The Buzzard-Diamond-Jarvis conjecture for unitary groups

TL;DR

This paper proves the weight part of Serre's conjecture for rank two unramified unitary groups at primes greater than 2, confirming that all predicted weights occur and completing prior analyses.

Contribution

It establishes the weight part of Serre's conjecture for unramified rank two unitary groups using purely local methods, confirming predicted Serre weights.

Findings

Confirmed all predicted Serre weights occur for the case

Used (phi,Ghat)-modules to analyze crystalline representations

Completed the proof of the Buzzard-Diamond-Jarvis conjecture in this setting

Abstract

Let p > 2 be prime. We prove the weight part of Serre's conjecture for rank two unitary groups for mod p representations in the unramified case (that is, the Buzzard-Diamond-Jarvis conjecture for unitary groups), by proving that any Serre weight which occurs is a predicted weight. This completes the analysis begun in [BLGG11], which proved that all predicted Serre weights occur. Our methods are purely local, using the theory of (phi,Ghat)-modules to determine the possible reductions of certain two-dimensional crystalline representations.