Extremal weights and a tameness criterion for mod $p$ Galois representations

We study the weight part of Serre’s conjecture for generic n -dimensional mod p Galois representations. We first generalize Herzig’s conjecture to the case where the field is ramified at p and prove the weight elimination direction of the conjecture. We then introduce a new class of weights associated to n -dimensional local mod p representations which we call extremal weights . Using a “Levi reduction” property of certain potentially crystalline Galois deformation spaces, we prove the modularity of these weights. As a consequence, we deduce the weight part of Serre’s conjecture for unit groups of some division algebras in generic situations.