Mazur 13 port: forget_eq_one_iff_eq_one_or_eq_sign

Proved

MazurProof.EvenSexticNormPair.forget_eq_one_iff_eq_one_or_eq_sign

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

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

If the base group has only two square roots of one, the forgetting kernel has at most the identity and one sign class.

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.EvenSexticNormPair
variable {A B : Type*} [CommGroup A] [CommGroup B]

Formal statement

theorem MazurProof.EvenSexticNormPair.forget_eq_one_iff_eq_one_or_eq_sign (N : A →* B) (e : B →* A) (norm_scalar : ∀ q : B, N (e q) = q ^ 6) (σ : B) (hσ : σ ^ 2 = 1) (sqOne : ∀ ε : B, ε ^ 2 = 1 → ε = 1 ∨ ε = σ) (z : FullTarget N e norm_scalar) : forget N e norm_scalar z = 1 ↔ z = 1 ∨ z = QuotientGroup.mk' (fullGauge N e norm_scalar) (signPair N σ hσ) := by sorry

Source

Xiang Huang, FLT fork, https://github.com/xiangyazi24/FLT/blob/51bbb4f/FLT/Assumptions/MazurProof/EvenSexticNormPair.lean#L332

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