General Serre weight conjectures

Abstract

We formulate a number of related generalisations of the weight part of Serre’s conjecture to the case of GL${}_n$ over an arbitrary number field, motivated by the formalism of the Breuil–Mézard conjecture. We give evidence for these conjectures, and discuss their relationship to previous work. We generalise one of these conjectures to the case of connected reductive groups which are unramified over ${\mathbb{Q}}_p$, and we also generalise the second author’s previous conjecture for GL${}_n/\mathbb{Q}$ to this setting, and show that the two conjectures are generically in agreement.