General Serre weight conjectures

TL;DR

This paper proposes and discusses generalizations of Serre's conjecture on modular forms to higher dimensions and arbitrary number fields, supported by evidence and connections to existing conjectures.

Contribution

It formulates new conjectures extending Serre's weight conjectures to GL(n) over number fields and unramified reductive groups, and explores their relationships and evidence.

Findings

Evidence supports the proposed conjectures.

Generalizations unify previous conjectures in a broader setting.

Conjectures are shown to be consistent in generic cases.

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-M\'ezard 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 Q_p, and we also generalise the second author's previous conjecture for GL(n)/Q to this setting, and show that the two conjectures are generically in agreement.