We prove the weight elimination direction of the Serre weight conjectures as formulated by Herzig for forms of which are compact at infinity and split at places dividing in generic situations. That is, we show that all modular weights for a mod Galois representation are contained in the set predicted by Herzig. Under some additional hypotheses, we also show modularity of all the “obvious” weights.
"Weight elimination in Serre-type conjectures." Duke Math. J. 168 (13) 2433 - 2506, 15 September 2019. https://doi.org/10.1215/00127094-2019-0015