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Project-declaredLean 4.32.0 ยท mathlib@249c48c2

Hecke Operator L tensor

TotallyDefiniteQuaternionAlgebra.WeightTwoAutomorphicForm.heckeOperatorL_tensor

number theoryarithmetic geometryFermat's Last Theoremconcrete

Plain-language statement

Hecke operators are preserved under the identification ๐’ฎยฒ(U, ฯ‡; M โŠ— N) โ‰ƒ M โŠ— ๐’ฎยฒ(U, ฯ‡; N).

Exact Lean statement

lemma heckeOperatorL_tensor [โ„’.toStruct.IsSufficientlySmall D] [โ„’.toStruct.IsFinite D]
    {S N T : Type*} [CommRing S] [Module S M] [CommRing T] [Module T M] [Algebra R S] [Algebra S T]
    [AddCommGroup N] [Module S N] [Module R N]
    [IsScalarTower R S M] [IsScalarTower S T M] [SMulCommClass R T M] [IsScalarTower R S N]
    (v : HeightOneSpectrum (๐“ž F)) (hv : โ„’.ฯ‡ v = 1)
    (g : GLโ‚‚(v.adicCompletion F)) [SMulCommClass R S M] :
    โ„’.heckeOperatorL D T (M โŠ—[S] N) v hv g = (โ„’.toStruct.formTensor D _ _ _ _).toLinearMap โˆ˜โ‚—
      AlgebraTensorModule.lTensor T M (โ„’.heckeOperatorL D S N v hv g) โˆ˜โ‚—
      (โ„’.toStruct.formTensor D _ _ _ _).symm.toLinearMap

Formal artifact

Lean source

Canonical source
Full Lean sourceLean 4
lemma heckeOperatorL_tensor [โ„’.toStruct.IsSufficientlySmall D] [โ„’.toStruct.IsFinite D]    {S N T : Type*} [CommRing S] [Module S M] [CommRing T] [Module T M] [Algebra R S] [Algebra S T]    [AddCommGroup N] [Module S N] [Module R N]    [IsScalarTower R S M] [IsScalarTower S T M] [SMulCommClass R T M] [IsScalarTower R S N]    (v : HeightOneSpectrum (๐“ž F)) (hv : โ„’.ฯ‡ v = 1)    (g : GLโ‚‚(v.adicCompletion F)) [SMulCommClass R S M] :    โ„’.heckeOperatorL D T (M โŠ—[S] N) v hv g = (โ„’.toStruct.formTensor D _ _ _ _).toLinearMap โˆ˜โ‚—      AlgebraTensorModule.lTensor T M (โ„’.heckeOperatorL D S N v hv g) โˆ˜โ‚—      (โ„’.toStruct.formTensor D _ _ _ _).symm.toLinearMap := by  rw [โ† LinearMap.comp_assoc, LinearEquiv.eq_comp_toLinearMap_symm]  ext x y : 3  suffices โ„’.heckeOperator D (M โŠ—[S] N) v hv g (โ„’.toStruct.formTensor D M S T N (x โŠ—โ‚œ[S] y)) =      โ„’.toStruct.formTensor D M S T N (x โŠ—โ‚œ[S] โ„’.heckeOperator D N v hv g y) by    simpa  ext a  obtain โŸจs, hsโŸฉ := QuotientGroup.exists_bijOn_mk_image_mul_singleton _    (โ„’.isOpen_US v) (โ„’.isCompact_US v) g  simp [โ„’.heckeOperator_eq_finsetSum _ _ _ s hs, tmul_sum, adicCompletion_smul_def]
Project
Fermat's Last Theorem
License
Apache-2.0
Commit
8dd808888295
Source
FLT/AutomorphicForm/QuaternionAlgebra/HeckeOperators/Concrete.lean:328-345

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Plain-language statement

If a is fixed by V then โˆ‘แถ  g โˆˆ s, g โ€ข a is independent of the choice s of coset representatives in G for a subset of G โงธ V

number theoryarithmetic geometryFermat's Last Theorem

Source project: Fermat's Last Theorem

Person-level attribution pending.

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Project-declaredLean 4.32.0

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Plain-language statement

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Source project: Fermat's Last Theorem

Person-level attribution pending.

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