Research proof index

The matrix reps of φ and f φ agree.

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)

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

Each coset in U1diagU1 is of the form unipotent_mul_diagU1 for some t ∈ O_v.

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

Let R₀ be a topological ring, topologically of finite type (over ℤ). Consider a family of (cardinality) finite rings R i with the discrete topology whose cardinalites are unifomly bounded. Given a family of continuous ring homs f i : R →+* R i, the lift R →+* 𝒰(Rᵢ) is also continuous.