Title: Categorical p-adic Langlands correspondence for GL2(Qp) and applications.

Abstract: In joint work with Emerton and Gee we have constructed a categorical version of the p-adic local Langlands correspondence for GL2(Qp), taking the form of a fully faithful embedding of the derived category of smooth p-power torsion representations of this group into a suitable category of Ind-coherent sheaves on a stack of (phi, Gamma)-modules. I will present our construction, and discuss some motivating applications to local-global compatibility for the completed cohomology of modular curves.

Title: Finite length for unramified GL2: beyond multiplicity one

Abstract: For K a finite unramified extension of Qp, natural places to look for a conjectural mod p Langlands correspondence for GL2(K) are Hecke eigenspaces in the mod p cohomology of a tower of Shimura curves. These are hoped to be a direct sum of copies of purely local representation of GL2(K) (i.e. depending only on the restriction at p of the Galois representation attached to the system of Hecke eigenvalues). The number of copies (called the multiplicity) should then be computed by counting modular Serre weights in the socle. In the multiplicity one case, a recent work of Breuil, Herzig, Hu, Morra and Schraen establishes the finite length of the above Hecke eigenspaces, under mild genericity assumptions. In this talk we will discuss how to extend this finite length result when the multiplicity one assumption is removed.

Title: Looking for a mod p local Langlands correspondence

Abstract: The mod p local Langlands program aims at a parametrization of smooth mod p representations of a p-adic GLn in terms of mod p continuous n-dimensional Galois representations of the p-adic field.

This was realized in the case of GL2(Qp) (and continuous 2-dimensional representations of Gal_{Qp}) by work of Colmez, following work of Breuil, and its compatibility with the mod p cohomology of (a tower above p of) modular curves was established by Emerton, all in the early 2000.

The situation for GL2 over an extension of Qp, even an unramified extension, remains unclear, even though many partial results show that such a local correspondence should exist. In particular, one of the main questions is whether the mod p Hecke eigenspaces of a tower of Shimura curves at p are "purely local", i.e. only depend on the underlying local Galois representations and not on the global context. Indeed, such a locality would give strong evidence for the existence of a mod p local Langlands correspondence outside of GL2(Qp) compatible with global constructions.

In this talk I will report on a joint work in progress, where we aim at constructing natural candidates for such a correspondence using tools from perfectoid geometry.

This is joint work in progress with C. Breuil, F. Herzig, Y. Hu, K. Koziol, B. Schraen and S-W. Shin.

Title: On p-ordinary, mod p local Langlands correspondences.

Abstract: The now established p-adic correspondence for GL2(Qp) associates to a local Galois representation a representation of GL2(Qp) 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 purely local framework which should generalise Breuil and Herzig's mod p results, in particular allowing for the non-generic case. We will examine our construction in a maximally non-split example highlighting the representations that can occur in the graded pieces.

Title: Resolving moduli spaces of crystalline representations and modularity

Abstract: In 2004, Kisin proved modularity lifting theorems for two-dimensional Barsotti-Tate representations of totally real fields. A key ingredient in his proof was the construction of resolutions of moduli spaces of crystalline representations of finite extensions of Qp using p-adic Hodge-theoretic data.

In this talk I will discuss recent joint work with Bao Le Hung and Brandon Levin which extends these results to three-dimensional Galois representations of minimal regular weight, with a focus on the role played by certain affine Springer loci inside the affine Grassmannian. In particular, I will indicate how sufficient control of the singularities of these loci, which we obtain for the quasi-minuscule coweight (2,1,0), largely reduces the problem to a dimension estimate.

Title: Local-global compatibility at l=p for automorphic Galois representations over CM fields

Abstract: In joint work with Hevesi, Thorne and Whitmore we prove that the Galois representations associated with cohomological cuspidal automorphic representations over CM fields are potentially semi-stable and compatible with the local Langlands correspondence, up to semisimplification. The novelty of our work is that we make no assumptions on residual Galois representation. Our method relies on a bound on the torsion in the cohomology of certain Shimura varieties, which can be seen as a generalisation of the Caraiani-Scholze vanishing theorem.

Title: Counting points on local Shimura varieties

Abstract: Kottwitz's classical point counting formula expresses the Frobenius-Hecke traces on the cohomology of Shimura varieties in terms of certain (twisted) orbital integrals. This forms a key step in Langlands' strategy to relate the L-function of Shimura varieties to automorphic L-functions. In this talk, I will present a new local analogue of Kottwitz's formula, which relates the cohomology of local Shimura varieties to twisted orbital integrals. This local formula bridges Kottwitz's global formula with the point-counting formula for Igusa varieties. As an application of our formula, we explain a new purely local approach, based on categorical Langlands, towards Rapoport's vanishing conjecture for certain twisted orbital integrals. This conjecture is itself a key ingredient in the global Langlands-Kottwitz method at a non-quasi-split prime. This is based on joint work in progress with Yihang Zhu.