Extremal weights and a tameness criterion for mod Galois representations

  • Daniel Le

    Purdue University, West Lafayette, USA
  • Bao Viet Le Hung

    Northwestern University, Evanston, USA
  • Brandon Levin

    Rice University, Houston, USA
  • Stefano Morra

    Université Paris 8, Villetaneuse, France
Extremal weights and a tameness criterion for mod $p$ Galois representations cover
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Abstract

We study the weight part of Serre’s conjecture for generic -dimensional mod Galois representations. We first generalize Herzig’s conjecture to the case where the field is ramified at and prove the weight elimination direction of the conjecture. We then introduce a new class of weights associated to -dimensional local mod 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.

Cite this article

Daniel Le, Bao Viet Le Hung, Brandon Levin, Stefano Morra, Extremal weights and a tameness criterion for mod Galois representations. J. Eur. Math. Soc. (2024), published online first

DOI 10.4171/JEMS/1513