ_rm ell(z+t;t,s)^5, U_Γ=F(1/3). This convention agrees with the checked primary authors' copy of Bergeron--C…
Exact mathematical passage
\theta_0(z;a)&=\prod_{n\ge0}(1-e^{2\pi i(z+na)})(1-e^{2\pi i((n+1)a-z)}),\label{eq:theta-product}\\ \Gamma_{\rm ell}(z;a,b)&=\prod_{j,k\ge0} \frac{1-e^{2\pi i((j+1)a+(k+1)b-z)}}{1-e^{2\pi i(ja+kb+z)}}.\label{eq:gamma-product}
\Gamma_{\rm Euler}Use in source · 1 passage
Search aliases: Γ_rm Euler Gamma rm
Use in source · stark.tex, line 60
X=fracΩ^2wp_Λ_0(1/J), Λ_0= Z+ Zω, Ω=e^π i/6fracΓ_rm Euler(1/3)^32π. tagA1.3
Exact mathematical passage
X=\frac{\Omega^2}{\wp_{\Lambda_0}(1/J)},\qquad \Lambda_0=\mathbb Z+\mathbb Z\omega,\qquad \Omega=e^{\pi i/6}\frac{\Gamma_{\rm Euler}(1/3)^3}{2\pi}. \tag{A1.3}
\deltaUse in source · 34 passages
Search aliases: δ delta
Use in source · H12_Cubic_Divisor_Traditional.tex, line 61
Fixed data, conventions, and the claim boundary. Fix throughout ω · =e^2π i/3, · β · =7^1/3ω^2, · ϵ · =2-β, · ρ · =ϵ^-1, τ · =(β^2+2β)/15, · t · =-τ, · s · =-(β+2)/15, · δ · =(τ+2s+4)/7. The real cube root 7^1/3 is positive; this fixes the lower complex embedding of β. We have Im t,Im s,Imδ>0, Imτ<0 and |ρ|<1. The last follows from N(ϵ)=1 and 0<2-7^1/3<1 at the real embedding. The numbers 1,t,s are rationally independent: a rational linear relation becomes a polynomial of degree at most two in β…
Exact mathematical passage
\label{eq:fixed-data} \begin{aligned} \omega&=e^{2\pi i/3},&\beta&=7^{1/3}\omega^2,&\epsilon&=2-\beta,&\rho&=\epsilon^{-1},\\ \tau&=(\beta^2+2\beta)/15,&t&=-\tau,&s&=-(\beta+2)/15,& \delta&=(\tau+2s+4)/7. \end{aligned}
\delta_MUse in source · 6 passages
Search aliases: δ_M delta
Use in source · rigidity.tex, line 24
The convergent transfer proof. For functions on T^3, let S_Nh(v)=∑_κ∈ker Nh(v+κ), C_Nh(v)=h(Nv), L_Nh(v)=∑_Nw=vh(w), δ_Mh=h-h∘ M. The transfer L_N is unnormalized. For h=∑ c_ke_k, S_Nh=∑_k∈ B Z^37c_ke_k, L_Nh=∑_k7c_Bke_k, S_Nh=(L_Nh)∘ N, L_NC_Nh=7h. Since M permutes ker N and commutes with N, the required operators commute with composition by M.
Exact mathematical passage
S_Nh(v)=\sum_{\kappa\in\ker N}h(v+\kappa),\quad C_Nh(v)=h(Nv),\quad L_Nh(v)=\sum_{Nw=v}h(w),\quad \delta_Mh=h-h\circ M.
\epsilonUse in source · 26 passages
Search aliases: ϵ epsilon
Use in source · H12_Cubic_Divisor_Traditional.tex, line 61
Fixed data, conventions, and the claim boundary. Fix throughout ω · =e^2π i/3, · β · =7^1/3ω^2, · ϵ · =2-β, · ρ · =ϵ^-1, τ · =(β^2+2β)/15, · t · =-τ, · s · =-(β+2)/15, · δ · =(τ+2s+4)/7. The real cube root 7^1/3 is positive; this fixes the lower complex embedding of β. We have Im t,Im s,Imδ>0, Imτ<0 and |ρ|<1. The last follows from N(ϵ)=1 and 0<2-7^1/3<1 at the real embedding. The numbers 1,t,s are rationally independent: a rational linear relation becomes a polynomial of degree at most two in β…
Exact mathematical passage
\label{eq:fixed-data} \begin{aligned} \omega&=e^{2\pi i/3},&\beta&=7^{1/3}\omega^2,&\epsilon&=2-\beta,&\rho&=\epsilon^{-1},\\ \tau&=(\beta^2+2\beta)/15,&t&=-\tau,&s&=-(\beta+2)/15,& \delta&=(\tau+2s+4)/7. \end{aligned}
\epsilon^{-1}Use in source · 2 passages
Search aliases: ϵ^-1 epsilon
Use in source · H12_Cubic_Divisor_Traditional.tex, line 61
Fixed data, conventions, and the claim boundary. Fix throughout ω · =e^2π i/3, · β · =7^1/3ω^2, · ϵ · =2-β, · ρ · =ϵ^-1, τ · =(β^2+2β)/15, · t · =-τ, · s · =-(β+2)/15, · δ · =(τ+2s+4)/7. The real cube root 7^1/3 is positive; this fixes the lower complex embedding of β. We have Im t,Im s,Imδ>0, Imτ<0 and |ρ|<1. The last follows from N(ϵ)=1 and 0<2-7^1/3<1 at the real embedding. The numbers 1,t,s are rationally independent: a rational linear relation becomes a polynomial of degree at most two in β…
Exact mathematical passage
\label{eq:fixed-data} \begin{aligned} \omega&=e^{2\pi i/3},&\beta&=7^{1/3}\omega^2,&\epsilon&=2-\beta,&\rho&=\epsilon^{-1},\\ \tau&=(\beta^2+2\beta)/15,&t&=-\tau,&s&=-(\beta+2)/15,& \delta&=(\tau+2s+4)/7. \end{aligned}
\epsilon^2Use in source · 2 passages
Search aliases: ϵ^2 epsilon
Use in source · H12_Cubic_Divisor_Traditional.tex, line 90
Put q_a(z)=θ_0(z;a)^5/θ_0(5z;5a). Direct product identities and the inherited modular normalization give F(z+1) · =F(z), · fracF(z+τ)F(z) · =q_s(z), · fracF(z+s)F(z) · =q_t(z)^-1, q_s(z) · =q_t(ρ z), · F(0) · =ϵ^2, · F(1/3)F(2/3) · =1. The two elliptic lattices are homothetic, ρΛ_s=Λ_t, with Λ_a= Z+ Za. This is not an endomorphism of the single elliptic lattice Λ_t. Section~[source reference] proves the exact rational fifth-torsion normalization and explains the covariance without omitting its s…
Exact mathematical passage
\label{eq:F-identities} \begin{aligned} F(z+1)&=F(z),&\frac{F(z+\tau)}{F(z)}&=q_s(z),& \frac{F(z+s)}{F(z)}&=q_t(z)^{-1},\\ q_s(z)&=q_t(\rho z),&F(0)&=\epsilon^2,&F(1/3)F(2/3)&=1. \end{aligned}
\epsilon^3Use in source · 3 passages
Search aliases: ϵ^3 epsilon
Use in source · polarization.tex, line 269
|U_Γ|^3=|ϵ^3a_2a_1^5|.
Exact mathematical passage
|U_\Gamma|^3=|\epsilon^3a_2a_1^5|.
\epsilon_0Use in source · 1 passage
Search aliases: ϵ_0 epsilon
Use in source · polarization.tex, line 51
ϵ_j=2-ω^jβ, ϵ=ϵ_0, ϵ_0ϵ_1ϵ_2=1.
Exact mathematical passage
\epsilon_j=2-\omega^j\beta,\qquad \epsilon=\epsilon_0, \qquad \epsilon_0\epsilon_1\epsilon_2=1.
\epsilon_1Use in source · 7 passages
Search aliases: ϵ_1 epsilon
Use in source · polarization.tex, line 51
ϵ_j=2-ω^jβ, ϵ=ϵ_0, ϵ_0ϵ_1ϵ_2=1.
Exact mathematical passage
\epsilon_j=2-\omega^j\beta,\qquad \epsilon=\epsilon_0, \qquad \epsilon_0\epsilon_1\epsilon_2=1.
\epsilon_2Use in source · 7 passages
Search aliases: ϵ_2 epsilon
Use in source · polarization.tex, line 51
ϵ_j=2-ω^jβ, ϵ=ϵ_0, ϵ_0ϵ_1ϵ_2=1.
Exact mathematical passage
\epsilon_j=2-\omega^j\beta,\qquad \epsilon=\epsilon_0, \qquad \epsilon_0\epsilon_1\epsilon_2=1.
\epsilon_jUse in source · 1 passage
Search aliases: ϵ_j epsilon
Use in source · polarization.tex, line 51
ϵ_j=2-ω^jβ, ϵ=ϵ_0, ϵ_0ϵ_1ϵ_2=1.
Exact mathematical passage
\epsilon_j=2-\omega^j\beta,\qquad \epsilon=\epsilon_0, \qquad \epsilon_0\epsilon_1\epsilon_2=1.
\zetaUse in source · 12 passages
Search aliases: ζ zeta
Use in source · stark.tex, line 157
The even characters make no contribution to the derivative at zero. The nontrivial even characters have at least two archimedean zeros. The trivial character has the zero of ζ_K, together with the extra zero of 1-3^-s. This also agrees with BCG, section 3.1.
Exact mathematical passage
\zeta_K
\zeta_{\mathfrak f}'Use in source · 6 passages
Search aliases: ζ_ f' zeta mathfrak
Use in source · stark.tex, line 274
ζ_ f'(c^e s_0^k,0) =(-1)^e(frac R6-ℓ_k-1). tagA3.5
Exact mathematical passage
\boxed{\zeta_{\mathfrak f}'(c^e s_0^k,0) =(-1)^e\left(\frac{\mathcal R}{6}-\ell_{k-1}\right).} \tag{A3.5}
\zeta_{\mathbb Q}Use in source · 2 passages
Search aliases: ζ_ Q zeta mathbb
Use in source · stark.tex, line 173
L_F_0(s,ϱ)=ζ_K(s)/ζ_ Q(s).
Exact mathematical passage
L_{F_0}(s,\varrho)=\zeta_K(s)/\zeta_{\mathbb Q}(s).
\zeta_{F_0}Use in source · 1 passage
Search aliases: ζ_F_0 zeta
Use in source · stark.tex, line 254
ζ_F_0(s)(1-3^-s)(1-7^-s),
Exact mathematical passage
\zeta_{F_0}(s)(1-3^{-s})(1-7^{-s}),
\zeta_KUse in source · 2 passages
Search aliases: ζ_K zeta
Use in source · stark.tex, line 157
The even characters make no contribution to the derivative at zero. The nontrivial even characters have at least two archimedean zeros. The trivial character has the zero of ζ_K, together with the extra zero of 1-3^-s. This also agrees with BCG, section 3.1.
Exact mathematical passage
\zeta_K
\etaUse in source · 14 passages
Search aliases: η eta
Use in source · H12_Cubic_Divisor_Traditional.tex, line 99
We use F_0= Q(ω) for the imaginary-quadratic field, to separate it from the scalar gamma function F. The eta quotient is H(a)=η(5a)/η(a)^5; the integer step function H(n) used in the divisor formula is a different object. The symbol R_0 is the scalar anchor, while R_* is a counting cutoff radius. In the arithmetic block ϱ is a ray character and a radical lift, as explicitly indicated there; it is distinct from the contracting period multiplier ρ.
Exact mathematical passage
\mathcal H(a)=\eta(5a)/\eta(a)^5
\eta^2Use in source · 3 passages
Search aliases: η^2 eta
Use in source · stark.tex, line 264
Conjugation of the base field by c sends ϱη^2 to ϱ^-1η^2=λ^2, so
Exact mathematical passage
\varrho\eta^2
\eta^jUse in source · 1 passage
Search aliases: η^j eta
Use in source · stark.tex, line 141
L_K(s,χ_j) =L_ Q(s,χ_-3χ_7^j) L_F_0(s,ϱη^j). tagA2.2
Exact mathematical passage
\boxed{L_K(s,\chi_j) =L_{\mathbb Q}(s,\chi_{-3}\chi_7^j)\, L_{F_0}(s,\varrho\eta^j).} \tag{A2.2}
\eta_{\rm Ded}Use in source · 1 passage
Search aliases: η_rm Ded eta rm
Use in source · stark.tex, line 677
H_τ(z)=e^-π(Im z)^2/y |fracθ_1(π z;e^π iτ)η_rm Ded(τ)|.
Exact mathematical passage
H_\tau(z)=e^{-\pi(\Im z)^2/y} \left|\frac{\theta_1(\pi z;e^{\pi i\tau})}{\eta_{\rm Ded}(\tau)}\right|.
\ThetaUse in source · 1 passage
Search aliases: Θ Theta
Use in source · reconstruction.tex, line 13
the P points have order minus one. Rational independence makes the parameter map injective. The pole indices are distinct because 5(k-k')=r-r' forces both sides to vanish; the zero and pole families are disjoint by c modulo five. [Two-sided cubic growth] Let n_+(r) and n_-(r) count the zeros and poles of R in |z|≤ r, with multiplicity. Then n_+(r)=Θ(r^3), n_-(r)=Θ(r^3), n_D(r):=n_+(r)+n_-(r)=Θ(r^3). In particular, for K_0=(|τ|+|s|+2)/|β|, n_+(K_0n)≥4n^3, n_-(K_0n)≥4n^3. These are unsigned counts…
Exact mathematical passage
n_+(r)=\Theta(r^3),\qquad n_-(r)=\Theta(r^3),\qquad n_D(r):=n_+(r)+n_-(r)=\Theta(r^3).
\thetaUse in source · 13 passages
Search aliases: θ theta
Use in source · H12_Cubic_Divisor_Traditional.tex, line 72
For positive imaginary periods use the additive product convention θ_0(z;a) · =∏_n≥0(1-e^2π i(z+na))(1-e^2π i((n+1)a-z)), Γ_rm ell(z;a,b) · =∏_j,k≥0 frac1-e^2π i((j+1)a+(k+1)b-z)1-e^2π i(ja+kb+z). The negative-first-period continuation is Γ_rm ell(z;-t,s)=Γ_rm ell(z+t;t,s)^-1. Consequently the main scalar function is F(z)=fracΓ_rm ell(z;τ,s)^5Γ_rm ell(5z;5τ,5s) =fracΓ_rm ell(5(z+t);5t,5s)Γ_rm ell(z+t;t,s)^5, U_Γ=F(1/3). This convention agrees with the checked primary authors' copy of Bergeron--C…
Exact mathematical passage
\theta_0(z;a)&=\prod_{n\ge0}(1-e^{2\pi i(z+na)})(1-e^{2\pi i((n+1)a-z)}),\label{eq:theta-product}\\ \Gamma_{\rm ell}(z;a,b)&=\prod_{j,k\ge0} \frac{1-e^{2\pi i((j+1)a+(k+1)b-z)}}{1-e^{2\pi i(ja+kb+z)}}.\label{eq:gamma-product}
\theta^2Use in source · 1 passage
Search aliases: θ^2 theta
Use in source · stark.tex, line 106
r:β↦ωβ, c:ω↦ω^2, s_0:θ↦θ^2-2,
Exact mathematical passage
r:\beta\mapsto\omega\beta,\qquad c:\omega\mapsto\omega^2,\qquad s_0:\theta\mapsto\theta^2-2,
\theta_0Use in source · 8 passages
Search aliases: θ_0 theta
Use in source · H12_Cubic_Divisor_Traditional.tex, line 72
For positive imaginary periods use the additive product convention θ_0(z;a) · =∏_n≥0(1-e^2π i(z+na))(1-e^2π i((n+1)a-z)), Γ_rm ell(z;a,b) · =∏_j,k≥0 frac1-e^2π i((j+1)a+(k+1)b-z)1-e^2π i(ja+kb+z). The negative-first-period continuation is Γ_rm ell(z;-t,s)=Γ_rm ell(z+t;t,s)^-1. Consequently the main scalar function is F(z)=fracΓ_rm ell(z;τ,s)^5Γ_rm ell(5z;5τ,5s) =fracΓ_rm ell(5(z+t);5t,5s)Γ_rm ell(z+t;t,s)^5, U_Γ=F(1/3). This convention agrees with the checked primary authors' copy of Bergeron--C…
Exact mathematical passage
\theta_0(z;a)&=\prod_{n\ge0}(1-e^{2\pi i(z+na)})(1-e^{2\pi i((n+1)a-z)}),\label{eq:theta-product}\\ \Gamma_{\rm ell}(z;a,b)&=\prod_{j,k\ge0} \frac{1-e^{2\pi i((j+1)a+(k+1)b-z)}}{1-e^{2\pi i(ja+kb+z)}}.\label{eq:gamma-product}
\theta_1Use in source · 1 passage
Search aliases: θ_1 theta
Use in source · stark.tex, line 677
H_τ(z)=e^-π(Im z)^2/y |fracθ_1(π z;e^π iτ)η_rm Ded(τ)|.
Exact mathematical passage
H_\tau(z)=e^{-\pi(\Im z)^2/y} \left|\frac{\theta_1(\pi z;e^{\pi i\tau})}{\eta_{\rm Ded}(\tau)}\right|.
\iotaUse in source · 11 passages
Search aliases: ι iota
Use in source · polarization.tex, line 61
ι(β)=β, ι(ω)=ω^2, ι(ξ)=β/ξ,
Exact mathematical passage
\iota(\beta)=\beta,\quad \iota(\omega)=\omega^2,\quad \iota(\xi)=\beta/\xi,
\LambdaUse in source · 24 passages
Search aliases: Λ Lambda
Use in source · H12_Cubic_Divisor_Traditional.tex, line 97
q_s(z) · =q_t(ρ z), · F(0) · =ϵ^2, · F(1/3)F(2/3) · =1. The two elliptic lattices are homothetic, ρΛ_s=Λ_t, with Λ_a= Z+ Za. This is not an endomorphism of the single elliptic lattice Λ_t. Section~[source reference] proves the exact rational fifth-torsion normalization and explains the covariance without omitting its scalar.
Exact mathematical passage
\rho\Lambda_s=\Lambda_t
\lambdaUse in source · 35 passages
Search aliases: λ lambda
Use in source · principal.tex, line 70
0 · -5 · 4. Its quotient generator is (1,2,20)/7, representing δ. Thus β^-1Λ_5=Λ_5+ Zδ. Write a potential singular point as λ(a,b,c)=fracaτ+bs+c/5β, p=lceilfrac3c-2b7rceil, k=7p+2b-3c∈{0,…,6}. Exactly one argument λ+kδ lies in Λ_5. Its integral coordinates are (widehat a,widehat b,widehat c)=(p,2p-a,20p+5b-8c). Indeed N_5(widehat a,widehat b,widehat c)^T=(a,b-8k,c-5k)^T, which expresses β(λ+kδ)=βλ+k(-8s-1). The other six digits fail integrality. Also widehat c-2c=5(4p+b-2c), w(widehat c)=w(c). T…
Exact mathematical passage
\label{eq:lambda} \lambda(a,b,c)=\frac{a\tau+bs+c/5}{\beta},\qquad p=\left\lceil\frac{3c-2b}{7}\right\rceil,\qquad k=7p+2b-3c\in\{0,\ldots,6\}.
\lambda^2Use in source · 8 passages
Search aliases: λ^2 lambda
Use in source · reconstruction.tex, line 166
The period-one transition fixes the cubic coefficient. The points 0,1,1/3 are regular and nonzero for R and W_D. For 1, [source reference] has nonzero factors at zero. For 1/3, β/3=-5s-2/3, whose rational coordinate is not a rational fifth; rational independence excludes a singular lattice point. Define B_D(1)=(log W_D)''(1) =∑_λ∈ Dm_λ[-frac1(λ-1)^2 +frac1λ^2+frac2λ^3]. The combined summand is O(|λ|^-4), so this is an absolutely convergent sum. Its terms must remain combined; the separately dive…
Exact mathematical passage
\label{eq:BD} B_D(1)=(\log W_D)''(1) =\sum_{\lambda\in D}m_\lambda\left[-\frac1{(\lambda-1)^2} +\frac1{\lambda^2}+\frac2{\lambda^3}\right].
\lambda^3Use in source · 4 passages
Search aliases: λ^3 lambda
Use in source · reconstruction.tex, line 88
The lower bound n_D(r)≥ c r^3 yields divergence of ∑_D|m_λ|/|λ|^3: partial summation contains 3∫ n_D(r)r^-4 dr, which diverges logarithmically. For a fixed z≠0, log E_2(z/λ)=-fracz^33λ^3+O(|λ|^-4). Its weighted absolute logarithm sum diverges, because the fourth-power remainder is summable. This proves failure of ungrouped absolute genus two, without a claim about cancellations in specially grouped signed products. [Explicit analytic tail] If n_D(r)=∑_|λ|≤ r|m_λ|≤ Ar^3 for all r≥ R_*, then for T…
Exact mathematical passage
\log E_2(z/\lambda)=-\frac{z^3}{3\lambda^3}+O(|\lambda|^{-4}).
\lambda^4Use in source · 1 passage
Search aliases: λ^4 lambda
Use in source · theta.tex, line 334
g_2=λ^4(b_2^2/12-2b_4), g_3=λ^6(-b_2^3/216+b_2b_4/6-u^2).
Exact mathematical passage
g_2=\lambda^4(b_2^2/12-2b_4),\quad g_3=\lambda^6(-b_2^3/216+b_2b_4/6-u^2).
\lambda^6Use in source · 1 passage
Search aliases: λ^6 lambda
Use in source · theta.tex, line 334
g_2=λ^4(b_2^2/12-2b_4), g_3=λ^6(-b_2^3/216+b_2b_4/6-u^2).
Exact mathematical passage
g_2=\lambda^4(b_2^2/12-2b_4),\quad g_3=\lambda^6(-b_2^3/216+b_2b_4/6-u^2).
\Lambda_0Use in source · 8 passages
Search aliases: Λ_0 Lambda
Use in source · stark.tex, line 60
X=fracΩ^2wp_Λ_0(1/J), Λ_0= Z+ Zω, Ω=e^π i/6fracΓ_rm Euler(1/3)^32π. tagA1.3
Exact mathematical passage
X=\frac{\Omega^2}{\wp_{\Lambda_0}(1/J)},\qquad \Lambda_0=\mathbb Z+\mathbb Z\omega,\qquad \Omega=e^{\pi i/6}\frac{\Gamma_{\rm Euler}(1/3)^3}{2\pi}. \tag{A1.3}
\Lambda_5Use in source · 6 passages
Search aliases: Λ_5 Lambda
Use in source · principal.tex, line 12
The seven-point product. Use the fixed periods of Section~[source reference]. Define S_5(z;a,b)=fracΓ_rm ell(z;a,b)^5 Γ_rm ell(5z;5a,5b), F(z)=S_5(z+t;t,s)^-1. All identities below are meromorphic identities; their quotients at apparent exceptional points mean meromorphic continuation. Let Λ_5= Zτ+ Zs+tfrac15 Z, δ=fracτ+2s+47. The period identities, obtained from β^3=7, are βτ · =2τ+4s+1, · β s · =-τ, · β · =-15s-2,
Exact mathematical passage
\Lambda_5=\mathbb Z\tau+\mathbb Zs+\tfrac15\mathbb Z, \qquad \delta=\frac{\tau+2s+4}{7}.
\Lambda_aUse in source · 1 passage
Search aliases: Λ_a Lambda
Use in source · H12_Cubic_Divisor_Traditional.tex, line 97
q_s(z) · =q_t(ρ z), · F(0) · =ϵ^2, · F(1/3)F(2/3) · =1. The two elliptic lattices are homothetic, ρΛ_s=Λ_t, with Λ_a= Z+ Za. This is not an endomorphism of the single elliptic lattice Λ_t. Section~[source reference] proves the exact rational fifth-torsion normalization and explains the covariance without omitting its scalar.
Exact mathematical passage
\Lambda_a=\mathbb Z+\mathbb Za
\Lambda_sUse in source · 2 passages
Search aliases: Λ_s Lambda
Use in source · H12_Cubic_Divisor_Traditional.tex, line 97
q_s(z) · =q_t(ρ z), · F(0) · =ϵ^2, · F(1/3)F(2/3) · =1. The two elliptic lattices are homothetic, ρΛ_s=Λ_t, with Λ_a= Z+ Za. This is not an endomorphism of the single elliptic lattice Λ_t. Section~[source reference] proves the exact rational fifth-torsion normalization and explains the covariance without omitting its scalar.
Exact mathematical passage
\rho\Lambda_s=\Lambda_t
\Lambda_tUse in source · 7 passages
Search aliases: Λ_t Lambda
Use in source · H12_Cubic_Divisor_Traditional.tex, line 97
q_s(z) · =q_t(ρ z), · F(0) · =ϵ^2, · F(1/3)F(2/3) · =1. The two elliptic lattices are homothetic, ρΛ_s=Λ_t, with Λ_a= Z+ Za. This is not an endomorphism of the single elliptic lattice Λ_t. Section~[source reference] proves the exact rational fifth-torsion normalization and explains the covariance without omitting its scalar.
Exact mathematical passage
\rho\Lambda_s=\Lambda_t
\Lambda_t^+Use in source · 1 passage
Search aliases: Λ_t^+ Lambda
Use in source · norms.tex, line 56
The fifth-isogeny differential retains its constant. Let a=wp_t(1/5), b=wp_t(2/5) and L_t=(q_t'/q_t)'. The enlarged lattice is Λ_t^+=tfrac15 Z+ Zt, with wp_+(z)=25wp_5t(5z). The degree-five quotient satisfies L_t(z)=wp_+(z)-5wp_t(z)+2(a+b). At zero L_t has principal part -4/z^2; at the four nonzero kernel points it has +1/(z-P_j)^2, with no simple-pole terms. Thus L_t-wp_++5wp_t is a constant elliptic function. The canonical formula gives q_t(z)=(a-b)^2z^4[1+(a+b)z^2+O(z^4)], fixing that constan…
Exact mathematical passage
\Lambda_t^+=\tfrac15\mathbb Z+\mathbb Zt
\muUse in source · 2 passages
Search aliases: μ mu
Use in source · polarization.tex, line 285
The right side has modulus one but is not in μ_6: the rational quotient calculation proves its sixth power differs from one. Its value is approximately
Exact mathematical passage
\mu_6
\mu_6Use in source · 2 passages
Search aliases: μ_6 mu
Use in source · polarization.tex, line 285
The right side has modulus one but is not in μ_6: the rational quotient calculation proves its sixth power differs from one. Its value is approximately
Exact mathematical passage
\mu_6
\nuUse in source · 2 passages
Search aliases: ν nu
Use in source · H12_Cubic_Divisor_Traditional.tex, line 108
The arithmetic candidate is fixed independently. Let J=2+3ω and choose the Gauss branch X=β^-1∑_a=1^6χ_7(a)e^6π ia/7, (χ_7(1),…,χ_7(6))=(1,ω,ω^2,ω^2,ω,1), X^3=J. Then E= Q(ω,β,X) has degree eighteen. The retained arithmetic construction defines ν · =(X+β/X-β^2)/3, · v · =-β^2ν-2ν^2, Y · =(4β^2+β-14)v^2+(11β-23)v-β^2+3β-3, Z · =frac(-2β^2+7β-8)v^2+(-10β^2+11β+17)v3 +7-β-β^2, U_rm alg · =fracY-(1+2ω)Z2. These formulas choose no coefficient or branch by fitting U_Γ. The original desired identity is…
Exact mathematical passage
\nu&=(X+\beta/X-\beta^2)/3,&v&=-\beta^2\nu-2\nu^2,\\ Y&=(4\beta^2+\beta-14)v^2+(11\beta-23)v-\beta^2+3\beta-3,\\ Z&=\frac{(-2\beta^2+7\beta-8)v^2+(-10\beta^2+11\beta+17)v}{3} +7-\beta-\beta^2,\\ U_{\rm alg}&=\frac{Y-(1+2\omega)Z}{2}.
\nu^2Use in source · 2 passages
Search aliases: ν^2 nu
Use in source · H12_Cubic_Divisor_Traditional.tex, line 108
The arithmetic candidate is fixed independently. Let J=2+3ω and choose the Gauss branch X=β^-1∑_a=1^6χ_7(a)e^6π ia/7, (χ_7(1),…,χ_7(6))=(1,ω,ω^2,ω^2,ω,1), X^3=J. Then E= Q(ω,β,X) has degree eighteen. The retained arithmetic construction defines ν · =(X+β/X-β^2)/3, · v · =-β^2ν-2ν^2, Y · =(4β^2+β-14)v^2+(11β-23)v-β^2+3β-3, Z · =frac(-2β^2+7β-8)v^2+(-10β^2+11β+17)v3 +7-β-β^2, U_rm alg · =fracY-(1+2ω)Z2. These formulas choose no coefficient or branch by fitting U_Γ. The original desired identity is…
Exact mathematical passage
\nu&=(X+\beta/X-\beta^2)/3,&v&=-\beta^2\nu-2\nu^2,\\ Y&=(4\beta^2+\beta-14)v^2+(11\beta-23)v-\beta^2+3\beta-3,\\ Z&=\frac{(-2\beta^2+7\beta-8)v^2+(-10\beta^2+11\beta+17)v}{3} +7-\beta-\beta^2,\\ U_{\rm alg}&=\frac{Y-(1+2\omega)Z}{2}.
\xiUse in source · 14 passages
Search aliases: ξ xi
Use in source · polarization.tex, line 10
ω^2+ω+1=0, J=2+3ω, β^3=7, ξ^3=J.
Exact mathematical passage
\omega^2+\omega+1=0,\quad J=2+3\omega,\quad \beta^3=7,\quad \xi^3=J.
\xi^2Use in source · 3 passages
Search aliases: ξ^2 xi
Use in source · polarization.tex, line 37
a_0=(1+ω)ξ^2-ξ+1, a_1=-ξ^2+(1+ω)ξ+1, a_2=-ωξ^2-ωξ+1.
Exact mathematical passage
a_0=(1+\omega)\xi^2-\xi+1,\quad a_1=-\xi^2+(1+\omega)\xi+1,\quad a_2=-\omega\xi^2-\omega\xi+1.
\xi^3Use in source · 2 passages
Search aliases: ξ^3 xi
Use in source · polarization.tex, line 10
ω^2+ω+1=0, J=2+3ω, β^3=7, ξ^3=J.
Exact mathematical passage
\omega^2+\omega+1=0,\quad J=2+3\omega,\quad \beta^3=7,\quad \xi^3=J.
\rhoUse in source · 15 passages
Search aliases: ρ rho
Use in source · H12_Cubic_Divisor_Traditional.tex, line 61
Fixed data, conventions, and the claim boundary. Fix throughout ω · =e^2π i/3, · β · =7^1/3ω^2, · ϵ · =2-β, · ρ · =ϵ^-1, τ · =(β^2+2β)/15, · t · =-τ, · s · =-(β+2)/15, · δ · =(τ+2s+4)/7. The real cube root 7^1/3 is positive; this fixes the lower complex embedding of β. We have Im t,Im s,Imδ>0, Imτ<0 and |ρ|<1. The last follows from N(ϵ)=1 and 0<2-7^1/3<1 at the real embedding. The numbers 1,t,s are rationally independent: a rational linear relation becomes a polynomial of degree at most two in β…
Exact mathematical passage
\label{eq:fixed-data} \begin{aligned} \omega&=e^{2\pi i/3},&\beta&=7^{1/3}\omega^2,&\epsilon&=2-\beta,&\rho&=\epsilon^{-1},\\ \tau&=(\beta^2+2\beta)/15,&t&=-\tau,&s&=-(\beta+2)/15,& \delta&=(\tau+2s+4)/7. \end{aligned}
\varrhoUse in source · 12 passages
Search aliases: ϱ varrho
Use in source · H12_Cubic_Divisor_Traditional.tex, line 99
We use F_0= Q(ω) for the imaginary-quadratic field, to separate it from the scalar gamma function F. The eta quotient is H(a)=η(5a)/η(a)^5; the integer step function H(n) used in the divisor formula is a different object. The symbol R_0 is the scalar anchor, while R_* is a counting cutoff radius. In the arithmetic block ϱ is a ray character and a radical lift, as explicitly indicated there; it is distinct from the contracting period multiplier ρ.
Exact mathematical passage
\varrho
\varrho^{-1}Use in source · 1 passage
Search aliases: ϱ^-1 varrho
Use in source · stark.tex, line 265
Conjugation of the base field by c sends ϱη^2 to ϱ^-1η^2=λ^2, so
Exact mathematical passage
\varrho^{-1}\eta^2=\lambda^2
\rho^2Use in source · 4 passages
Search aliases: ρ^2 rho
Use in source · reconstruction.tex, line 150
Let L_t=(q_t'/q_t)' and S_t=wp_t(1/5)+wp_t(2/5). The normalized theta formula gives L_t(z)=-4z^-2+2S_t+O(z^2), S_s=ρ^2S_t. Since 7δ-2s=-t+4, evenness and t-periodicity pair the six translated L_t terms. Using s=u_0+3δ-2, their sum is 2∑_j=0^2L_t(u_j). The second logarithmic derivative of [source reference] cancels all z^-2 terms, leaving (log T)''(0)=2(1-β^2)(1-ρ^2)S_t-2∑_j=0^2L_t(u_j). The left side is 4α_2. Therefore α_2=frac12[(1-β^2)(1-ρ^2)S_t- ∑_j=0^2L_t(u_j)]. The factor 1/2 is fixed by re…
Exact mathematical passage
L_t(z)=-4z^{-2}+2S_t+O(z^2),\qquad S_s=\rho^2S_t.
\rho^4Use in source · 1 passage
Search aliases: ρ^4 rho
Use in source · theta.tex, line 368
a_s=ρ^2a_t, b_s=ρ^2b_t, c_s=ρ^4c_t,
Exact mathematical passage
a_s=\rho^2a_t,\quad b_s=\rho^2b_t,\quad c_s=\rho^4c_t,
\varrho^6Use in source · 1 passage
Search aliases: ϱ^6 varrho
Use in source · stark.tex, line 561
For t^6=W, the lift ϱ(t)=t^5/U_rm alg commutes with t↦ c_6t, and satisfies ϱ^6(t)=c_6^4t. Irreducibility is checked at the residue assignment (β,v,ω)=(32,99,48) over F_181, where W=23 and T^6-23 is irreducible. The earlier local-integrality argument validates this reduction. The resulting abelian group is C_2× C_18.
Exact mathematical passage
\varrho^6(t)=c_6^4t
\rho^mUse in source · 1 passage
Search aliases: ρ^m rho
Use in source · rigidity.tex, line 13
The kernel on X has representatives j(1,2,4)^T/7, 0≤ j≤6. The conductor point p_3=e_2/3 obeys Mp_3-p_3=(0,5,1)^T and Np_3-p_3=(0,-5,-1)^T. The dual character e_k(v)=e^2π ik^Tv satisfies e_k(Mv)=e_Ak(v). Frequency transport is distinct from the scalar return sequence F(ρ^m/3).
Exact mathematical passage
F(\rho^m/3)
\sigmaUse in source · 18 passages
Search aliases: σ sigma
Use in source · stark.tex, line 69
The arithmetic Frobenius σ at the norm-two prime acts by
Exact mathematical passage
\sigma
\sigma^2Use in source · 1 passage
Search aliases: σ^2 sigma
Use in source · stark.tex, line 556
W^21=c_6U_rm alg^25σ(U_rm alg)^5σ^2(U_rm alg).
Exact mathematical passage
W^{21}=c_6U_{\rm alg}^{25}\sigma(U_{\rm alg})^5\sigma^2(U_{\rm alg}).
\sigma^3Use in source · 1 passage
Search aliases: σ^3 sigma
Use in source · stark.tex, line 89
For the last formula, the conjugate Gauss resolvent is 7/G, and barβ=ω^2β. Formula (A1.5) follows directly. It must not be replaced by the involution σ^3, which fixes K.
Exact mathematical passage
\sigma^3
\sigma^5Use in source · 2 passages
Search aliases: σ^5 sigma
Use in source · stark.tex, line 587
log|U_Γ|^2 =ζ_ f'(1,0)-5ζ_ f'(σ^5,0). tagA7.1
Exact mathematical passage
\log|U_\Gamma|^2 =\zeta_{\mathfrak f}'(1,0)-5\zeta_{\mathfrak f}'(\sigma^5,0). \tag{A7.1}
\sigma^hUse in source · 3 passages
Search aliases: σ^h sigma
Use in source · stark.tex, line 499
N_κ(σ^hW) =N_κ(ϵ a_h+1^2)^(-1)^h, 0≤ h<6. tagA6.2
Exact mathematical passage
\boxed{N_\kappa(\sigma^hW) =N_\kappa(\epsilon a_{h+1}^2)^{(-1)^h},\quad 0\le h<6.} \tag{A6.2}
\sigma^{h+5}Use in source · 1 passage
Search aliases: σ^h+5 sigma
Use in source · stark.tex, line 520
log|σ^h W|=(-1)^h(frac R2+2log|a_h+1|) =-3ζ_ f'(σ^h+5,0). tagA6.3
Exact mathematical passage
\log|\sigma^h W|=(-1)^h\left(\frac{\mathcal R}{2}+2\log|a_{h+1}|\right) =-3\zeta_{\mathfrak f}'(\sigma^{h+5},0). \tag{A6.3}
\sigma_{\mathfrak a}Use in source · 1 passage
Search aliases: σ_ a sigma mathfrak
Use in source · stark.tex, line 632
|Γ_ a,h( f b^-1)|^6 =|(σ_ bW_rm St)^N( a)-σ_ a|. tagA7.4
Exact mathematical passage
|\mathbf\Gamma_{\mathfrak a,h}(\mathfrak f\mathfrak b^{-1})|^6 =\left|(\sigma_{\mathfrak b}W_{\rm St})^{N(\mathfrak a)-\sigma_{\mathfrak a}}\right|. \tag{A7.4}
\sigma_{\mathfrak b}Use in source · 1 passage
Search aliases: σ_ b sigma mathfrak
Use in source · stark.tex, line 632
|Γ_ a,h( f b^-1)|^6 =|(σ_ bW_rm St)^N( a)-σ_ a|. tagA7.4
Exact mathematical passage
|\mathbf\Gamma_{\mathfrak a,h}(\mathfrak f\mathfrak b^{-1})|^6 =\left|(\sigma_{\mathfrak b}W_{\rm St})^{N(\mathfrak a)-\sigma_{\mathfrak a}}\right|. \tag{A7.4}
\sigma_\LambdaUse in source · 1 passage
Search aliases: σ_Λ sigma Lambda
Use in source · stark.tex, line 399
fracσ_Λ(T+z)σ_Λ(T-z) σ_Λ(T)^2σ_Λ(z)^2 =wp_Λ(z)-wp_Λ(T). tagA5.4
Exact mathematical passage
\frac{\sigma_\Lambda(T+z)\sigma_\Lambda(T-z)}{ \sigma_\Lambda(T)^2\sigma_\Lambda(z)^2} =\wp_\Lambda(z)-\wp_\Lambda(T). \tag{A5.4}
\tauUse in source · 35 passages
Search aliases: τ tau
Use in source · H12_Cubic_Divisor_Traditional.tex, line 61
Fixed data, conventions, and the claim boundary. Fix throughout ω · =e^2π i/3, · β · =7^1/3ω^2, · ϵ · =2-β, · ρ · =ϵ^-1, τ · =(β^2+2β)/15, · t · =-τ, · s · =-(β+2)/15, · δ · =(τ+2s+4)/7. The real cube root 7^1/3 is positive; this fixes the lower complex embedding of β. We have Im t,Im s,Imδ>0, Imτ<0 and |ρ|<1. The last follows from N(ϵ)=1 and 0<2-7^1/3<1 at the real embedding. The numbers 1,t,s are rationally independent: a rational linear relation becomes a polynomial of degree at most two in β…
Exact mathematical passage
\label{eq:fixed-data} \begin{aligned} \omega&=e^{2\pi i/3},&\beta&=7^{1/3}\omega^2,&\epsilon&=2-\beta,&\rho&=\epsilon^{-1},\\ \tau&=(\beta^2+2\beta)/15,&t&=-\tau,&s&=-(\beta+2)/15,& \delta&=(\tau+2s+4)/7. \end{aligned}
\phiUse in source · 2 passages
Search aliases: ϕ phi
Use in source · theta.tex, line 161
ϕ_2(X)=X^4+tfrac12g_2X^2+2g_3X+tfrac116g_2^2.
Exact mathematical passage
\phi_2(X)=X^4+\tfrac12g_2X^2+2g_3X+\tfrac1{16}g_2^2.
\phi_2Use in source · 2 passages
Search aliases: ϕ_2 phi
Use in source · theta.tex, line 161
ϕ_2(X)=X^4+tfrac12g_2X^2+2g_3X+tfrac116g_2^2.
Exact mathematical passage
\phi_2(X)=X^4+\tfrac12g_2X^2+2g_3X+\tfrac1{16}g_2^2.
\chiUse in source · 7 passages
Search aliases: χ chi
Use in source · H12_Cubic_Divisor_Traditional.tex, line 103
The arithmetic candidate is fixed independently. Let J=2+3ω and choose the Gauss branch X=β^-1∑_a=1^6χ_7(a)e^6π ia/7, (χ_7(1),…,χ_7(6))=(1,ω,ω^2,ω^2,ω,1), X^3=J. Then E= Q(ω,β,X) has degree eighteen. The retained arithmetic construction defines ν · =(X+β/X-β^2)/3, · v · =-β^2ν-2ν^2,
Exact mathematical passage
X=\beta^{-1}\sum_{a=1}^{6}\chi_7(a)e^{6\pi ia/7},\qquad (\chi_7(1),\ldots,\chi_7(6))=(1,\omega,\omega^2,\omega^2,\omega,1),\quad X^3=J.
\chi_{-3}Use in source · 3 passages
Search aliases: χ_-3 chi
Use in source · stark.tex, line 141
L_K(s,χ_j) =L_ Q(s,χ_-3χ_7^j) L_F_0(s,ϱη^j). tagA2.2
Exact mathematical passage
\boxed{L_K(s,\chi_j) =L_{\mathbb Q}(s,\chi_{-3}\chi_7^j)\, L_{F_0}(s,\varrho\eta^j).} \tag{A2.2}
\chi_7Use in source · 5 passages
Search aliases: χ_7 chi
Use in source · H12_Cubic_Divisor_Traditional.tex, line 103
The arithmetic candidate is fixed independently. Let J=2+3ω and choose the Gauss branch X=β^-1∑_a=1^6χ_7(a)e^6π ia/7, (χ_7(1),…,χ_7(6))=(1,ω,ω^2,ω^2,ω,1), X^3=J. Then E= Q(ω,β,X) has degree eighteen. The retained arithmetic construction defines ν · =(X+β/X-β^2)/3, · v · =-β^2ν-2ν^2,
Exact mathematical passage
X=\beta^{-1}\sum_{a=1}^{6}\chi_7(a)e^{6\pi ia/7},\qquad (\chi_7(1),\ldots,\chi_7(6))=(1,\omega,\omega^2,\omega^2,\omega,1),\quad X^3=J.
\chi_7^jUse in source · 1 passage
Search aliases: χ_7^j chi
Use in source · stark.tex, line 141
L_K(s,χ_j) =L_ Q(s,χ_-3χ_7^j) L_F_0(s,ϱη^j). tagA2.2
Exact mathematical passage
\boxed{L_K(s,\chi_j) =L_{\mathbb Q}(s,\chi_{-3}\chi_7^j)\, L_{F_0}(s,\varrho\eta^j).} \tag{A2.2}
\chi_jUse in source · 2 passages
Search aliases: χ_j chi
Use in source · stark.tex, line 126
χ_j(c)=-1, χ_j(s_0)=ω^j, j=0,1,2.
Exact mathematical passage
\chi_j(c)=-1,\qquad \chi_j(s_0)=\omega^j,\qquad j=0,1,2.
\psiUse in source · 12 passages
Search aliases: ψ psi
Use in source · norms.tex, line 17
The point S=sbmodΛ_t has infinite order by rational independence. Put ψ(w)=fracq_t(w)^2q_t(2w)q_t(2w+s). [All finite norms of the return kernel] For every integer n≥1, N_nψ(w)= 1, · 5mid n,
Exact mathematical passage
\psi(w)=\frac{q_t(w)^2}{q_t(2w)q_t(2w+s)}.
\psi_3Use in source · 7 passages
Search aliases: ψ_3 psi
Use in source · theta.tex, line 155
ψ_3(X)=3X^4-tfrac32g_2X^2-3g_3X-tfrac116g_2^2,
Exact mathematical passage
\psi_3(X)=3X^4-\tfrac32g_2X^2-3g_3X-\tfrac1{16}g_2^2,
\OmegaUse in source · 6 passages
Search aliases: Ω Omega
Use in source · reconstruction.tex, line 50
The precise finite-theta obstruction. [Fixed affine finite-theta class] Fix finitely many symmetric period matrices Ω_ℓ with positive-definite imaginary parts ImΩ_ℓ, affine maps z↦ A_ℓ z+b_ℓ, and finite derivatives of their Riemann theta restrictions. Let P,Q be nonzero entire functions formed by finite polynomial expressions in these restrictions, derivatives and polynomials in z. Let u be a single global nowhere-zero entire function. Every meromorphic function u(z)fracP(z)Q(z) has zero and pol…
Exact mathematical passage
\Omega_\ell
\omegaUse in source · 75 passages
Search aliases: ω omega
Use in source · H12_Cubic_Divisor_Traditional.tex, line 61
Fixed data, conventions, and the claim boundary. Fix throughout ω · =e^2π i/3, · β · =7^1/3ω^2, · ϵ · =2-β, · ρ · =ϵ^-1, τ · =(β^2+2β)/15, · t · =-τ, · s · =-(β+2)/15, · δ · =(τ+2s+4)/7. The real cube root 7^1/3 is positive; this fixes the lower complex embedding of β. We have Im t,Im s,Imδ>0, Imτ<0 and |ρ|<1. The last follows from N(ϵ)=1 and 0<2-7^1/3<1 at the real embedding. The numbers 1,t,s are rationally independent: a rational linear relation becomes a polynomial of degree at most two in β…
Exact mathematical passage
\label{eq:fixed-data} \begin{aligned} \omega&=e^{2\pi i/3},&\beta&=7^{1/3}\omega^2,&\epsilon&=2-\beta,&\rho&=\epsilon^{-1},\\ \tau&=(\beta^2+2\beta)/15,&t&=-\tau,&s&=-(\beta+2)/15,& \delta&=(\tau+2s+4)/7. \end{aligned}
\omega^{1-k}Use in source · 1 passage
Search aliases: ω^1-k omega
Use in source · stark.tex, line 282
-frac13(ω^1-kL_F_0'(0,λ) +ω^2-2kL_F_0'(0,λ^2)).
Exact mathematical passage
-\frac13\left(\omega^{1-k}L_{F_0}'(0,\lambda) +\omega^{2-2k}L_{F_0}'(0,\lambda^2)\right).
\Omega^2Use in source · 3 passages
Search aliases: Ω^2 Omega
Use in source · stark.tex, line 60
X=fracΩ^2wp_Λ_0(1/J), Λ_0= Z+ Zω, Ω=e^π i/6fracΓ_rm Euler(1/3)^32π. tagA1.3
Exact mathematical passage
X=\frac{\Omega^2}{\wp_{\Lambda_0}(1/J)},\qquad \Lambda_0=\mathbb Z+\mathbb Z\omega,\qquad \Omega=e^{\pi i/6}\frac{\Gamma_{\rm Euler}(1/3)^3}{2\pi}. \tag{A1.3}
\omega^2Use in source · 24 passages
Search aliases: ω^2 omega
Use in source · H12_Cubic_Divisor_Traditional.tex, line 61
Fixed data, conventions, and the claim boundary. Fix throughout ω · =e^2π i/3, · β · =7^1/3ω^2, · ϵ · =2-β, · ρ · =ϵ^-1, τ · =(β^2+2β)/15, · t · =-τ, · s · =-(β+2)/15, · δ · =(τ+2s+4)/7. The real cube root 7^1/3 is positive; this fixes the lower complex embedding of β. We have Im t,Im s,Imδ>0, Imτ<0 and |ρ|<1. The last follows from N(ϵ)=1 and 0<2-7^1/3<1 at the real embedding. The numbers 1,t,s are rationally independent: a rational linear relation becomes a polynomial of degree at most two in β…
Exact mathematical passage
\label{eq:fixed-data} \begin{aligned} \omega&=e^{2\pi i/3},&\beta&=7^{1/3}\omega^2,&\epsilon&=2-\beta,&\rho&=\epsilon^{-1},\\ \tau&=(\beta^2+2\beta)/15,&t&=-\tau,&s&=-(\beta+2)/15,& \delta&=(\tau+2s+4)/7. \end{aligned}
\omega^{2-2k}Use in source · 1 passage
Search aliases: ω^2-2k omega
Use in source · stark.tex, line 282
-frac13(ω^1-kL_F_0'(0,λ) +ω^2-2kL_F_0'(0,λ^2)).
Exact mathematical passage
-\frac13\left(\omega^{1-k}L_{F_0}'(0,\lambda) +\omega^{2-2k}L_{F_0}'(0,\lambda^2)\right).
\omega^iUse in source · 1 passage
Search aliases: ω^i omega
Use in source · polarization.tex, line 89
β↦ω^iβ, ξ↦ω^jξ.
Exact mathematical passage
\beta\mapsto\omega^i\beta,\qquad \xi\mapsto\omega^j\xi.
\omega^jUse in source · 3 passages
Search aliases: ω^j omega
Use in source · polarization.tex, line 51
ϵ_j=2-ω^jβ, ϵ=ϵ_0, ϵ_0ϵ_1ϵ_2=1.
Exact mathematical passage
\epsilon_j=2-\omega^j\beta,\qquad \epsilon=\epsilon_0, \qquad \epsilon_0\epsilon_1\epsilon_2=1.
\omega^kUse in source · 1 passage
Search aliases: ω^k omega
Use in source · stark.tex, line 512
A basis for the rational calculation is β^iX^jω^k, 0≤ i,j<3, 0≤ k<2. For odd h, multiply the two norms instead of dividing: the residual is their product minus one. Both supplied implementations obtain the exact zero polynomial in all six cases.
Exact mathematical passage
\beta^iX^j\omega^k
\Omega_\ellUse in source · 2 passages
Search aliases: Ω_ℓ Omega ell
Use in source · reconstruction.tex, line 50
The precise finite-theta obstruction. [Fixed affine finite-theta class] Fix finitely many symmetric period matrices Ω_ℓ with positive-definite imaginary parts ImΩ_ℓ, affine maps z↦ A_ℓ z+b_ℓ, and finite derivatives of their Riemann theta restrictions. Let P,Q be nonzero entire functions formed by finite polynomial expressions in these restrictions, derivatives and polynomials in z. Let u be a single global nowhere-zero entire function. Every meromorphic function u(z)fracP(z)Q(z) has zero and pol…
Exact mathematical passage
\Omega_\ell
\ldotsUse in source · 4 passages
Search aliases: … ldots
Use in source · H12_Cubic_Divisor_Traditional.tex, line 103
The arithmetic candidate is fixed independently. Let J=2+3ω and choose the Gauss branch X=β^-1∑_a=1^6χ_7(a)e^6π ia/7, (χ_7(1),…,χ_7(6))=(1,ω,ω^2,ω^2,ω,1), X^3=J. Then E= Q(ω,β,X) has degree eighteen. The retained arithmetic construction defines ν · =(X+β/X-β^2)/3, · v · =-β^2ν-2ν^2,
Exact mathematical passage
X=\beta^{-1}\sum_{a=1}^{6}\chi_7(a)e^{6\pi ia/7},\qquad (\chi_7(1),\ldots,\chi_7(6))=(1,\omega,\omega^2,\omega^2,\omega,1),\quad X^3=J.
\ellUse in source · 21 passages
Search aliases: ℓ ell
Use in source · reconstruction.tex, line 50
The precise finite-theta obstruction. [Fixed affine finite-theta class] Fix finitely many symmetric period matrices Ω_ℓ with positive-definite imaginary parts ImΩ_ℓ, affine maps z↦ A_ℓ z+b_ℓ, and finite derivatives of their Riemann theta restrictions. Let P,Q be nonzero entire functions formed by finite polynomial expressions in these restrictions, derivatives and polynomials in z. Let u be a single global nowhere-zero entire function. Every meromorphic function u(z)fracP(z)Q(z) has zero and pol…
Exact mathematical passage
\Omega_\ell
\ell^1Use in source · 1 passage
Search aliases: ℓ^1 ell
Use in source · rigidity.tex, line 57
l of X, so does g. Apply L_N/7 to [source reference]. The trace relation gives L_Nh-h=(1/7)δ_M L_Nr, and induction yields L_N^nh-h=frac17δ_M∑_j=1^nL_N^jr. Because B^3=7I, L_N^3nf=∑_k≠0343^nc_7^nke_k for a mean-zero f. For any C^q norm choose s>max(q,3). Smooth Fourier coefficients have finite weighted ℓ^1 norms of every order, and the displayed expression is bounded, up to a derivative convention constant, by 7^n(3-s)∑_k≠0 k ^s|c_k|. It tends geometrically to zero and is summable. The other two …
Exact mathematical passage
\ell^1
\ell^2Use in source · 2 passages
Search aliases: ℓ^2 ell
Use in source · theta.tex, line 128
ℓ^2=4(2a+b), m^2=4(a+2b),
Exact mathematical passage
\ell^2=4(2a+b),\quad m^2=4(a+2b),
\ell_0Use in source · 2 passages
Search aliases: ℓ_0 ell
Use in source · stark.tex, line 246
L_F_0'(0,λ)=ℓ_0+ωℓ_1+ω^2ℓ_2, L_F_0'(0,λ^2)=ℓ_0+ω^2ℓ_1+ωℓ_2. tagA3.3
Exact mathematical passage
L_{F_0}'(0,\lambda)=\ell_0+\omega\ell_1+\omega^2\ell_2, \qquad L_{F_0}'(0,\lambda^2)=\ell_0+\omega^2\ell_1+\omega\ell_2. \tag{A3.3}
\ell_1Use in source · 3 passages
Search aliases: ℓ_1 ell
Use in source · stark.tex, line 246
L_F_0'(0,λ)=ℓ_0+ωℓ_1+ω^2ℓ_2, L_F_0'(0,λ^2)=ℓ_0+ω^2ℓ_1+ωℓ_2. tagA3.3
Exact mathematical passage
L_{F_0}'(0,\lambda)=\ell_0+\omega\ell_1+\omega^2\ell_2, \qquad L_{F_0}'(0,\lambda^2)=\ell_0+\omega^2\ell_1+\omega\ell_2. \tag{A3.3}
\ell_2Use in source · 3 passages
Search aliases: ℓ_2 ell
Use in source · stark.tex, line 246
L_F_0'(0,λ)=ℓ_0+ωℓ_1+ω^2ℓ_2, L_F_0'(0,λ^2)=ℓ_0+ω^2ℓ_1+ωℓ_2. tagA3.3
Exact mathematical passage
L_{F_0}'(0,\lambda)=\ell_0+\omega\ell_1+\omega^2\ell_2, \qquad L_{F_0}'(0,\lambda^2)=\ell_0+\omega^2\ell_1+\omega\ell_2. \tag{A3.3}
\ell_{k-1}Use in source · 3 passages
Search aliases: ℓ_k-1 ell
Use in source · stark.tex, line 274
ζ_ f'(c^e s_0^k,0) =(-1)^e(frac R6-ℓ_k-1). tagA3.5
Exact mathematical passage
\boxed{\zeta_{\mathfrak f}'(c^e s_0^k,0) =(-1)^e\left(\frac{\mathcal R}{6}-\ell_{k-1}\right).} \tag{A3.5}
\ell_mUse in source · 3 passages
Search aliases: ℓ_m ell
Use in source · stark.tex, line 244
Let Z_m(s) be the sum of the two ray partial zeta functions in row m, and write ℓ_m=Z_m'(0). Then
Exact mathematical passage
\ell_m=Z_m'(0)
\toUse in source · 1 passage
Search aliases: → to
Use in source · theta.tex, line 191
Let z→0 in (N1.1). The fourth-order zero gives
Exact mathematical passage
z\to0
\mapstoUse in source · 8 passages
Search aliases: ↦ mapsto
Use in source · polarization.tex, line 89
β↦ω^iβ, ξ↦ω^jξ.
Exact mathematical passage
\beta\mapsto\omega^i\beta,\qquad \xi\mapsto\omega^j\xi.
\notinUse in source · 1 passage
Search aliases: ∉ notin
Use in source · stark.tex, line 302
For z=aω+b∉Λ_0, define the positive quantity
Exact mathematical passage
z=a\omega+b\notin\Lambda_0
\prodUse in source · 22 passages
Search aliases: ∏ prod
Use in source · H12_Cubic_Divisor_Traditional.tex, line 72
For positive imaginary periods use the additive product convention θ_0(z;a) · =∏_n≥0(1-e^2π i(z+na))(1-e^2π i((n+1)a-z)), Γ_rm ell(z;a,b) · =∏_j,k≥0 frac1-e^2π i((j+1)a+(k+1)b-z)1-e^2π i(ja+kb+z). The negative-first-period continuation is Γ_rm ell(z;-t,s)=Γ_rm ell(z+t;t,s)^-1. Consequently the main scalar function is F(z)=fracΓ_rm ell(z;τ,s)^5Γ_rm ell(5z;5τ,5s) =fracΓ_rm ell(5(z+t);5t,5s)Γ_rm ell(z+t;t,s)^5, U_Γ=F(1/3). This convention agrees with the checked primary authors' copy of Bergeron--C…
Exact mathematical passage
\theta_0(z;a)&=\prod_{n\ge0}(1-e^{2\pi i(z+na)})(1-e^{2\pi i((n+1)a-z)}),\label{eq:theta-product}\\ \Gamma_{\rm ell}(z;a,b)&=\prod_{j,k\ge0} \frac{1-e^{2\pi i((j+1)a+(k+1)b-z)}}{1-e^{2\pi i(ja+kb+z)}}.\label{eq:gamma-product}
\prod_{a=0}^{n-1}Use in source · 2 passages
Search aliases: ∏_a=0^n-1 prod
Use in source · norms.tex, line 10
1, · 5mid n. In the prime-to-five case the two product identities are ∏_a=0^n-1θ_0(z+a/n;t)=θ_0(nz;nt), ∏_b=0^n-1θ_0(nz+bt;nt)=θ_0(nz;t). Multiplication by five permutes the real residues in the smoothing denominator. When 5mid n, real averaging retains only frequencies already deleted by fifth smoothing. This proves the scalar one in both cases, rather than a norm formula up to a root of unity.
Exact mathematical passage
\prod_{a=0}^{n-1}\theta_0(z+a/n;t)=\theta_0(nz;nt),\qquad \prod_{b=0}^{n-1}\theta_0(nz+bt;nt)=\theta_0(nz;t).
\prod_{b=0}^{n-1}Use in source · 2 passages
Search aliases: ∏_b=0^n-1 prod
Use in source · norms.tex, line 10
1, · 5mid n. In the prime-to-five case the two product identities are ∏_a=0^n-1θ_0(z+a/n;t)=θ_0(nz;nt), ∏_b=0^n-1θ_0(nz+bt;nt)=θ_0(nz;t). Multiplication by five permutes the real residues in the smoothing denominator. When 5mid n, real averaging retains only frequencies already deleted by fifth smoothing. This proves the scalar one in both cases, rather than a norm formula up to a root of unity.
Exact mathematical passage
\prod_{a=0}^{n-1}\theta_0(z+a/n;t)=\theta_0(nz;nt),\qquad \prod_{b=0}^{n-1}\theta_0(nz+bt;nt)=\theta_0(nz;t).
\prod_{H\text{ a line}}Use in source · 1 passage
Search aliases: ∏_H a line prod text
Use in source · polarization.tex, line 103
∏_H a lineN_E/E^H(z) =z^3N_E/F_0(z). tagQ2
Exact mathematical passage
\boxed{ \prod_{H\text{ a line}}N_{E/E^H}(z) =z^3N_{E/F_0}(z). } \tag{Q2}
\prod_{j,k\ge0}Use in source · 1 passage
Search aliases: ∏_j,k≥0 prod ge
Use in source · H12_Cubic_Divisor_Traditional.tex, line 72
For positive imaginary periods use the additive product convention θ_0(z;a) · =∏_n≥0(1-e^2π i(z+na))(1-e^2π i((n+1)a-z)), Γ_rm ell(z;a,b) · =∏_j,k≥0 frac1-e^2π i((j+1)a+(k+1)b-z)1-e^2π i(ja+kb+z). The negative-first-period continuation is Γ_rm ell(z;-t,s)=Γ_rm ell(z+t;t,s)^-1. Consequently the main scalar function is F(z)=fracΓ_rm ell(z;τ,s)^5Γ_rm ell(5z;5τ,5s) =fracΓ_rm ell(5(z+t);5t,5s)Γ_rm ell(z+t;t,s)^5, U_Γ=F(1/3). This convention agrees with the checked primary authors' copy of Bergeron--C…
Exact mathematical passage
\theta_0(z;a)&=\prod_{n\ge0}(1-e^{2\pi i(z+na)})(1-e^{2\pi i((n+1)a-z)}),\label{eq:theta-product}\\ \Gamma_{\rm ell}(z;a,b)&=\prod_{j,k\ge0} \frac{1-e^{2\pi i((j+1)a+(k+1)b-z)}}{1-e^{2\pi i(ja+kb+z)}}.\label{eq:gamma-product}
\prod_{j=0}^{2}Use in source · 1 passage
Search aliases: ∏_j=0^2 prod
Use in source · principal.tex, line 128
[Three-theta anchor] R is regular and nonzero at zero, and R_0:=R(0)=∏_j=0^2q_t(u_j)^-1, u_j=fract+δ2+jδ. Let H(a)=η(5a)/η(a)^5. Direct product cancellation gives S_5(z;a,b)∼frac5(2π)^4 H(a) H(b)z^-4. The ratio in [source reference] therefore has value P_7(0)=[ H(s)/ H(t)]/S_5(s;t,δ). Since F(0)= H(s)/ H(t)=ϵ^2, the eta factors cancel and R_0=S_5(s;t,δ)^-1. The period relation is s=(t+δ)/2+3δ-2. At the central point, S_5((t+δ)/2;t,δ)=1 by termwise coincidence of the gamma numerator and denominat…
Exact mathematical passage
\label{eq:R0} R_0:=R(0)=\prod_{j=0}^{2}q_t(u_j)^{-1},\qquad u_j=\frac{t+\delta}{2}+j\delta.
\prod_{j=0}^{A_v-1}Use in source · 2 passages
Search aliases: ∏_j=0^A_v-1 prod
Use in source · theta.tex, line 397
F(ρ w_v)=F(w_Tv)B_v, B_v=∏_j=0^A_v-1q_t(w_T^2v+js), A_v=5+4b-2c+7a.
Exact mathematical passage
F(\rho w_v)=F(w_{Tv})B_v,\qquad B_v=\prod_{j=0}^{A_v-1}q_t(w_{T^2v}+js), \quad A_v=5+4b-2c+7a.
\prod_{j=0}^{k-1}Use in source · 2 passages
Search aliases: ∏_j=0^k-1 prod
Use in source · norms.tex, line 34
ks)^4, · 5nmid n, n=2k. For odd n, doubling permutes E_t[n]. For n=2k, it maps E_t[n] to E_t[k] with four lifts per point, so each denominator norm is a fourth power. Apply [source reference] and cancel the numerator. If 5mid n then all these norms are one, including the even case since 5mid k. For n=2k+1 prime to five, B_k(w)=∏_j=0^k-1q_t(2w+(2j+1)s) gives N_2k+1ψ(w)=ψ(w)fracB_k(w)B_k(w+s). For n=2k prime to five, C_k(w)=∏_j=0^k-1q_t(w+js)^4 gives N_2kψ(w)=q_t(w)^-6fracC_k(w)C_k(w+s). Finite pr…
Exact mathematical passage
B_k(w)=\prod_{j=0}^{k-1}q_t(2w+(2j+1)s)
\prod_{k=0}^{6}Use in source · 2 passages
Search aliases: ∏_k=0^6 prod
Use in source · principal.tex, line 24
βδ · =-8s-1, · 7δ · =τ+2s+4. The seven-point product and its correction are P_7(z)=∏_k=0^6F(z+kδ) =fracS_5(z;s,δ)S_5(z;t,δ)S_5(z+s;t,δ), R(z)=fracP_7(z)F(β z). The product identity is an actual gamma multiplication identity, not the assertion P_7(z)=F(β z). For completeness, its elementary product proof is retained from v0.12: rewrite t=2s+4-7δ, regard periods modulo integers, and perform real root averaging and vertical residue splitting in the defining gamma products. The resulting split facto…
Exact mathematical passage
\label{eq:p7} P_7(z)=\prod_{k=0}^{6}F(z+k\delta) =\frac{S_5(z;s,\delta)}{S_5(z;t,\delta)S_5(z+s;t,\delta)}, \qquad R(z)=\frac{P_7(z)}{F(\beta z)}.
\prod_{k=1}^{6}Use in source · 1 passage
Search aliases: ∏_k=1^6 prod
Use in source · reconstruction.tex, line 142
coefficient The product and continuation convention yield F(z)F(-z)=fracq_t(z)q_s(z), fracF(w+7δ)F(w)=fracq_s(w)q_t(w)q_t(w+s). Pair the labels k and 7-k in P_7(z)P_7(-z). For 1≤ k≤6, F(z+kδ)F(-z+(7-k)δ)=q_t(z+kδ-s)^-1. The unpaired k=0 term retains q_t(z)/q_s(z). Dividing by the reflected denominator gives R(z)R(-z)=T(z):= fracq_t(z)q_s(β z)q_s(z)q_t(β z) ∏_k=1^6q_t(z+kδ-s)^-1. Its apparent singularities at zero are removable, and T(0)=R_0^2.
Exact mathematical passage
\label{eq:reflection} R(z)R(-z)=T(z):= \frac{q_t(z)q_s(\beta z)}{q_s(z)q_t(\beta z)} \prod_{k=1}^{6}q_t(z+k\delta-s)^{-1}.
\prod_{nz=w}Use in source · 1 passage
Search aliases: ∏_nz=w prod
Use in source · norms.tex, line 5
Exactly normalized elliptic norms. On E_t= C/Λ_t define N_nf(w)=∏_nz=wf(z) for integers n≥1, with all n^2 preimages counted. The product is independent of the representative lift. Root-of-unity averaging and vertical residue splitting in [source reference] give N_nq_t(w)=q_t(w), · 5nmid n,
Exact mathematical passage
N_nf(w)=\prod_{nz=w}f(z)
\prod_{n\ge0}Use in source · 2 passages
Search aliases: ∏_n≥0 prod ge
Use in source · H12_Cubic_Divisor_Traditional.tex, line 72
For positive imaginary periods use the additive product convention θ_0(z;a) · =∏_n≥0(1-e^2π i(z+na))(1-e^2π i((n+1)a-z)), Γ_rm ell(z;a,b) · =∏_j,k≥0 frac1-e^2π i((j+1)a+(k+1)b-z)1-e^2π i(ja+kb+z). The negative-first-period continuation is Γ_rm ell(z;-t,s)=Γ_rm ell(z+t;t,s)^-1. Consequently the main scalar function is F(z)=fracΓ_rm ell(z;τ,s)^5Γ_rm ell(5z;5τ,5s) =fracΓ_rm ell(5(z+t);5t,5s)Γ_rm ell(z+t;t,s)^5, U_Γ=F(1/3). This convention agrees with the checked primary authors' copy of Bergeron--C…
Exact mathematical passage
\theta_0(z;a)&=\prod_{n\ge0}(1-e^{2\pi i(z+na)})(1-e^{2\pi i((n+1)a-z)}),\label{eq:theta-product}\\ \Gamma_{\rm ell}(z;a,b)&=\prod_{j,k\ge0} \frac{1-e^{2\pi i((j+1)a+(k+1)b-z)}}{1-e^{2\pi i(ja+kb+z)}}.\label{eq:gamma-product}
\prod_{n\ge1}Use in source · 1 passage
Search aliases: ∏_n≥1 prod ge
Use in source · stark.tex, line 304
H(z)=e^-πIm(ω)B_2(a) |1-e^2π iz| ∏_n≥1|(1-e^2π i(nω+z))(1-e^2π i(nω-z))|, tagA4.1
Exact mathematical passage
H(z)=e^{-\pi\operatorname{Im}(\omega)B_2(a)} |1-e^{2\pi iz}| \prod_{n\ge1}|(1-e^{2\pi i(n\omega+z)})(1-e^{2\pi i(n\omega-z)})|, \tag{A4.1}
\prod_{r=0}^{3}Use in source · 1 passage
Search aliases: ∏_r=0^3 prod
Use in source · principal.tex, line 106
All transition multipliers and the scalar anchor. [Theta transitions] The following hold meromorphically: K(z):=fracR(z+1)R(z) · =∏_r=1^15q_t(β z-rs)^-1, fracR(z+s)R(z) · =fracq_s(β z-τ)∏_k=0^6q_t(z+kδ), fracR(z+τ)R(z) · = frac∏_k=0^6q_s(z+kδ)∏_r=0^3q_t(β z+rs) q_s(β z)q_s(β z+τ), fracR(z+δ)R(z) · = fracq_s(z)q_t(z)q_t(z+s)∏_r=1^8q_t(β z-rs). Use F(z+1)=F(z), F(z+τ)/F(z)=q_s(z) and F(z+s)/F(z)=q_t(z)^-1. For [source reference], P_7 is period one and β=-15s-2, producing fifteen negative s shifts …
Exact mathematical passage
K(z):=\frac{R(z+1)}{R(z)}&=\prod_{r=1}^{15}q_t(\beta z-rs)^{-1},\label{eq:K}\\ \frac{R(z+s)}{R(z)}&=\frac{q_s(\beta z-\tau)}{\prod_{k=0}^{6}q_t(z+k\delta)},\label{eq:Rs}\\ \frac{R(z+\tau)}{R(z)}&= \frac{\prod_{k=0}^{6}q_s(z+k\delta)\prod_{r=0}^{3}q_t(\beta z+rs)} {q_s(\beta z)q_s(\beta z+\tau)},\label{eq:Rtau}\\ \frac{R(z+\delta)}{R(z)}&= \frac{q_s(z)}{q_t(z)q_t(z+s)\prod_{r=1}^{8}q_t(\beta z-rs)}.\label{eq:Rdelta}
\prod_{r=1}^{15}Use in source · 1 passage
Search aliases: ∏_r=1^15 prod
Use in source · principal.tex, line 106
All transition multipliers and the scalar anchor. [Theta transitions] The following hold meromorphically: K(z):=fracR(z+1)R(z) · =∏_r=1^15q_t(β z-rs)^-1, fracR(z+s)R(z) · =fracq_s(β z-τ)∏_k=0^6q_t(z+kδ), fracR(z+τ)R(z) · = frac∏_k=0^6q_s(z+kδ)∏_r=0^3q_t(β z+rs) q_s(β z)q_s(β z+τ), fracR(z+δ)R(z) · = fracq_s(z)q_t(z)q_t(z+s)∏_r=1^8q_t(β z-rs). Use F(z+1)=F(z), F(z+τ)/F(z)=q_s(z) and F(z+s)/F(z)=q_t(z)^-1. For [source reference], P_7 is period one and β=-15s-2, producing fifteen negative s shifts …
Exact mathematical passage
K(z):=\frac{R(z+1)}{R(z)}&=\prod_{r=1}^{15}q_t(\beta z-rs)^{-1},\label{eq:K}\\ \frac{R(z+s)}{R(z)}&=\frac{q_s(\beta z-\tau)}{\prod_{k=0}^{6}q_t(z+k\delta)},\label{eq:Rs}\\ \frac{R(z+\tau)}{R(z)}&= \frac{\prod_{k=0}^{6}q_s(z+k\delta)\prod_{r=0}^{3}q_t(\beta z+rs)} {q_s(\beta z)q_s(\beta z+\tau)},\label{eq:Rtau}\\ \frac{R(z+\delta)}{R(z)}&= \frac{q_s(z)}{q_t(z)q_t(z+s)\prod_{r=1}^{8}q_t(\beta z-rs)}.\label{eq:Rdelta}
\prod_{r=1}^{8}Use in source · 1 passage
Search aliases: ∏_r=1^8 prod
Use in source · principal.tex, line 106
All transition multipliers and the scalar anchor. [Theta transitions] The following hold meromorphically: K(z):=fracR(z+1)R(z) · =∏_r=1^15q_t(β z-rs)^-1, fracR(z+s)R(z) · =fracq_s(β z-τ)∏_k=0^6q_t(z+kδ), fracR(z+τ)R(z) · = frac∏_k=0^6q_s(z+kδ)∏_r=0^3q_t(β z+rs) q_s(β z)q_s(β z+τ), fracR(z+δ)R(z) · = fracq_s(z)q_t(z)q_t(z+s)∏_r=1^8q_t(β z-rs). Use F(z+1)=F(z), F(z+τ)/F(z)=q_s(z) and F(z+s)/F(z)=q_t(z)^-1. For [source reference], P_7 is period one and β=-15s-2, producing fifteen negative s shifts …
Exact mathematical passage
K(z):=\frac{R(z+1)}{R(z)}&=\prod_{r=1}^{15}q_t(\beta z-rs)^{-1},\label{eq:K}\\ \frac{R(z+s)}{R(z)}&=\frac{q_s(\beta z-\tau)}{\prod_{k=0}^{6}q_t(z+k\delta)},\label{eq:Rs}\\ \frac{R(z+\tau)}{R(z)}&= \frac{\prod_{k=0}^{6}q_s(z+k\delta)\prod_{r=0}^{3}q_t(\beta z+rs)} {q_s(\beta z)q_s(\beta z+\tau)},\label{eq:Rtau}\\ \frac{R(z+\delta)}{R(z)}&= \frac{q_s(z)}{q_t(z)q_t(z+s)\prod_{r=1}^{8}q_t(\beta z-rs)}.\label{eq:Rdelta}
\prod_{\eta\in E_t[2]\setminus\{0\}}Use in source · 1 passage
Search aliases: ∏_η∈ E_t[2]∖{0} prod eta in setminus
Use in source · theta.tex, line 204
∏_η∈ E_t[2]∖{0}Q_5(x(η)) =fracf(a)f(b)16.
Exact mathematical passage
\prod_{\eta\in E_t[2]\setminus\{0\}}Q_5(x(\eta)) =\frac{f(a)f(b)}{16}.
\prod_{\eta\in E_t[3]\setminus\{0\}}Use in source · 1 passage
Search aliases: ∏_η∈ E_t[3]∖{0} prod eta in setminus
Use in source · theta.tex, line 224
∏_η∈ E_t[3]∖{0}Q_5(x(η)) =(fracψ_3(a)ψ_3(b)9)^2.
Exact mathematical passage
\prod_{\eta\in E_t[3]\setminus\{0\}}Q_5(x(\eta)) =\left(\frac{\psi_3(a)\psi_3(b)}9\right)^2.
\prod_{\eta\in E_t[n]}Use in source · 1 passage
Search aliases: ∏_η∈ E_t[n] prod eta in
Use in source · theta.tex, line 23
∏_η∈ E_t[n]q_t(z+η)=q_t(nz). tagN1.1
Exact mathematical passage
\prod_{\eta\in E_t[n]}q_t(z+\eta)=q_t(nz). \tag{N1.1}
\prod_{\eta\in E_t[n]\setminus\{0\}}Use in source · 1 passage
Search aliases: ∏_η∈ E_t[n]∖{0} prod eta in setminus
Use in source · theta.tex, line 193
∏_η∈ E_t[n]∖{0}q_t(η)=n^4 (5nmid n). tagN4.1
Exact mathematical passage
\boxed{\prod_{\eta\in E_t[n]\setminus\{0\}}q_t(\eta)=n^4} \quad(5\nmid n). \tag{N4.1}
\prod_{\eta\ne0}Use in source · 1 passage
Search aliases: ∏_η≠0 prod eta ne
Use in source · theta.tex, line 211
c_t^3=16∏_η≠0Q_5(x(η))=f(a)f(b)=d^6. tagN4.2
Exact mathematical passage
c_t^3=16\prod_{\eta\ne0}Q_5(x(\eta))=f(a)f(b)=d^6. \tag{N4.2}
\prod_{\lambda\in D}Use in source · 1 passage
Search aliases: ∏_λ∈ D prod lambda in
Use in source · reconstruction.tex, line 67
The genus-three divisor product. Let D={λ:m_λ≠0}; zero is not in D. Define E_3(w)=(1-w)e^w+w^2/2+w^3/3, W_D(z)=∏_λ∈ DE_3(z/λ)^m_λ. Negative exponents mean the quotient of the separate positive and negative entire products. [Canonical regularizer and absolute genus] The products defining W_D converge normally on compact sets away from their poles, and divW_D=divR, W_D(0)=1, (log W_D)^(j)(0)=0 (1≤ j≤3). Genus three is sufficient and minimal for this ungrouped absolute construction. Special signed …
Exact mathematical passage
\label{eq:WD} E_3(w)=(1-w)e^{w+w^2/2+w^3/3},\qquad W_D(z)=\prod_{\lambda\in D}E_3(z/\lambda)^{m_\lambda}.
\sumUse in source · 33 passages
Search aliases: ∑ sum
Use in source · H12_Cubic_Divisor_Traditional.tex, line 103
The arithmetic candidate is fixed independently. Let J=2+3ω and choose the Gauss branch X=β^-1∑_a=1^6χ_7(a)e^6π ia/7, (χ_7(1),…,χ_7(6))=(1,ω,ω^2,ω^2,ω,1), X^3=J. Then E= Q(ω,β,X) has degree eighteen. The retained arithmetic construction defines ν · =(X+β/X-β^2)/3, · v · =-β^2ν-2ν^2,
Exact mathematical passage
X=\beta^{-1}\sum_{a=1}^{6}\chi_7(a)e^{6\pi ia/7},\qquad (\chi_7(1),\ldots,\chi_7(6))=(1,\omega,\omega^2,\omega^2,\omega,1),\quad X^3=J.
\sum_{a=1}^{21}Use in source · 1 passage
Search aliases: ∑_a=1^21 sum
Use in source · stark.tex, line 169
For example, the latter two use -21^-1∑_a=1^21aχ_-3(a)χ_7(a)^j.
Exact mathematical passage
-21^{-1}\sum_{a=1}^{21}a\chi_{-3}(a)\chi_7(a)^j
\sum_{a=1}^{6}Use in source · 2 passages
Search aliases: ∑_a=1^6 sum
Use in source · H12_Cubic_Divisor_Traditional.tex, line 103
The arithmetic candidate is fixed independently. Let J=2+3ω and choose the Gauss branch X=β^-1∑_a=1^6χ_7(a)e^6π ia/7, (χ_7(1),…,χ_7(6))=(1,ω,ω^2,ω^2,ω,1), X^3=J. Then E= Q(ω,β,X) has degree eighteen. The retained arithmetic construction defines ν · =(X+β/X-β^2)/3, · v · =-β^2ν-2ν^2,
Exact mathematical passage
X=\beta^{-1}\sum_{a=1}^{6}\chi_7(a)e^{6\pi ia/7},\qquad (\chi_7(1),\ldots,\chi_7(6))=(1,\omega,\omega^2,\omega^2,\omega,1),\quad X^3=J.
\sum_DUse in source · 2 passages
Search aliases: ∑_D sum
Use in source · reconstruction.tex, line 81
thm:regularizer The products defining W_D converge normally on compact sets away from their poles, and divW_D=divR, W_D(0)=1, (log W_D)^(j)(0)=0 (1≤ j≤3). Genus three is sufficient and minimal for this ungrouped absolute construction. Special signed groupings are not excluded. The cubic upper bound gives ∑_D|m_λ|/|λ|^4<∞, by partial summation or dyadic shells. For |w|<1, log E_3(w)=-∑_k≥4fracw^kk. On each fixed compact set the tail logarithms are thus absolutely and uniformly summable. The finit…
Exact mathematical passage
\sum_D|m_\lambda|/|\lambda|^4<\infty
\sum_{i<j}Use in source · 1 passage
Search aliases: ∑_i<j sum
Use in source · stark.tex, line 444
a_0+a_1+a_2=3, ∑_i<ja_i a_j=6ω, a_0a_1a_2=1.
Exact mathematical passage
a_0+a_1+a_2=3,\qquad \sum_{i<j}a_i a_j=6\omega,\qquad a_0a_1a_2=1.
\sum_{j=0}^{2}Use in source · 3 passages
Search aliases: ∑_j=0^2 sum
Use in source · reconstruction.tex, line 153
Let L_t=(q_t'/q_t)' and S_t=wp_t(1/5)+wp_t(2/5). The normalized theta formula gives L_t(z)=-4z^-2+2S_t+O(z^2), S_s=ρ^2S_t. Since 7δ-2s=-t+4, evenness and t-periodicity pair the six translated L_t terms. Using s=u_0+3δ-2, their sum is 2∑_j=0^2L_t(u_j). The second logarithmic derivative of [source reference] cancels all z^-2 terms, leaving (log T)''(0)=2(1-β^2)(1-ρ^2)S_t-2∑_j=0^2L_t(u_j). The left side is 4α_2. Therefore α_2=frac12[(1-β^2)(1-ρ^2)S_t- ∑_j=0^2L_t(u_j)]. The factor 1/2 is fixed by re…
Exact mathematical passage
2\sum_{j=0}^{2}L_t(u_j)
\sum_{j=1}^{3}Use in source · 1 passage
Search aliases: ∑_j=1^3 sum
Use in source · reconstruction.tex, line 201
The repaired conjugation quotient and its boundary. For f^#(z)=overlinef(overline z), the canonical product W_D^# has the conjugate divisor. Hence fracR/R^#W_D/W_D^# =fracR_0overlineR_0 exp(∑_j=1^3(α_j-overlineα_j)z^j). This is entire and nowhere zero, with modulus one on the real axis. It cancels all unmatched divisor orders by a canonical meromorphic factor. That factor is not a holomorphic-unit gauge or a finite theta section. Regularity on the scalar plane does not establish periodic descent…
Exact mathematical passage
\label{eq:regularized-conjugation} \frac{R/R^\#}{W_D/W_D^\#} =\frac{R_0}{\overline{R_0}} \exp\left(\sum_{j=1}^{3}(\alpha_j-\overline{\alpha_j})z^j\right).
\sum_{j=1}^{4}Use in source · 1 passage
Search aliases: ∑_j=1^4 sum
Use in source · norms.tex, line 61
tial L_t(z)=wp_+(z)-5wp_t(z)+2(a+b). At zero L_t has principal part -4/z^2; at the four nonzero kernel points it has +1/(z-P_j)^2, with no simple-pole terms. Thus L_t-wp_++5wp_t is a constant elliptic function. The canonical formula gives q_t(z)=(a-b)^2z^4[1+(a+b)z^2+O(z^4)], fixing that constant as 2(a+b). The degree-five quotient is equivalently the paired sum wp_+(z)=wp_t(z)+∑_j=1^4[wp_t(z+j/5)-wp_t(j/5)]. The analytic identity determines a rational isogeny map once elliptic coordinates are c…
Exact mathematical passage
\wp_+(z)=\wp_t(z)+\sum_{j=1}^{4}[\wp_t(z+j/5)-\wp_t(j/5)].
\sum_{j=1}^{n}Use in source · 1 passage
Search aliases: ∑_j=1^n sum
Use in source · rigidity.tex, line 50
eginequation S_Nh-h∘ N=δ_Mr, then h=δ_Mg for a smooth mean-zero g, with convergent formula g=-frac17∑_j≥1L_N^jr. If h,r extend holomorphically to all of X, so does g. Apply L_N/7 to [source reference]. The trace relation gives L_Nh-h=(1/7)δ_M L_Nr, and induction yields L_N^nh-h=frac17δ_M∑_j=1^nL_N^jr. Because B^3=7I, L_N^3nf=∑_k≠0343^nc_7^nke_k for a mean-zero f. For any C^q norm choose s>max(q,3). Smooth Fourier coefficients have finite weighted ℓ^1 norms of every order, and the displayed expre…
Exact mathematical passage
L_N^nh-h=\frac17\delta_M\sum_{j=1}^{n}L_N^jr.
\sum_{j\ge1}Use in source · 1 passage
Search aliases: ∑_j≥1 sum ge
Use in source · rigidity.tex, line 43
[Principal-norm rigidity] Let h,r be smooth mean-zero complex functions on T^3. If S_Nh-h∘ N=δ_Mr, then h=δ_Mg for a smooth mean-zero g, with convergent formula g=-frac17∑_j≥1L_N^jr. If h,r extend holomorphically to all of X, so does g. Apply L_N/7 to [source reference]. The trace relation gives L_Nh-h=(1/7)δ_M L_Nr, and induction yields L_N^nh-h=frac17δ_M∑_j=1^nL_N^jr. Because B^3=7I, L_N^3nf=∑_k≠0343^nc_7^nke_k for a mean-zero f. For any C^q norm choose s>max(q,3). Smooth Fourier coefficients …
Exact mathematical passage
g=-\frac17\sum_{j\ge1}L_N^jr.
\sum_kUse in source · 1 passage
Search aliases: ∑_k sum
Use in source · rigidity.tex, line 30
The convergent transfer proof. For functions on T^3, let S_Nh(v)=∑_κ∈ker Nh(v+κ), C_Nh(v)=h(Nv), L_Nh(v)=∑_Nw=vh(w), δ_Mh=h-h∘ M. The transfer L_N is unnormalized. For h=∑ c_ke_k, S_Nh=∑_k∈ B Z^37c_ke_k, L_Nh=∑_k7c_Bke_k, S_Nh=(L_Nh)∘ N, L_NC_Nh=7h. Since M permutes ker N and commutes with N, the required operators commute with composition by M.
Exact mathematical passage
S_Nh=\sum_{k\in B\mathbb Z^3}7c_ke_k,\qquad L_Nh=\sum_k7c_{Bk}e_k,\qquad S_Nh=(L_Nh)\circ N, \qquad L_NC_Nh=7h.
\sum_{k=0}^{6}Use in source · 3 passages
Search aliases: ∑_k=0^6 sum
Use in source · product_proof.tex, line 9
5nmid n fracP^nZ^n-Q^nZ^-nn(1-P^n)(1-Q^n), where A_0=e^2π ia,B_0=e^2π ib in the first line. At one surviving frequency suppress the index n. Summing seven shifts multiplies the two coefficients by ∑_k=0^6D^k and ∑_k=0^6D^-k. The exact identities are fracP∑_k=0^6D^k(1-P)(1-Q) · =frac1+Q(1-P)(1-D)-frac1(1-Q)(1-D),
Exact mathematical passage
\sum_{k=0}^{6}D^k
\sum_{k=1}^6Use in source · 1 passage
Search aliases: ∑_k=1^6 sum
Use in source · polarization.tex, line 18
ξ=β^-1∑_k=1^6χ_7(k)e^6π i k/7, (χ_7(1),…,χ_7(6))=(1,ω,ω^2,ω^2,ω,1).
Exact mathematical passage
\xi=\beta^{-1}\sum_{k=1}^6\chi_7(k)e^{6\pi i k/7}, \qquad (\chi_7(1),\ldots,\chi_7(6))=(1,\omega,\omega^2,\omega^2,\omega,1).
\sum_{k\in B\mathbb Z^3}Use in source · 1 passage
Search aliases: ∑_k∈ B Z^3 sum in mathbb
Use in source · rigidity.tex, line 30
The convergent transfer proof. For functions on T^3, let S_Nh(v)=∑_κ∈ker Nh(v+κ), C_Nh(v)=h(Nv), L_Nh(v)=∑_Nw=vh(w), δ_Mh=h-h∘ M. The transfer L_N is unnormalized. For h=∑ c_ke_k, S_Nh=∑_k∈ B Z^37c_ke_k, L_Nh=∑_k7c_Bke_k, S_Nh=(L_Nh)∘ N, L_NC_Nh=7h. Since M permutes ker N and commutes with N, the required operators commute with composition by M.
Exact mathematical passage
S_Nh=\sum_{k\in B\mathbb Z^3}7c_ke_k,\qquad L_Nh=\sum_k7c_{Bk}e_k,\qquad S_Nh=(L_Nh)\circ N, \qquad L_NC_Nh=7h.
\sum_{k\ne0}Use in source · 2 passages
Search aliases: ∑_k≠0 sum ne
Use in source · rigidity.tex, line 54
equation then h=δ_Mg for a smooth mean-zero g, with convergent formula g=-frac17∑_j≥1L_N^jr. If h,r extend holomorphically to all of X, so does g. Apply L_N/7 to [source reference]. The trace relation gives L_Nh-h=(1/7)δ_M L_Nr, and induction yields L_N^nh-h=frac17δ_M∑_j=1^nL_N^jr. Because B^3=7I, L_N^3nf=∑_k≠0343^nc_7^nke_k for a mean-zero f. For any C^q norm choose s>max(q,3). Smooth Fourier coefficients have finite weighted ℓ^1 norms of every order, and the displayed expression is bounded, up…
Exact mathematical passage
L_N^{3n}f=\sum_{k\ne0}343^nc_{7^nk}e_k
\sum_{k\ge4}Use in source · 1 passage
Search aliases: ∑_k≥4 sum ge
Use in source · reconstruction.tex, line 82
ir poles, and divW_D=divR, W_D(0)=1, (log W_D)^(j)(0)=0 (1≤ j≤3). Genus three is sufficient and minimal for this ungrouped absolute construction. Special signed groupings are not excluded. The cubic upper bound gives ∑_D|m_λ|/|λ|^4<∞, by partial summation or dyadic shells. For |w|<1, log E_3(w)=-∑_k≥4fracw^kk. On each fixed compact set the tail logarithms are thus absolutely and uniformly summable. The finitely many remaining factors supply exactly the stated orders, and each factor has its firs…
Exact mathematical passage
\log E_3(w)=-\sum_{k\ge4}\frac{w^k}{k}.
\sum_{m,n\in\mathbb Z}Use in source · 1 passage
Search aliases: ∑_m,n∈ Z sum in mathbb
Use in source · stark.tex, line 317
E_z(s)=∑_m,n∈ Z|m+nω+z|^-2s,
Exact mathematical passage
\mathcal E_z(s)=\sum_{m,n\in\mathbb Z}|m+n\omega+z|^{-2s},
\sum_{n,m\in\mathbb Z}Use in source · 1 passage
Search aliases: ∑_n,m∈ Z sum in mathbb
Use in source · stark.tex, line 649
E_z(s)=∑_n,m∈ Z ((m+b+(n+a)x)^2+(n+a)^2y^2)^-s.
Exact mathematical passage
\mathcal E_z(s)=\sum_{n,m\in\mathbb Z} \left((m+b+(n+a)x)^2+(n+a)^2y^2\right)^{-s}.
\sum_{Nw=v}Use in source · 1 passage
Search aliases: ∑_Nw=v sum
Use in source · rigidity.tex, line 24
The convergent transfer proof. For functions on T^3, let S_Nh(v)=∑_κ∈ker Nh(v+κ), C_Nh(v)=h(Nv), L_Nh(v)=∑_Nw=vh(w), δ_Mh=h-h∘ M. The transfer L_N is unnormalized. For h=∑ c_ke_k, S_Nh=∑_k∈ B Z^37c_ke_k, L_Nh=∑_k7c_Bke_k, S_Nh=(L_Nh)∘ N, L_NC_Nh=7h. Since M permutes ker N and commutes with N, the required operators commute with composition by M.
Exact mathematical passage
S_Nh(v)=\sum_{\kappa\in\ker N}h(v+\kappa),\quad C_Nh(v)=h(Nv),\quad L_Nh(v)=\sum_{Nw=v}h(w),\quad \delta_Mh=h-h\circ M.
\sum_{n\in\mathbb Z}Use in source · 1 passage
Search aliases: ∑_n∈ Z sum in mathbb
Use in source · reconstruction.tex, line 120
beginequation R(z)=R_0exp(α_1z+α_2z^2+α_3z^3)W_D(z), α_j=frac(log R)^(j)(0)j!. The logarithmic derivatives are branch independent. The notation (log R)^(4) means the third derivative of R'/R, a single-valued meromorphic function. The sine partial fractions [source reference] give fracd^4dw^4log(1-e^2π iw) =-6∑_n∈ Z(w-n)^-4. The linear exponential factor relating 1-e^2π iw to sinπ w has zero fourth logarithmic derivative. Apply this identity to each factor of the defining double gamma products. T…
Exact mathematical passage
\frac{d^4}{dw^4}\log(1-e^{2\pi iw}) =-6\sum_{n\in\mathbb Z}(w-n)^{-4}.
\sum_{n\ge1}Use in source · 1 passage
Search aliases: ∑_n≥1 sum ge
Use in source · stark.tex, line 669
-2log|1-e^2π iz| -2∑_n≥1log|(1-e^2π i(nτ+z))(1-e^2π i(nτ-z))|.
Exact mathematical passage
-2\log|1-e^{2\pi iz}| -2\sum_{n\ge1}\log|(1-e^{2\pi i(n\tau+z)})(1-e^{2\pi i(n\tau-z)})|.
\sum_{r=1}^{15}Use in source · 1 passage
Search aliases: ∑_r=1^15 sum
Use in source · reconstruction.tex, line 176
2λ^3]. The combined summand is O(|λ|^-4), so this is an absolutely convergent sum. Its terms must remain combined; the separately divergent lower-power sums are not substituted. Taking two logarithmic derivatives of [source reference] at the transition [source reference] gives (log K)''(0)=6α_3+B_D(1). Since L_t is even, α_3=-frac16[β^2∑_r=1^15L_t(rs)+B_D(1)]. Evaluating the period-one ratio also determines e^α_1=fracK(0)e^-α_2-α_3W_D(1). This evaluates the exponential, without choosing an unjus…
Exact mathematical passage
\label{eq:alpha3} \boxed{\alpha_3=-\frac16\left[\beta^2\sum_{r=1}^{15}L_t(rs)+B_D(1)\right].}
\sum_{\substack{n\ge1\\5\nmid n}}Use in source · 1 passage
Search aliases: ∑_substackn≥1 5nmid n sum substack ge nmid
Use in source · product_proof.tex, line 3
Full seven-product series proof. For positive imaginary periods t,s,d with t+7d=2s+4, put P=e^2π it, Q=e^2π is, D=e^2π id and Z=e^2π iz. Then PD^7=Q^2. In a common convergence region the fifth-smoothed logarithms are log S_5(z;a,b) · =5∑_substackn≥1 5nmid n fracZ^n-(A_0B_0/Z)^nn(1-A_0^n)(1-B_0^n), log F_t,s(z) · =-5∑_substackn≥1 5nmid n fracP^nZ^n-Q^nZ^-nn(1-P^n)(1-Q^n), where A_0=e^2π ia,B_0=e^2π ib in the first line. At one surviving frequency suppress the index n. Summing seven shifts multipl…
Exact mathematical passage
\log S_5(z;a,b)&=5\sum_{\substack{n\ge1\\5\nmid n}} \frac{Z^n-(A_0B_0/Z)^n}{n(1-A_0^n)(1-B_0^n)},\\ \log F_{t,s}(z)&=-5\sum_{\substack{n\ge1\\5\nmid n}} \frac{P^nZ^n-Q^nZ^{-n}}{n(1-P^n)(1-Q^n)},
\sum_{|\lambda|>T}Use in source · 1 passage
Search aliases: ∑_|λ|>T sum lambda
Use in source · reconstruction.tex, line 95
logarithm sum diverges, because the fourth-power remainder is summable. This proves failure of ungrouped absolute genus two, without a claim about cancellations in specially grouped signed products. [Explicit analytic tail] If n_D(r)=∑_|λ|≤ r|m_λ|≤ Ar^3 for all r≥ R_*, then for T≥ R_* and |z|≤ T/2, ∑_|λ|>Tfrac|m_λ||λ|^4≤frac4AT, |log W_D,>T(z)|≤frac2A|z|^4T. Partial summation gives the first upper bound by dropping its nonpositive boundary term and using 4∫_T^∞ Ar^3r^-5 dr=4A/T. For the tail fac…
Exact mathematical passage
\label{eq:tail} \sum_{|\lambda|>T}\frac{|m_\lambda|}{|\lambda|^4}\le\frac{4A}{T}, \qquad |\log W_{D,>T}(z)|\le\frac{2A|z|^4}{T}.
\sum_{|\lambda|\le r}Use in source · 1 passage
Search aliases: ∑_|λ|≤ r sum lambda le
Use in source · reconstruction.tex, line 94
thmically. For a fixed z≠0, log E_2(z/λ)=-fracz^33λ^3+O(|λ|^-4). Its weighted absolute logarithm sum diverges, because the fourth-power remainder is summable. This proves failure of ungrouped absolute genus two, without a claim about cancellations in specially grouped signed products. [Explicit analytic tail] If n_D(r)=∑_|λ|≤ r|m_λ|≤ Ar^3 for all r≥ R_*, then for T≥ R_* and |z|≤ T/2, ∑_|λ|>Tfrac|m_λ||λ|^4≤frac4AT, |log W_D,>T(z)|≤frac2A|z|^4T. Partial summation gives the first upper bound by dro…
Exact mathematical passage
n_D(r)=\sum_{|\lambda|\le r}|m_\lambda|\le Ar^3
\sum_{\kappa\in\ker N}Use in source · 1 passage
Search aliases: ∑_κ∈ker N sum kappa in ker
Use in source · rigidity.tex, line 24
The convergent transfer proof. For functions on T^3, let S_Nh(v)=∑_κ∈ker Nh(v+κ), C_Nh(v)=h(Nv), L_Nh(v)=∑_Nw=vh(w), δ_Mh=h-h∘ M. The transfer L_N is unnormalized. For h=∑ c_ke_k, S_Nh=∑_k∈ B Z^37c_ke_k, L_Nh=∑_k7c_Bke_k, S_Nh=(L_Nh)∘ N, L_NC_Nh=7h. Since M permutes ker N and commutes with N, the required operators commute with composition by M.
Exact mathematical passage
S_Nh(v)=\sum_{\kappa\in\ker N}h(v+\kappa),\quad C_Nh(v)=h(Nv),\quad L_Nh(v)=\sum_{Nw=v}h(w),\quad \delta_Mh=h-h\circ M.
\sum_{\lambda\in D}Use in source · 2 passages
Search aliases: ∑_λ∈ D sum lambda in
Use in source · reconstruction.tex, line 108
The fourth logarithmic derivative fixes the remaining degree. [Cubic-exponential reconstruction] For the actual correction R, (log R)^(4)(z)=-6∑_λ∈ Dfracm_λ(z-λ)^4, locally normally away from D. Consequently R(z)=R_0exp(α_1z+α_2z^2+α_3z^3)W_D(z), α_j=frac(log R)^(j)(0)j!. The logarithmic derivatives are branch independent. The notation (log R)^(4) means the third derivative of R'/R, a single-valued meromorphic function. The sine partial fractions [source reference] give fracd^4dw^4log(1-e^2π iw)…
Exact mathematical passage
\label{eq:fourth} (\log R)^{(4)}(z)=-6\sum_{\lambda\in D}\frac{m_\lambda}{(z-\lambda)^4},
\setminusUse in source · 5 passages
Search aliases: ∖ setminus
Use in source · principal.tex, line 96
Apply Theorem~[source reference] at the unique integral numerator lift [source reference] and subtract the denominator order at βλ. The weights agree. To bound the bracket, when a≤0,p>0 it is H(b); when a>0,p≤0 it is H(b)-1. In the other cases it is a difference of two zero-one values. Thus it lies in {-1,0,1}. In particular, ord_nτR=4 (n≥1), ord_nsR=-4 (n∈ Z∖{0}), ord_-nR=4 (n≥1). For these assertions the triples are (2n,4n,5n), (-n,0,0) and (0,15n,10n) respectively. At zero the order is zero. …
Exact mathematical passage
\operatorname{ord}_{n\tau}R=4\quad(n\ge1),\qquad \operatorname{ord}_{ns}R=-4\quad(n\in\mathbb Z\setminus\{0\}),\qquad \operatorname{ord}_{-n}R=4\quad(n\ge1).
\inftyUse in source · 2 passages
Search aliases: ∞ infty
Use in source · reconstruction.tex, line 81
thm:regularizer The products defining W_D converge normally on compact sets away from their poles, and divW_D=divR, W_D(0)=1, (log W_D)^(j)(0)=0 (1≤ j≤3). Genus three is sufficient and minimal for this ungrouped absolute construction. Special signed groupings are not excluded. The cubic upper bound gives ∑_D|m_λ|/|λ|^4<∞, by partial summation or dyadic shells. For |w|<1, log E_3(w)=-∑_k≥4fracw^kk. On each fixed compact set the tail logarithms are thus absolutely and uniformly summable. The finit…
Exact mathematical passage
\sum_D|m_\lambda|/|\lambda|^4<\infty
\intUse in source · 2 passages
Search aliases: ∫ int
Use in source · reconstruction.tex, line 87
The lower bound n_D(r)≥ c r^3 yields divergence of ∑_D|m_λ|/|λ|^3: partial summation contains 3∫ n_D(r)r^-4 dr, which diverges logarithmically. For a fixed z≠0, log E_2(z/λ)=-fracz^33λ^3+O(|λ|^-4). Its weighted absolute logarithm sum diverges, because the fourth-power remainder is summable. This proves failure of ungrouped absolute genus two, without a claim about cancellations in specially grouped signed products. [Explicit analytic tail] If n_D(r)=∑_|λ|≤ r|m_λ|≤ Ar^3 for all r≥ R_*, then for T…
Exact mathematical passage
3\int n_D(r)r^{-4}\,dr
\int_T^\inftyUse in source · 1 passage
Search aliases: ∫_T^∞ int infty
Use in source · reconstruction.tex, line 101
n_D(r)=∑_|λ|≤ r|m_λ|≤ Ar^3 for all r≥ R_*, then for T≥ R_* and |z|≤ T/2, ∑_|λ|>Tfrac|m_λ||λ|^4≤frac4AT, |log W_D,>T(z)|≤frac2A|z|^4T. Partial summation gives the first upper bound by dropping its nonpositive boundary term and using 4∫_T^∞ Ar^3r^-5 dr=4A/T. For the tail factors the canonical branch satisfies |log E_3(w)|≤ |w|^4/[4(1-|w|)]≤ |w|^4/2, proving the second inequality. The condition |z|≤ T/2 keeps these tail factors away from zeros and fixes the logarithm through its power series. This …
Exact mathematical passage
4\int_T^\infty Ar^3r^{-5}\,dr=4A/T
\simUse in source · 2 passages
Search aliases: ∼ sim
Use in source · principal.tex, line 135
[Three-theta anchor] R is regular and nonzero at zero, and R_0:=R(0)=∏_j=0^2q_t(u_j)^-1, u_j=fract+δ2+jδ. Let H(a)=η(5a)/η(a)^5. Direct product cancellation gives S_5(z;a,b)∼frac5(2π)^4 H(a) H(b)z^-4. The ratio in [source reference] therefore has value P_7(0)=[ H(s)/ H(t)]/S_5(s;t,δ). Since F(0)= H(s)/ H(t)=ϵ^2, the eta factors cancel and R_0=S_5(s;t,δ)^-1. The period relation is s=(t+δ)/2+3δ-2. At the central point, S_5((t+δ)/2;t,δ)=1 by termwise coincidence of the gamma numerator and denominat…
Exact mathematical passage
S_5(z;a,b)\sim\frac5{(2\pi)^4}\mathcal H(a)\mathcal H(b)z^{-4}.
\simeqUse in source · 1 passage
Search aliases: ≃ simeq
Use in source · rigidity.tex, line 5
The actual two commuting actions. On X= C^3/ Z^3≃( C^*)^3 and its real torus T^3, use M=0 · 1 · 0
Exact mathematical passage
X=\mathbb C^3/\mathbb Z^3\simeq(\mathbb C^*)^3
\approxUse in source · 2 passages
Search aliases: ≈ approx
Use in source · polarization.tex, line 252
C≈13.80641265880838653-46.68563220754137422i,
Exact mathematical passage
C\approx13.80641265880838653-46.68563220754137422i,
\neUse in source · 7 passages
Search aliases: ≠ ne
Use in source · reconstruction.tex, line 26
For the upper bound first consider F. A nonzero coefficient in [source reference] has H(a)≠ H(b), so the imaginary parts of aτ and bs have the same sign. Therefore |Im(aτ+bs)|=|a|Im t+|b|Im s. A bounded complex radius bounds |a| and |b| by O(r); the real part then bounds |c| by O(r). There are O((1+r)^3) triples, and absolute weights are at most four. The seven translated numerator factors and the scaled denominator in [source reference] give a finite union of such cone supports and cannot incre…
Exact mathematical passage
H(a)\ne H(b)
\equivUse in source · 7 passages
Search aliases: ≡ equiv
Use in source · rigidity.tex, line 16
tic polynomial (x-2)^3+7 gives a stable real eigenvalue a=2-7^1/3∈(0,1) and a conjugate unstable pair of modulus a^-1/2>1. Every nonzero integer A-orbit escapes in both directions: otherwise an integer vector would tend to zero along positive or negative iterates. Fourier orbit sums are therefore absolutely convergent for smooth coefficients. The dual degree-seven image is exactly k∈ B Z^3 Longleftrightarrow ℓ(k):=k_0+2k_1+4k_2≡0pmod7. Indeed k-ℓ(k)e_0=B(-k_1,k_2,-8k_2)^T and B(0,-2,15)^T=7e_0. …
Exact mathematical passage
k\in B\mathbb Z^3\quad\Longleftrightarrow\quad \ell(k):=k_0+2k_1+4k_2\equiv0\pmod7.
\leUse in source · 24 passages
Search aliases: ≤ le
Use in source · principal.tex, line 50
[Base half-space divisor] The divisor of F is supported on Λ_5. At its uniquely parametrized point, ord_aτ+bs+c/5F=w(c) [H(a)-H(b)]. For positive periods t,s, the defining gamma product has a simple zero at jt+ks+ Z for j,k≥1, a simple pole for j,k≤0, and no divisor in the mixed cones. Rational independence of 1,t,s prevents collisions. The argument z+t has coordinates (1-a,b) in the (t,s) basis, giving gamma order H(b)-H(a). In F(z)=fracΓ_rm ell(5(z+t);5t,5s) Γ_rm ell(z+t;t,s)^5, the numerator …
Exact mathematical passage
j,k\le0
\geUse in source · 16 passages
Search aliases: ≥ ge
Use in source · H12_Cubic_Divisor_Traditional.tex, line 72
For positive imaginary periods use the additive product convention θ_0(z;a) · =∏_n≥0(1-e^2π i(z+na))(1-e^2π i((n+1)a-z)), Γ_rm ell(z;a,b) · =∏_j,k≥0 frac1-e^2π i((j+1)a+(k+1)b-z)1-e^2π i(ja+kb+z). The negative-first-period continuation is Γ_rm ell(z;-t,s)=Γ_rm ell(z+t;t,s)^-1. Consequently the main scalar function is F(z)=fracΓ_rm ell(z;τ,s)^5Γ_rm ell(5z;5τ,5s) =fracΓ_rm ell(5(z+t);5t,5s)Γ_rm ell(z+t;t,s)^5, U_Γ=F(1/3). This convention agrees with the checked primary authors' copy of Bergeron--C…
Exact mathematical passage
\theta_0(z;a)&=\prod_{n\ge0}(1-e^{2\pi i(z+na)})(1-e^{2\pi i((n+1)a-z)}),\label{eq:theta-product}\\ \Gamma_{\rm ell}(z;a,b)&=\prod_{j,k\ge0} \frac{1-e^{2\pi i((j+1)a+(k+1)b-z)}}{1-e^{2\pi i(ja+kb+z)}}.\label{eq:gamma-product}
\subsetUse in source · 1 passage
Search aliases: ⊂ subset
Use in source · principal.tex, line 57
z+t has coordinates (1-a,b) in the (t,s) basis, giving gamma order H(b)-H(a). In F(z)=fracΓ_rm ell(5(z+t);5t,5s) Γ_rm ell(z+t;t,s)^5, the numerator contributes for every integer c and the denominator only when 5mid c. Their difference is [source reference]. The same product lists all possible zeros and poles. The rank-three additive group Λ_5⊂ C is nondiscrete. This does not make the divisor dense: its selected nonzero coefficients have locally finite support.
Exact mathematical passage
\Lambda_5\subset\mathbb C
\oplusUse in source · 1 passage
Search aliases: ⊕ oplus
Use in source · stark.tex, line 132
Ind_C_2^S_3(sgn) =sgn⊕Std, Std=Ind_C_3^S_3(ϱ)
Exact mathematical passage
\operatorname{Ind}_{C_2}^{S_3}(\operatorname{sgn}) =\operatorname{sgn}\oplus\operatorname{Std}, \qquad \operatorname{Std}=\operatorname{Ind}_{C_3}^{S_3}(\varrho)
\langleUse in source · 1 passage
Search aliases: ⟨ langle
Use in source · stark.tex, line 116
Gal(E/K)=⟨ c,s_0⟩, σ=cs_0, Gal(E/F_0)=⟨ r,s_0⟩.
Exact mathematical passage
\operatorname{Gal}(E/K)=\langle c,s_0\rangle, \qquad \sigma=cs_0, \qquad \operatorname{Gal}(E/F_0)=\langle r,s_0\rangle.
\rangleUse in source · 1 passage
Search aliases: ⟩ rangle
Use in source · stark.tex, line 116
Gal(E/K)=⟨ c,s_0⟩, σ=cs_0, Gal(E/F_0)=⟨ r,s_0⟩.
Exact mathematical passage
\operatorname{Gal}(E/K)=\langle c,s_0\rangle, \qquad \sigma=cs_0, \qquad \operatorname{Gal}(E/F_0)=\langle r,s_0\rangle.
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- research/hilbert12-cubic-divisor/README.md · 0 mathematical passages
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- research/hilbert12-cubic-divisor/traditional/H12_Cubic_Divisor_Traditional.tex · 33 mathematical passages
- research/hilbert12-cubic-divisor/traditional/sections/bibliography.tex · 1 mathematical passage
- research/hilbert12-cubic-divisor/traditional/sections/history.tex · 10 mathematical passages
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- research/hilbert12-cubic-divisor/traditional/sections/principal.tex · 109 mathematical passages
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- research/hilbert12-cubic-divisor/traditional/sections/reconstruction.tex · 175 mathematical passages
- research/hilbert12-cubic-divisor/traditional/sections/remaining.tex · 8 mathematical passages
- research/hilbert12-cubic-divisor/traditional/sections/rigidity.tex · 84 mathematical passages
- research/hilbert12-cubic-divisor/traditional/sections/stark.tex · 221 mathematical passages
- research/hilbert12-cubic-divisor/traditional/sections/theta.tex · 93 mathematical passages
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