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 innovations are the use of outer forms of \(\text{ GL}_2\) to avoid the parity difficulties with inner forms, and the use of the lifting techniques and “change of weight” results introduced in Barnet-Lamb et al. [J Am Math Soc 24(2):411–469, 2011; Duke Math J 161(8):1521–1580, 2012; Potential automorphy and change of weight, Preprint, 2010].