$ell$-modular Deligne representations of the Weil group, local Langlands correspondence and local constants (Automorphic forms, automorphic representations and related topics)

Abstract

This is an expositary note on our papers [6] and [7] joint with R. Kurinczuk, on which I gave a talk at the RIMS conference "Automorphic forms, automorphic representations and related topics" in January 2019. Let F be a non-archimedean local field of residual characteristic p and WF its Weil group. For l a prime number different from p, we classify equivalence classes of WF-semi-simple Deligne l-modular representations in terms of isomorphism classes of irreducible l-modular representations of WF. After extending to the l-modular setting the constructions of local constants by Jacquet, Piatetski-Shapiro and Shalika on the reductive group side, and by Artin and Deligne on the Weil group side, we define a variant of the l-modular local Langlands correspondence of Vignéras which satisfies a preservation of local constants property for pairs of generic representations.