John Voight (Vermont)

Computing automorphic forms on Shimura curves

Modular symbols allow a detailed investigation of classical modular forms. Automorphic forms on more general Shimura varieties have proved no less interesting to study, and yet many aspects of the theory are not as well understood. In joint work with Matt Greenberg, we present an algorithm to compute automorphic forms on Shimura curves working with explicit group cohomology; via the Jacquet-Langlands correspondence, these methods also allow the computation of Hilbert modular forms over totally real fields of odd degree. Our methods work without reference to cusps or a canonical moduli interpretation of the Shimura curve, as these features of the classical situation are absent.