Non-paritious Hilbert modular forms

• Autores: Dembélé, Lassina; Loeffler, David; Pacetti, Ariel Martín
• Año de publicación: 2019
• Idioma: inglés
• Tipo de recurso: artículo
• Materia: GALOIS REPRESENTATIONS; HILBERT MODULAR FORMS
• Institución: Consejo Nacional de Investigaciones Científicas y Técnicas

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 projectiveℓ-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 GL2 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 an example.
Fil: Dembélé, Lassina. Institute Max Planck For Mathematics; Alemania
Fil: Loeffler, David. University of Warwick; Reino Unido
Fil: Pacetti, Ariel Martín. University of Warwick; Reino Unido. Universidad Nacional de Córdoba. Facultad de Matemática, Astronomía y Física; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Córdoba. Centro de Investigación y Estudios de Matemática. Universidad Nacional de Córdoba. Centro de Investigación y Estudios de Matemática; Argentina