Mazur 13 port: normalizedKummerInteger_multiplicity_even_of_simpleRoot

Proved

MazurProof.N13GlobalKummerSimpleRootParity.normalizedKummerInteger_multiplicity_even_of_simpleRoot

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

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

At every prime away from the different, the simple-root branch gives even multiplicity of the normalized Kummer integer, with no condition on the denominator-clearing scale.

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.N13GlobalKummerSimpleRootParity
open Polynomial
open IsDedekindDomain
open IsDedekindDomain.HeightOneSpectrum
open scoped nonZeroDivisors
open N13GlobalKummerNormalization
open N13GlobalKummerIdealSquare
open N13GoodPrimeSimpleRoot
attribute [local instance] MazurProof.N13GlobalKummerSimpleRootParity.fieldL
attribute [local instance] MazurProof.N13GlobalKummerSimpleRootParity.dedekindO
attribute [local instance] MazurProof.N13GlobalKummerSimpleRootParity.fractionRingOL
attribute [local instance] MazurProof.N13GlobalKummerSimpleRootParity.charZeroL
attribute [local instance] MazurProof.N13GlobalKummerSimpleRootParity.charZeroCompletion

Formal statement

theorem MazurProof.N13GlobalKummerSimpleRootParity.normalizedKummerInteger_multiplicity_even_of_simpleRoot (D : N13LowDegreeKummerHom.LowRep) (P : HeightOneSpectrum O) (hmem : normalizedKummerInteger D ∈ P.asIdeal) (hdifferent : integralEval N13SexticIrreducible.fInt.derivative ∉ P.asIdeal) (hsimple : IsUnit ((localU D P).derivative.eval (localTheta P))) : Even (multiplicity P.asIdeal (Ideal.span {normalizedKummerInteger D})) := by sorry

Source

Xiang Huang, FLT fork, https://github.com/xiangyazi24/FLT/blob/51bbb4f/FLT/Assumptions/MazurProof/N13GlobalKummerSimpleRootParity.lean#L383

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