Miriam Norris

On p-ordinary mod p local Langlands correspondences.

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 goal of the Langlands programme is to classify the image of automorphic Galois representations in the set of all Galois representations, establishing a correspondence. When n = 2 and the number field is the rationals, a correspondence was built combining mod p and p-adic correspondences with local-global compatibility results. In particular, the p-adic correspondence in this case is a representation of GL2(Qp), associated to a local Galois representation, which occurs 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 of myself and Shu Sasaki in which we construct a framework which should generalise Breuil and Herzig’s mod p construction, in particular allowing for the non-generic case.

This is the last Heilbronn Number Theory seminar of the term. The seminar will provisionally resume on 14 January 2026.