Mazur 13 port: normalizedKummerInteger_eq_unit_mul_sq

Proved

MazurProof.N13GaussianDifferentSupport.normalizedKummerInteger_eq_unit_mul_sq

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

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

The normalized Kummer integer is already a unit times a square in the maximal order. The proof first leaves only the two primes supporting the different, and then separates their two parity bits by the absolute ideal norm at 2 and 13.

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

set_option backward.isDefEq.respectTransparency false
set_option backward.isDefEq.respectTransparency.types false
open MazurProof MazurProof.N13GaussianDifferentSupport
open Algebra Module Polynomial
open scoped nonZeroDivisors
open N13GaussianFieldEquiv
open N13GlobalKummerNormalization
open N13GlobalKummerIdealSquare
open IsDedekindDomain
attribute [local instance] MazurProof.N13GaussianDifferentSupport.fieldL
attribute [local instance] MazurProof.N13GaussianDifferentSupport.numberFieldL
attribute [local instance] MazurProof.N13GaussianDifferentSupport.dedekindO
attribute [local instance] MazurProof.N13GaussianDifferentSupport.fractionRingOL
attribute [local instance] MazurProof.N13GaussianDifferentSupport.intAlgebraO
attribute [local instance] MazurProof.N13GaussianDifferentSupport.dedekindRelativeO
attribute [local instance] MazurProof.N13GaussianDifferentSupport.torsionFreeGIL
attribute [local instance] MazurProof.N13GaussianDifferentSupport.torsionFreeRelativeO

Formal statement

theorem MazurProof.N13GaussianDifferentSupport.normalizedKummerInteger_eq_unit_mul_sq (D : N13LowDegreeKummerHom.LowRep) : ∃ ε : Oˣ, ∃ y : O, normalizedKummerInteger D = (ε : O) * y ^ 2 := by sorry

Source

Xiang Huang, FLT fork, https://github.com/xiangyazi24/FLT/blob/51bbb4f/FLT/Assumptions/MazurProof/N13GaussianDifferentSupport.lean#L949

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