On the computation of algebraic modular forms on compact inner forms of ${\mathrm{GSp}}_4$

Abstract

In this paper, we describe an algorithm for computing algebraic modular forms on compact inner forms of ${\mathrm{GSp}}_4$ over totally real number fields. By analogues of the Jacquet-Langlands correspondence for ${\mathrm{GL}}_2$, this algorithm in fact computes Hecke eigensystems of Hilbert-Siegel modular forms of genus 2. We give some examples of such eigensystems over $\Q(\sqrt{2})$.