Crystalline extensions and the weight part of Serre's conjecture

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 Serre weight which occurs is a predicted weight. This completes the analysis begun by Barnet-Lamb, Gee, and Geraghty, who proved that all predicted Serre weights occur. Our methods are a mixture of local and global techniques, and in the course of the proof we use global techniques (as well as local arguments) to establish some purely local results on crystalline extension classes. We also apply these local results to prove similar theorems for the weight part of Serre's conjecture for Hilbert modular forms in the totally ramified case.