A Serre weight conjecture for geometric Hilbert modular forms in characteristic
Fred Diamond
King’s College London, UKShu Sasaki
Queen Mary University of London, UK
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Abstract
Let be a prime and a totally real field in which is unramified. We consider mod Hilbert modular forms for , defined as sections of automorphic line bundles on Hilbert modular varieties of level prime to in characteristic . For a mod Hilbert modular Hecke eigenform of arbitrary weight (without parity hypotheses), we associate a two-dimensional representation of the absolute Galois group of , and we give a conjectural description of the set of weights of all eigenforms from which it arises. This conjecture can be viewed as a “geometric” variant of the “algebraic” Serre weight conjecture of Buzzard–Diamond–Jarvis, in the spirit of Edixhoven's variant of Serre's original conjecture in the case . We develop techniques for studying the set of weights giving rise to a fixed Galois representation, and prove results in support of the conjecture, including cases of partial weight 1.
Cite this article
Fred Diamond, Shu Sasaki, A Serre weight conjecture for geometric Hilbert modular forms in characteristic . J. Eur. Math. Soc. 25 (2023), no. 9, pp. 3453–3536
DOI 10.4171/JEMS/1265