Mazur 13 port: orientedMumfordFullClass_eq_one_iff_exists_gauge

Proved

MazurProof.N13MumfordFullKummerIdentityFiber.orientedMumfordFullClass_eq_one_iff_exists_gauge

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

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

Equality to one in the full target is exactly membership in the product of the square/norm and scalar/cubic gauges.

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_L4

open MazurProof MazurProof.N13MumfordFullKummerIdentityFiber
open Polynomial
open SexticMumford
open scoped nonZeroDivisors

Formal statement

theorem MazurProof.N13MumfordFullKummerIdentityFiber.orientedMumfordFullClass_eq_one_iff_exists_gauge (D : LowRep) : N13MumfordOrientedFullKummer.orientedMumfordFullClass D = 1 ↔ ∃ β : Lˣ, ∃ q : ℚˣ, N13MumfordOrientedFullKummer.orientedMumfordNormPair D = EvenSexticNormPair.chi N13FullNormPair.normUnits β * EvenSexticNormPair.iota N13FullNormPair.normUnits N13FullNormPair.scalarUnits N13FullNormPair.normUnits_scalarUnits q := by sorry

Source

Xiang Huang, FLT fork, https://github.com/xiangyazi24/FLT/blob/51bbb4f/FLT/Assumptions/MazurProof/N13MumfordFullKummerIdentityFiber.lean#L57

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