Mazur 13 port: graphDivisor_eq_special_of_mumfordIdeal_eq

Proved

MazurProof.N13SpecialGraphDivisor.graphDivisor_eq_special_of_mumfordIdeal_eq

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

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

The literal special graph ideal determines the selected effective divisor. The proof recovers the two roots and evaluates the graph there; it does not enumerate the special curve.

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.N13SpecialGraphDivisor
open Polynomial
open scoped Sym2
open MazurProof.N13GoodCoordinateRingTwo
attribute [local instance] MazurProof.N13SpecialGraphDivisor.instFactPrimeOfNatNat_fLT

Formal statement

theorem MazurProof.N13SpecialGraphDivisor.graphDivisor_eq_special_of_mumfordIdeal_eq (D : SemiMumford) (hdeg : D.u.natDegree = 2) (hideal : mumfordIdeal D.u D.v = N13SpecialQuotientBasis.specialIdeal) : graphDivisor D hdeg = N13AbelChartBase.specialBaseDivisor := by sorry

Source

Xiang Huang, FLT fork, https://github.com/xiangyazi24/FLT/blob/51bbb4f/FLT/Assumptions/MazurProof/N13SpecialGraphDivisor.lean#L374

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