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

TL;DR

This paper develops algorithms to compute automorphic forms on Shimura curves over totally real fields with arbitrary class number, enabling the calculation of Hecke eigenvalues for Hilbert modular forms of any level.

Contribution

It extends existing methods to handle Shimura curves over fields with arbitrary class number, broadening computational capabilities in automorphic forms.

Findings

Successfully computed systems of Hecke eigenvalues for Hilbert modular forms.

Demonstrated the effectiveness of the algorithms with two illustrative examples.

Abstract

We extend methods of Greenberg and the author to compute in the cohomology of a Shimura curve defined over a totally real field with arbitrary class number. Via the Jacquet-Langlands correspondence, we thereby compute systems of Hecke eigenvalues associated to Hilbert modular forms of arbitrary level over a totally real field of odd degree. We conclude with two examples which illustrate the effectiveness of our algorithms.