Weight elimination in serre-type conjectures

We prove the weight elimination direction of the Serre weight conjectures as formulated by Herzig for forms of U.n/ which are compact at infinity and split at places dividing p in generic situations. That is, we show that all modular weights for a mod p Galois representation are contained in the set predicted by Herzig. Under some additional hypotheses, we also show modularity of all the "obvious" weights.