Mazur 13 port: affinePointIdeal_scalarSaturated

Proved

MazurProof.N13EscapingPointPicardRealization.affinePointIdeal_scalarSaturated

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

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

The affine closure of an integral infinity-chart point is saturated by every nonzero two-adic scalar. The infinity point ideal is a monic graph, hence saturated. Saturation survives localization to the overlap, and affinePointIdeal_xUnitMod shows that contracting from the overlap recovers the original affine ideal.

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

set_option backward.isDefEq.respectTransparency false
set_option backward.isDefEq.respectTransparency.types false
open MazurProof MazurProof.N13EscapingPointPicardRealization
open scoped nonZeroDivisors
attribute [local instance] MazurProof.N13EscapingPointPicardRealization.instFactPrimeOfNatNat_fLT

Formal statement

theorem MazurProof.N13EscapingPointPicardRealization.affinePointIdeal_scalarSaturated (P : N13IntegralInfinityPointSpread.IntegralInfinityPoint) (r : N13IntegralModelContraction.R₂) (hr : r ≠ 0) (z : N13IntegralModelContraction.IntegralRing) (hz : algebraMap N13IntegralModelContraction.R₂ N13IntegralModelContraction.IntegralRing r * z ∈ N13IntegralInfinityPointSpread.affinePointIdeal P) : z ∈ N13IntegralInfinityPointSpread.affinePointIdeal P := by sorry

Source

Xiang Huang, FLT fork, https://github.com/xiangyazi24/FLT/blob/51bbb4f/FLT/Assumptions/MazurProof/N13EscapingPointPicardRealization.lean#L74

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