Mazur 13 port: root_jet

Proved

MazurProof.N13GoodPointFirstJet.root_jet

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

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

Supporting lemma root_jet (namespace MazurProof.N13GoodPointFirstJet).

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.N13GoodPointFirstJet
open Polynomial MulOpposite
universe u
variable {K : Type u} [Field K]

Formal statement

theorem MazurProof.N13GoodPointFirstJet.root_jet (x y s : K) (hc : y ^ 2 + (x ^ 3 + x + 1) * y - (x ^ 5 + x ^ 4) = 0) (hd : (2 * y + (x ^ 3 + x + 1)) * s + (3 * x ^ 2 + 1) * y - (5 * x ^ 4 + 4 * x ^ 3) = 0) : (N13GeneralizedMumfordIntegral.curvePoly (R := K)).eval₂ (polynomialJet x) (y, s) = 0 := by sorry

Source

Xiang Huang, FLT fork, https://github.com/xiangyazi24/FLT/blob/51bbb4f/FLT/Assumptions/MazurProof/N13GoodPointFirstJet.lean#L44

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