On $p$-ordinary mod $p$ local Langlands correspondences.

Abstract: To a suitably “nice” automorphic representation, we can attach a $p$-adic representation of the absolute Galois group of a number field. We call a Galois representation arising in this way automorphic. One of the goals of the Langlands programme is to classify the image of automorphic Galois representations within the set of all Galois representations, thereby establishing a correspondence.

When $n=2$ and the number field is $\mathbb{Q}$, such a correspondence has been constructed by combining mod $p$ and $p$-adic correspondences with local–global compatibility results. In particular, the $p$-adic correspondence is a representation of $GL_2(\mathbb{Q}_p)$, associated to a local Galois representation, which appears in the cohomology of the modular curve.

In work of Breuil and Herzig, a candidate for a more general correspondence for $p$-ordinary local Galois representations was constructed. In this talk I will discuss joint work with Shu Sasaki in which we construct a framework that should generalise Breuil and Herzig's construction, in particular allowing for the non-generic case.

Mathematics

Audience: researchers in the topic

ANTLR seminar

Series comments: This is the Algebra, Number Theory, Logic and Representation theory seminar.