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

TL;DR

This paper presents an algorithm to compute algebraic modular forms on compact inner forms of GSp_4 over totally real fields, enabling the calculation of Hecke eigensystems of Hilbert-Siegel modular forms of genus 2.

Contribution

It introduces a novel algorithm for computing algebraic modular forms on GSp_4 inner forms, extending computational capabilities for Hilbert-Siegel modular forms.

Findings

Successfully computed Hecke eigensystems over Q(√2)

Demonstrated the algorithm's effectiveness on specific examples

Extended computational methods to totally real fields

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