Mazur 13 port: overlap_unitPointIdeal

Proved

MazurProof.N13SpecialDivisorCharts.overlap_unitPointIdeal

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

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

On the overlap, the point ideals written in the coordinates x=t⁻¹ and y=x³v agree at x=t=1. The first generators differ by the units x and t. Once those generators are identified, the difference between the second generators is a multiple of x³-1 or t³-1, respectively.

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.N13SpecialDivisorCharts
open Polynomial
open scoped Sym2
attribute [local instance] MazurProof.N13SpecialDivisorCharts.instFactPrimeOfNatNat_fLT

Formal statement

theorem MazurProof.N13SpecialDivisorCharts.overlap_unitPointIdeal (c : SpecialOverlap) : Ideal.span {N13SpecialCurveOverlap.xOverlap - 1, N13SpecialCurveOverlap.xOverlap ^ 3 * N13SpecialCurveOverlap.vOverlap - c} = Ideal.span {N13SpecialCurveOverlap.tOverlap - 1, N13SpecialCurveOverlap.vOverlap - c} := by sorry

Source

Xiang Huang, FLT fork, https://github.com/xiangyazi24/FLT/blob/51bbb4f/FLT/Assumptions/MazurProof/N13SpecialDivisorCharts.lean#L125

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