Mazur 13 port: gaussianTheta_gaussian_cubic

Proved

MazurProof.N13GaussianFieldEquiv.gaussianTheta_gaussian_cubic

by xuanji · Oct 9, 2026 · Mathlib 0df444a (Lean v4.33.1)

huang-flt-portmazur-full-torsionmazur-n13-portorder13

Supporting lemma gaussianTheta_gaussian_cubic (namespace MazurProof.N13GaussianFieldEquiv).

This lemma is one step of a machine-checked proof that the genus-two curve

Y^2 = X^6+4X^5+6X^4+2X^3+X^2+2X+1

(a model of $X_1(13)$ ) has no rational affine points other than the cusps $X\in\{0,-1\}$ , which gives the case $N=13$ of Mazur's torsion theorem. The proof is Xiang Huang's Lean development, ported to this Mathlib and split into one node per large lemma; definitions live in the layered entries MazurN13_L0, MazurN13_L1, ….

Preamble

import Mathlib
import Definitions.Def_MazurN13_L3

set_option backward.isDefEq.respectTransparency false
set_option backward.isDefEq.respectTransparency.types false
open MazurProof MazurProof.N13GaussianFieldEquiv
open Algebra Module Polynomial
open N13GaussianGlobalArithmetic
attribute [local instance] MazurProof.N13GaussianFieldEquiv.fieldLg
attribute [local instance] MazurProof.N13GaussianFieldEquiv.fieldLs
attribute [local instance] MazurProof.N13GaussianFieldEquiv.finiteKL
attribute [local instance] MazurProof.N13GaussianFieldEquiv.finiteQL

Formal statement

theorem MazurProof.N13GaussianFieldEquiv.gaussianTheta_gaussian_cubic : gaussianTheta ^ 3 + 2 * gaussianTheta ^ 2 - gaussianTheta - 1 - gaussianI * (2 * gaussianTheta * (gaussianTheta + 1)) = 0 := by sorry

Source

Xiang Huang, FLT fork, https://github.com/xiangyazi24/FLT/blob/51bbb4f/FLT/Assumptions/MazurProof/N13GaussianFieldEquiv.lean#L178

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