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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.