Serre weights for rank two unitary groups

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].