Lecture 1:

Computing Hilbert modular forms: overview and examples

Lecture 2:

Computing fundamental domains for cofinite Fuchsian groups

Lecture 3:

Computing automorphic forms on Shimura curves over fields with arbitrary class number

John Voight, University of Vermont 

In these lectures, we exhibit an algorithm to compute spaces of Hilbert modular forms as Hecke modules.  Our algorithm uses the Jacquet-Langlands correspondence to locate systems of Hecke eigenvalues in the cohomology of Shimura varieties of dimensions 0 and 1, associated to quaternion algebras.  In the first lecture, we given an overview of this method, focusing on the case of Shimura curves (dimension 1), and provide several examples.  In the second lecture, we describe the key components of the algorithm in detail, including the computation of a (Dirichlet) fundamental domain and some parts of the algorithmic theory of quaternion orders.  Finally, in the third lecture, we show how to make the adelic theory amenable to explicit computation and thereby extend the method to compute in the cohomology of a Shimura curve defined over a totally real field with arbitrary class number.