Serre weights for rank two unitary groups

Abstract

We study the weight part of (a generalisation of) Serre's conjecture for mod l Galois representations associated to automorphic representations on rank two unitary groups for odd primes l. We propose a conjectural set of Serre weights, agreeing with all conjectures in the literature, and under a mild assumption on the image of the mod l Galois representation we are able to show that any modular representation is modular of each conjectured weight. We make no assumptions on the ramification or inertial degrees of l. Our main innovation is to make use of the lifting techniques introduced in [BLGG11], [BLGG10] and [BLGGT10].

Crystalline extensions and the weight part of Serre’s conjecture

“…(r ). It had been previously supposed that this was the easier direction; indeed, just as in the classical case, the results of [Barnet-Lamb et al 2011] reduce the weight part of Serre's conjecture for these unitary groups to a purely local problem in p-adic Hodge theory. However, this problem has proved to be difficult, and so far only fragmentary results are known.…”

“…Conjectural descriptions of the set of Serre weights were made in increasing generality in [Buzzard et al 2010;Schein 2008;, and cases of these conjectures were proved in [Gee 2011;Gee and Savitt 2011a]. Most recently, significant progress was made towards completely establishing the conjecture for rank-two unitary groups in [Barnet-Lamb et al 2011]. We briefly recall this result.…”

“…We briefly recall this result. Let p > 2 be prime, F a CM field, and r : G F → GL 2 ‫ކ(‬ p ) a modular representation (see [Barnet-Lamb et al 2011] for the precise definition of "modular", which is in terms of automorphic forms on compact unitary groups). There is a conjectural set W ?…”

“…(r ) of Serre weights in whichr is predicted to be modular, which is defined in Section 2, following [Gee et al 2012]. Then the main result of [Barnet-Lamb et al 2011] is that under mild technical hypotheses,r is modular of every weight in W ? (r ).…”

“…While [Barnet-Lamb et al 2011] reduced this result to a purely local problem, our methods are not purely local; in fact we use the main result of [ibid. ], together with potential automorphy theorems, as part of our proof.…”