Mazur 13 port: affineEquation_of_infinityEquation

Proved

MazurProof.N13OrdinaryOverlapCore.affineEquation_of_infinityEquation

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

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

The infinity equation implies the affine equation after substituting x = t⁻¹ and y = t⁻³v.

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

set_option backward.isDefEq.respectTransparency false
set_option backward.isDefEq.respectTransparency.types false

Formal statement

theorem MazurProof.N13OrdinaryOverlapCore.affineEquation_of_infinityEquation {S : Type*} [CommRing S] (t x v : S) (htx : t * x = 1) (hInfinity : v ^ 2 + (1 + t ^ 2 + t ^ 3) * v = t + t ^ 2) : (x ^ 3 * v) ^ 2 + (x ^ 3 + x + 1) * (x ^ 3 * v) = x ^ 5 + x ^ 4 := by sorry

Source

Xiang Huang, FLT fork, https://github.com/xiangyazi24/FLT/blob/51bbb4f/FLT/Assumptions/MazurProof/N13OrdinaryOverlapCore.lean#L22

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