Mazur 13 port: norm_aeval_eq_resultant

Proved

MazurProof.PowerBasisDiscriminant.norm_aeval_eq_resultant

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

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

The norm of a polynomial in a power-basis generator is the resultant with the generator's minimal polynomial.

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
open Polynomial
open scoped Polynomial BigOperators

Formal statement

theorem MazurProof.PowerBasisDiscriminant.norm_aeval_eq_resultant {K L : Type*} [Field K] [Field L] [Algebra K L] [FiniteDimensional K L] [Algebra.IsSeparable K L] (B : PowerBasis K L) (q : K[X]) : Algebra.norm K (Polynomial.aeval B.gen q) = (minpoly K B.gen).resultant q := by sorry

Source

Xiang Huang, FLT fork, https://github.com/xiangyazi24/FLT/blob/51bbb4f/FLT/Assumptions/MazurProof/PowerBasisDiscriminant.lean#L32

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