Computing Hilbert modular forms over fields with nontrivial class group

TL;DR

This paper extends algorithms for computing Hilbert modular forms to all totally real fields of even degree with nontrivial class group, enabling new insights into modular abelian varieties and the Eichler-Shimura conjecture.

Contribution

It introduces an algorithm applicable to all such fields, broadening computational capabilities and providing new examples related to the Eichler-Shimura conjecture.

Findings

New instances of the Eichler-Shimura conjecture over specific fields

Identification of modular abelian varieties with good reduction everywhere

Extension of computational methods to more complex number fields

Abstract

In previous work, the first author developed an algorithm for the computation of Hilbert modular forms. In this paper, we extend this to all totally real number fields of even degree and nontrivial class group. Using the algorithm over $\Q(\sqrt{10})$ and $\Q(\sqrt{85})$ and their Hilbert class fields, we present some new instances of the conjectural Eichler-Shimura construction for totally real fields, and in particular find new examples of modular abelian varieties with everywhere good reduction.