The original geometric arithmetic class detects equality of points of the same rational Picard scheme
OpenMazurTransfer.Order13ActualPicardGeometricArithmeticGroup.geometricArithmeticClassOver_injectivealgebraic-geometryarithmetic-geometrymazur-theorem
Let K be a perfect characteristic-zero field and choose a two-affine cover of the actual curve over K. For K-valued points a,b of the same original rational represented Picard family, viewed in the over category,
The map is the original arithmetic class from a full degree-zero divisor presentation of the actual geometric line bundle. The result gives the faithful point comparison needed to detect the diamond and nineteen-norm scheme identities.
Preamble
/- Copyright (c) 2026 Vas and contributors. Released under Apache-2.0. Design boundary: the actual geometric arithmetic group map on points of the same existing rational represented Picard is injective. Named downstream consumer: detection of the actual diamond cyclotomic and scheme norm identities. Complete original MazurTheorem geometric arithmetic group proof; official Anthropic FLT Poincare identity inputs at 6e837e75355538c7f80bab5b956861e86c4eacc2, Apache-2.0. -/ import Definitions.Def_MazurTransfer_Order13ActualPicardGeometricArithmeticGroupData import Definitions.Def_AlgebraicGeometry_RelPicardAlgEquivZeroGroupCut noncomputable section open CategoryTheory CategoryTheory.Limits MonoidalCategory AlgebraicGeometry AlgebraicGeometry.RelPicard AlgebraicCurve open NeronModelInfra GoodReductionJacobian open MazurTorsion.XOneThirteenProjectiveCurve open GoodCharacteristicGeometryPublicProof.MazurTransfer.Order13GoodCharacteristicCurveModel open MazurTransfer.Order13ActualPicardDiamondSchemeEndomorphism open MazurTransfer.Order13ActualPicardGeometricDiamondLineBundle open MazurTransfer.Order13ActualPicardGeometricArithmeticClass open MazurTransfer.Order13ActualPicardGeometricArithmeticGroup open MazurTransfer.Order13GeometricRelativeCurveChange open scoped CategoryTheory.MonObj variable (K : Type) [Field K] [CharZero K] [PerfectField K] local instance : Fact ((104 : K) ≠ 0) := ⟨by norm_num⟩ local instance : Fact ((104 : ℚ) ≠ 0) := ⟨by norm_num⟩ local instance : Algebra K (actualCurveFunctionField K) := actualCurveFunctionFieldAlgebra K variable (𝒱 : (curveScheme K).TwoAffineOpenCover)
Formal statement
theorem MazurTransfer.Order13ActualPicardGeometricArithmeticGroup.geometricArithmeticClassOver_injective : Function.Injective (geometricArithmeticClassOver K 𝒱) := by sorry
Source
Complete original MazurTheorem Order13ActualPicardGeometricArithmeticGroup.geometricArithmeticClassOver_injective, retaining all three original unit and kernel helpers and using the accepted exact geometric group law, zero-class criterion and line-bundle point detection. Official Anthropic FLT Poincare identity inputs at 6e837e75355538c7f80bab5b956861e86c4eacc2, Apache-2.0.