Non-paritious Hilbert modular forms

LOEFFLER, DAVID; DEMBÉLÉ, LASSINA; PACETTI, ARIEL

MATHEMATISCHE ZEITSCHRIFT

SPRINGER

Lugar: Berlin; Año: 2019 vol. 292 p. 361 - 385

0025-5874

The arithmetic of Hilbert modular forms has been extensively studied under the assumptionthat the forms concerned are ?paritious??all the components of the weight are congruentmodulo 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 thebalance somewhat by studying the arithmetic of non-paritious Hilbert modular eigenforms.On the theoretical side, our starting point is a theorem of Patrikis, which associatesprojective-adic Galois representations to these forms. We show that a general conjecture of Buzzardand Gee actually predicts that a strengthening of Patrikis? result should hold, giving Galoisrepresentations into certain groups intermediate between GL2and PGL2; and we verify thatthe predicted Galois representations do indeed exist. On the computational side, we give analgorithm 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 endthe article with an example