Mazur 13 port: curveFactor_bezout

Proved

MazurProof.N13MumfordFullKummerIdentityFiber.curveFactor_bezout

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

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

If f - L² = a w, the complementary graph ideal is contained in (a,Y-L) whenever a ∣ w. Cantor's product formula then supplies the missing generator Y-L of the square. Thus no factorization of a is needed.

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_L0

open MazurProof MazurProof.N13MumfordFullKummerIdentityFiber
open Polynomial
open SexticMumford
open scoped nonZeroDivisors
universe u
variable {K : Type u} [Field K]

Formal statement

theorem MazurProof.N13MumfordFullKummerIdentityFiber.curveFactor_bezout (C : SexticMumford.Model K) (u₀ w L₀ : K[X]) (hu₀ : u₀ ≠ 0) (hcurve : C.f - L₀ ^ 2 = u₀ * w) : ∃ A B E : K[X], A * u₀ + B * (2 * L₀) + E * w = 1 := by sorry

Source

Xiang Huang, FLT fork, https://github.com/xiangyazi24/FLT/blob/51bbb4f/FLT/Assumptions/MazurProof/N13MumfordFullKummerIdentityFiber.lean#L1474

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