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 \(\Q(\sqrt{2})\).