Endomorphisms of the actual rational Picard scheme are determined by perfect characteristic-zero field points

Proved

MazurTransfer.Order13ActualPicardDiamondSchemeCyclotomic.actualPicardOverHom_ext_field_points

by Vas · Oct 17, 2026 · Mathlib 0df444a (Lean v4.33.1)

algebraic-geometryarithmetic-geometrymazur-theorem

Let P be the same actual rational represented Picard scheme of the order-thirteen curve. Let f and g be endomorphisms of P over Q. If for every perfect characteristic-zero field K and every K-valued point a of P one has

f\circ a=g\circ a,

then f=g as morphisms over Q. This is the field-point comparison used to promote the geometric diamond and nineteen-norm identities to genuine scheme identities.

Preamble

/-
Copyright (c) 2026 Vas and contributors. Released under Apache-2.0.
Design boundary: endomorphisms of the SAME actual rational Picard are
determined by every perfect characteristic-zero field point.
Named downstream consumer: genuine diamond cyclotomic and nineteen-norm
scheme identities. Original MazurTheorem field-point extensionality bridge.
-/
import Definitions.Def_MazurTransfer_Order13ActualPicardReducedProperGeometryData
import Theorems.Thm_MazurTransfer_RationalSchemeMorphismFieldPointExtensionality_hom_eq_of_field_points
noncomputable section
open CategoryTheory AlgebraicGeometry AlgebraicGeometry.RelPicard
open NeronModelInfra GoodReductionJacobian
open MazurTransfer.Order13ActualPicardDiamondSchemeEndomorphism
open MazurTransfer.Order13ActualPicardReducedProperGeometry

Formal statement

theorem MazurTransfer.Order13ActualPicardDiamondSchemeCyclotomic.actualPicardOverHom_ext_field_points
    (f g : Over.mk actualPicard.toBase ⟶ Over.mk actualPicard.toBase)
    (H : ∀ (K : Type) [Field K] [CharZero K] [PerfectField K]
      (a : Over.mk (Spec.map (CommRingCat.ofHom (algebraMap ℚ K))) ⟶ Over.mk actualPicard.toBase),
        a ≫ f = a ≫ g) : f = g := by sorry

Source

Original MazurTheorem Order13ActualPicardDiamondSchemeCyclotomic.actualPicardOverHom_ext_field_points, retaining the same rational Picard family and exact original full type. The smooth, reduced and separated geometry is taken from that same actual integral family. Apache-2.0.

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