Serre weights for quaternion algebras

TL;DR

This paper classifies the possible weights of certain mod p Galois representations arising from automorphic forms on quaternion algebras over totally real fields, providing explicit lists in most cases.

Contribution

It determines the exact set of weights for irreducible mod p Galois representations associated with automorphic forms on definite quaternion algebras, extending previous partial results.

Findings

Complete classification of weights in most cases

Explicit list of possible weights with exceptions

Enhanced understanding of mod p Galois representations

Abstract

We study the possible weights of an irreducible two-dimensional mod p representation of the absolute Galois group of F which is modular in the sense of that it comes from an automorphic form on a definite quaternion algebra with centre F which is ramified at all places dividing p, where F is a totally real field. In most cases we determine the precise list of possible weights; in the remaining cases we determine the possible weights up to a short and explicit list of exceptions.