Mazur 13 port: affineEquation_iff_fixed

Proved

MazurProof.N13GoodModelTwo.affineEquation_iff_fixed

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

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

In a field with four-power Frobenius, an affine point has both coordinates in the prime subfield. The non-prime-field branch is excluded by the Artin--Schreier identity, not by a finite table.

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_L0

set_option backward.isDefEq.respectTransparency false
set_option backward.isDefEq.respectTransparency.types false
open MazurProof MazurProof.N13GoodModelTwo
open scoped CharTwo
open Polynomial
universe u
variable {R : Type u} [CommRing R]
variable {K : Type u} [Field K] [CharP K 2]

Formal statement

theorem MazurProof.N13GoodModelTwo.affineEquation_iff_fixed (hfour : ∀ z : K, z ^ 4 = z) (x y : K) : AffineEquation x y ↔ x ^ 2 = x ∧ y ^ 2 = y := by sorry

Source

Xiang Huang, FLT fork, https://github.com/xiangyazi24/FLT/blob/51bbb4f/FLT/Assumptions/MazurProof/N13GoodModelTwo.lean#L195

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