Finite double Coset
TotallyDefiniteQuaternionAlgebra.finite_doubleCoset
Plain-language statement
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)
Exact Lean statement
lemma finite_doubleCoset
(hU : IsOpen (U : Set (GL (Fin 2) (FiniteAdeleRing (𝓞 F) F)))) :
Finite (Dˣ＼GL₂(𝔸 F)／U)Formal artifact
Lean source
lemma finite_doubleCoset (hU : IsOpen (U : Set (GL (Fin 2) (FiniteAdeleRing (𝓞 F) F)))) : Finite (Dˣ＼GL₂(𝔸 F)／U) := (NumberField.FiniteAdeleRing.DivisionAlgebra.finiteDoubleCoset F D (U := U.comap (Units.map (WithRigidification.algEquiv ..).toMonoidHom)) (hU.preimage (Units.continuous_map (by exact IsModuleTopology.continuous_of_linearMap (WithRigidification.algEquiv F D).toLinearMap)))).of_equiv _ (DoubleCoset.quotientEquiv _ _ _ _ (Units.mapEquiv (WithRigidification.algEquiv ..).toMulEquiv) (by rw [← MulEquiv.coe_toEquiv, Equiv.preimage_eq_iff_eq_image, ← Set.range_comp] dsimp congr 1 ext x : 2 simp [WithRigidification.algEquiv]) rfl)- Project
- Fermat's Last Theorem
- License
- Apache-2.0
- Commit
- 8dd808888295
- Source
- FLT/AutomorphicForm/QuaternionAlgebra/FiniteDimensional.lean:52-66
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