Crystalline extensions and the weight part of Serre's conjecture

TL;DR

This paper proves the weight part of Serre's conjecture for rank two unitary groups in the totally ramified case, confirming that all predicted weights actually occur, using a combination of local and global methods.

Contribution

It completes the proof of the weight part of Serre's conjecture in a specific ramified case, introducing new local results on crystalline extension classes.

Findings

All predicted weights occur for the considered case.

Established new local results on crystalline extension classes.

Confirmed the conjecture in the totally ramified setting.

Abstract

Let p>2 be prime. We complete the proof of the weight part of Serre's conjecture for rank two unitary groups for mod p representations in the totally ramified case, by proving that any weight which occurs is a predicted weight. Our methods are a mixture of local and global techniques, and in the course of the proof we establish some purely local results on crystalline extension classes.