Automorphic forms

We recall some of the definitions of spaces of automorphic forms, from the quaternion algebra project, section Miniproject: Quaternion algebras.

We fix a totally real field \(F\), a totally definite quaternion algebra \(D/F\), and a coefficient (additive abelian) group \(R\). Set \(G=(D\otimes_F\mathbb{A}_F^\infty)^\times\) as in the previous section. Note that \(G\) naturally contains copies of \(D^\times\) and \((\mathbb{A}_F^\infty)^\times\). Recall from definition TotallyDefiniteQuaternionAlgebra.WeightTwoAutomorphicForm that an \(R\)-valued weight 2 automorphic form is a function \(f:G\to R\) satisfying the following axioms:

  1. \(f(dg)=f(g)\) for all \(d\in D^\times\subseteq(D\otimes_F\mathbb{A}_F^\times)\);
  2. There exists a compact open subgroup of \(U\) (the level of \(f\)) such that \(f(gu)=f(g)\) for all \(g\in G\) and \(u\in U\);
  3. \(f(gz)=f(g)\) for all \(z\in(\mathbb{A}_F^\infty)^\times\).

It can be checked that the collection of all such forms is an additive abelian group, and if \(R\) is a ring then it is naturally an \(R\)-module. Let's call this group \(A\) for short. Then \(A\) has a left action of \(G\), with \(g\cdot f\) defined via \((g\cdot f)(x):=f(xg)\). Recall from definition TotallyDefiniteQuaternionAlgebra.WeightTwoAutomorphicFormOfLevel that a weight 2 automorphic form of level \(U\) is simply an element of the fixed points \(A^U\). In other words, the forms of level \(U\) are the forms satisfying the three axioms defining an automorphic form but with the compact open subgroup in the second axiom being \(U\).

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