Computing systems of Hecke eigenvalues associated to Hilbert modular forms

TL;DR

This paper presents algorithms for computing systems of Hecke eigenvalues linked to Hilbert modular forms over totally real fields, leveraging cohomology of Shimura curves and the Jacquet-Langlands correspondence.

Contribution

It introduces effective algorithms combining Shimura curve cohomology and Jacquet-Langlands correspondence to compute Hecke eigenvalues for Hilbert modular forms.

Findings

Successful computation of Hecke eigenvalues for specific cases

Demonstration of the effectiveness of the algorithms

Potential applications to number theory and automorphic forms

Abstract

We utilize effective algorithms for computing in the cohomology of a Shimura curve together with the Jacquet-Langlands correspondence to compute systems of Hecke eigenvalues associated to Hilbert modular forms over a totally real field.