Weight elimination in Serre-type conjectures

TL;DR

This paper proves the weight elimination part of Serre's conjectures for certain unitary groups, confirming that all modular weights align with Herzig's predictions in generic cases.

Contribution

It establishes the weight elimination direction of Serre's conjectures for $U(n)$, confirming the predicted set of modular weights in generic situations.

Findings

All modular weights for the considered Galois representations are contained in Herzig's predicted set.

Under additional hypotheses, all 'obvious' weights are shown to be modular.

The results apply to forms of $U(n)$ that are compact at infinity and split at places dividing $p$.

Abstract

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.