Non-paritious Hilbert modular forms

TL;DR

This paper explores the arithmetic of non-paritious Hilbert modular forms, providing new theoretical insights into associated Galois representations and introducing a novel computational algorithm using quaternion algebras.

Contribution

It establishes the existence of predicted Galois representations for non-paritious forms and presents the first general algorithm for computing these forms.

Findings

Verified the existence of certain Galois representations predicted by conjectures.

Developed a new algorithm for computing non-paritious Hilbert modular forms.

Provided examples illustrating the theoretical and computational results.

Abstract

The arithmetic of Hilbert modular forms has been extensively studied under the assumption that the forms concerned are "paritious" -- all the components of the weight are congruent modulo 2. In contrast, non-paritious Hilbert modular forms have been relatively little studied, both from a theoretical and a computational standpoint. In this article, we aim to redress the balance somewhat by studying the arithmetic of non-paritious Hilbert modular eigenforms. On the theoretical side, our starting point is a theorem of Patrikis, which associates projective l-adic Galois representations to these forms. We show that a general conjecture of Buzzard and Gee actually predicts that a strengthening of Patrikis' result should hold, giving Galois representations into certain groups intermediate between GL(2) and PGL(2), and we verify that the predicted Galois representations do indeed exist. On the computational side, we give an algorithm to compute non-paritious Hilbert modular forms using definite quaternion algebras. To our knowledge, this is the first time such a general method has been presented. We end the article with a selection of examples.