Mazur 13 port: isDouble_of_finiteIdealSquareRoot

Proved

MazurProof.N13MumfordFullKummerIdentityFiber.isDouble_of_finiteIdealSquareRoot

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

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

A finite fractional-ideal square is sufficient for N13. Whatever integer remains at infinity is a multiple of infinityClass, and the already constructed N13 half of that class absorbs it.

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_L2

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

Formal statement

theorem MazurProof.N13MumfordFullKummerIdentityFiber.isDouble_of_finiteIdealSquareRoot (D : LowRep) (hfinite : ∃ I : InvFrac M, ∃ α : (FunctionField M)ˣ, mumfordIdealUnit M D.toSemi * toPrincipalIdeal (CoordinateRing M) (FunctionField M) α = I ^ 2) : ∃ Q : G, N13LowDegreeKummerHom.lowClass D = 2 • Q := by sorry

Source

Xiang Huang, FLT fork, https://github.com/xiangyazi24/FLT/blob/51bbb4f/FLT/Assumptions/MazurProof/N13MumfordFullKummerIdentityFiber.lean#L2230

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