FiniteDimensional.lean`

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Finite-dimensionality of spaces of weight-2 automorphic forms

We prove that the space of weight-2 automorphic forms on a totally definite quaternion algebra over a totally real number field, of fixed level U, is a finitely generated R-module.

Main results

WeightTwoAutomorphicForm.finiteDimensional: the space of weight-2 forms is module-finite over the coefficient ring.

Imports (5)

Imported by 2 files

lemma TotallyDefiniteQuaternionAlgebra.finite_doubleCoset For any open U ⊆ GL₂(𝔸_F), Dˣ\GL₂(𝔸_F)/U is finite line 53 source
Source documentation
For any open U ⊆ GL₂(𝔸_F), Dˣ\GL₂(𝔸_F)/U is finite. (where Dˣ is viewed as a subgroup of GL₂(𝔸_F) under the identification M₂(𝔸_F) ≃ D ⊗ 𝔸_F)

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Field, NumberField, DivisionRing, Algebra, Module.Finite, Algebra.IsCentral, IsQuaternionAlgebra.NumberField.WithRigidification, Subgroup, Matrix.GeneralLinearGroup, Fin, OfNat.ofNat, Nat, IsDedekindDomain.FiniteAdeleRing, NumberField.RingOfIntegers, Matrix, IsOpen, Finite, DoubleCoset.Quotient, MonoidHom.range, Units and 1 more
Proof uses: Finite.of_equiv, NumberField.AdeleRing.DivisionAlgebra.Aux.Dfx, NumberField.AdeleRing.DivisionAlgebra.Aux.Df, Set.range, MonoidHom, NumberField.AdeleRing.DivisionAlgebra.Aux.incl₁, Subgroup.comap, Units.map, MulEquiv.toMonoidHom, AlgEquiv.toMulEquiv, TensorProduct, TensorProduct.RightActions.instAlgebra_fLT, IsQuaternionAlgebra.NumberField.WithRigidification.algEquiv, NumberField.FiniteAdeleRing.DivisionAlgebra.finiteDoubleCoset, IsOpen.preimage, TensorProduct.RightActions.instTopologicalSpaceOfFinite_fLT, Units.continuous_map, IsModuleTopology.continuous_of_linearMap, TensorProduct.RightActions.instIsModuleTopology_fLT, AlgEquiv.toLinearMap and 47 more
125 instances, coercions and auxiliary constants left out.

Tactic trace: 6 steps
Open to load the checked before-and-after goal states.
```
lemma finite_doubleCoset
    (hU : IsOpen (U : Set (GL (Fin 2) (FiniteAdeleRing (𝓞 F) F)))) :
    Finite (Dˣ＼GL₂(𝔸 F)／U) :=
…
```
instance TotallyDefiniteQuaternionAlgebra.(instance at line 69) line 69 source