Serre weights for mod p Hilbert modular forms: the totally ramified case

TL;DR

This paper characterizes the possible weights of certain mod p Galois representations over totally real fields that are totally ramified at p, providing explicit classifications in most cases and detailed exceptions.

Contribution

It offers a comprehensive classification of Serre weights for mod p Hilbert modular forms in the totally ramified case, including explicit exception lists.

Findings

Complete list of possible weights in most cases

Explicit exceptions identified for remaining cases

Enhanced understanding of Galois representations at ramified primes

Abstract

We study the possible weights of an irreducible 2-dimensional modular mod p representation of the absolute Galois group of F, where F is a totally real field which is totally ramified at p, and the representation is tamely ramified at the prime above p. 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.