Prove2Me

The original geometric arithmetic class detects equality of points of the same rational Picard scheme

Proved
MazurTransfer.Order13ActualPicardGeometricArithmeticGroup.geometricArithmeticClassOver_injective_explicitOriginalType20261017

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

algebraic-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,

cl⁡K(a)=cl⁡K(b)⟹a=b.\operatorname{cl}_K(a)=\operatorname{cl}_K(b)\quad\Longrightarrow\quad a=b.

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. The field and affine-cover parameters are explicit in the public statement; the complete original theorem type is preserved.

Preamble
import Definitions.Def_MazurTransfer_Order13ActualPicardGeometricDiamondGroupActionExplicitConstructorFacts20261017
Formal statement
theorem MazurTransfer.Order13ActualPicardGeometricArithmeticGroup.geometricArithmeticClassOver_injective_explicitOriginalType20261017
    (K : Type) [Field K] [CharZero K] [PerfectField K]
    (𝒱 : (MazurTorsion.XOneThirteenProjectiveCurve.curveScheme K).TwoAffineOpenCover) :
    Function.Injective (MazurTransfer.Order13ActualPicardGeometricArithmeticGroup.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.

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