Project-declaredLean 4.32.0 · mathlib@249c48c2

Hecke Operator eq l Tensor

TotallyDefiniteQuaternionAlgebra.WeightTwoAutomorphicForm.heckeOperator_eq_lTensor

number theory arithmetic geometry Fermat's Last Theorem concrete

Plain-language statement

Hecke operators are preserved under the identification 𝒮²(U, χ; M) ≃ M ⊗ 𝒮²(U, χ; R).

Exact Lean statement

lemma heckeOperator_eq_lTensor [ℒ.toStruct.IsSufficientlySmall D] [ℒ.toStruct.IsFinite D]
    (v : HeightOneSpectrum (𝓞 F)) (hv : ℒ.χ v = 1) (g : GL₂(v.adicCompletion F)) :
    ℒ.heckeOperator D M v hv g = (ℒ.toStruct.formTensorScalar D _ _).toLinearMap ∘ₗ
      AlgebraTensorModule.lTensor R M (ℒ.heckeOperator D R v hv g) ∘ₗ
      (ℒ.toStruct.formTensorScalar D _ _).symm.toLinearMap

Formal artifact

Lean source

Canonical source

Full Lean sourceLean 4

lemma heckeOperator_eq_lTensor [ℒ.toStruct.IsSufficientlySmall D] [ℒ.toStruct.IsFinite D]    (v : HeightOneSpectrum (𝓞 F)) (hv : ℒ.χ v = 1) (g : GL₂(v.adicCompletion F)) :    ℒ.heckeOperator D M v hv g = (ℒ.toStruct.formTensorScalar D _ _).toLinearMap ∘ₗ      AlgebraTensorModule.lTensor R M (ℒ.heckeOperator D R v hv g) ∘ₗ      (ℒ.toStruct.formTensorScalar D _ _).symm.toLinearMap := by  trans (ℒ.toStruct.formCongr D (AlgebraTensorModule.rid R R M)).toLinearMap ∘ₗ    ℒ.heckeOperatorL D R (M ⊗[R] R) v hv g ∘ₗ    (ℒ.toStruct.formCongr D (AlgebraTensorModule.rid R R M)).symm.toLinearMap  · ext f a    obtain ⟨s, hs⟩ := QuotientGroup.exists_bijOn_mk_image_mul_singleton _      (ℒ.isOpen_US v) (ℒ.isCompact_US v) g    simp [ℒ.heckeOperator_eq_finsetSum _ _ _ s hs, LevelStruct.formCongr, adicCompletion_smul_def]  · rw [heckeOperatorL_tensor]; rfl

Project: Fermat's Last Theorem
License: Apache-2.0
Commit: 8dd808888295
Source: FLT/AutomorphicForm/QuaternionAlgebra/HeckeOperators/Concrete.lean:349-361

Reuse this declaration

Bring the exact result into your workflow

The import identifies the source module. Your project still needs the pinned package dependency shown on this page.

What this badge means

This completion status comes from the project or community source. It has not yet been represented here as an independent rebuild and axiom audit.

Continue in this project

Related declarations

Project-declaredLean 4.32.0

Eq finsum quotient out of bij On

AbstractHeckeOperator.eq_finsum_quotient_out_of_bijOn'

Plain-language statement

If a is fixed by V then ∑ᶠ g ∈ s, g • a is independent of the choice s of coset representatives in G for a subset of G ⧸ V

number theoryarithmetic geometryFermat's Last Theorem

Source project: Fermat's Last Theorem

Person-level attribution pending.

View proof record

Project-declaredLean 4.32.0

Comm Group no compact automorphisms

CommGroup.no_compact_automorphisms

Plain-language statement

A connected compact Hausdorff abelian topological group does not admit a nontrivial compact group of automorphisms.

number theoryarithmetic geometryFermat's Last Theorem

Source project: Fermat's Last Theorem

Person-level attribution pending.

View proof record

Project-declaredLean 4.32.0

Fermat Last Theorem of p ge 5

FermatLastTheorem.of_p_ge_5

Plain-language statement

If Fermat's Last Theorem is true for primes p ≥ 5, then FLT is true.

number theoryarithmetic geometryFermat's Last Theorem

Source project: Fermat's Last Theorem

Person-level attribution pending.

View proof record