Mazur 13 port: gaussian_cubic_jet_relation

Proved

MazurProof.N13LocalDlogRegimes.gaussian_cubic_jet_relation

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

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

The Gaussian cubic relation remains valid in the first-jet quotient. This verifies directly that i ↦ 1+ε and θ ↦ α are compatible with the local presentation, rather than merely assigning two unrelated dual numbers.

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_L1

set_option backward.isDefEq.respectTransparency false
set_option backward.isDefEq.respectTransparency.types false
open MazurProof MazurProof.N13LocalDlogRegimes
open scoped CharTwo
open N13LocalDlogTwo
open TrivSqZeroExt

Formal statement

theorem MazurProof.N13LocalDlogRegimes.gaussian_cubic_jet_relation : thetaDual ^ 3 + 2 * thetaDual ^ 2 - thetaDual - 1 - gaussianIDual * (2 * thetaDual * (thetaDual + 1)) = 0 := by sorry

Source

Xiang Huang, FLT fork, https://github.com/xiangyazi24/FLT/blob/51bbb4f/FLT/Assumptions/MazurProof/N13LocalDlogRegimes.lean#L60

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