Explicit Methods for Hilbert Modular Forms of Weight 1

TL;DR

This paper introduces a novel algorithm for computing Hilbert modular forms of weight one, overcoming previous limitations and enabling calculations over finite fields in all characteristics.

Contribution

The authors develop an algorithm leveraging the algebraic structure of Hilbert modular forms to compute weight one forms, including non-liftable forms, in a unified framework.

Findings

Algorithm successfully computes weight one Hilbert modular forms.

First candidate of a non-liftable weight one form identified.

Conditions for non-liftability of the computed form established.

Abstract

In this article we present an algorithm that uses the graded algebra structure of Hilbert modular forms to compute the adelic $q$-expansion of Hilbert modular forms of weight one as the quotient of Hilbert modular forms of higher weight. The main improvement to existing methods is that our algorithm can be applied in weight $1$, which fills a gap left by standard computational methods. Additionally, the algorithm can be used to compute Hilbert modular forms over finite fields in all characteristic simultaneously. We use this algorithm to compute a first candidate of a Hilbert modular form of parallel weight $1$ that is non-liftable and specify the exact conditions under which our candidate $q$-expansion corresponds to a non-liftable Hilbert Modular form.