Related papers: Local-global compatibility of mod $p$ Langlands pr…

We construct global cohomology classes in the middle degree cohomology of the Shimura variety of the symplectic group $GSp_6$ compatible when one varies the level at $p$. These classes are expected constituents of an Euler system for the…

Let $F$ be a CM field. In this paper, we prove the local-global compatibility for cohomological cuspidal automorphic representations of $\mathrm{GL}_n(\mathbb{A}_F)$ at $p \neq l$ by using certain potential automorphy theorems in some cases…

We construct universal $G$-zips on good reductions of the Pappas-Rapoport splitting models for PEL-type Shimura varieties. We study the induced Ekedahl-Oort stratification, which sheds new light on the mod $p$ geometry of splitting models.…

Let $p$ be a prime number and $F$ a totally real number field unramified at places above $p$. Let $\bar{r}:\operatorname{Gal}(\bar F/F)\rightarrow\operatorname{GL}_2(\bar{\mathbb{F}_p})$ be a modular Galois representation which satisfies…

Let $p$ be a prime number, $n>2$ an integer, and $F$ a CM field in which $p$ splits completely. Assume that a continuous automorphic Galois representation…

We study the locally analytic theory of infinite level local Shimura varieties. As a main result, we prove that in the case of a duality of local Shimura varieties, the locally analytic vectors of different period sheaves at infinite level…

The main purpose of this article is to give the integral cohomology of classical principal congruence subgroups in SL(2,Z) as well as their analogues in the third braid group with local coefficients in symmetric powers of the natural…

The goal of these lecture notes is to survey progress on the global Langlands reciprocity conjecture for $\mathrm{GL}_n$ over number fields from the last decade and a half. We highlight results and conjectures on Shimura varieties and more…

In this paper we generalize the work of Harris-Soudry-Taylor and construct the compatible systems of two-dimensional Galois representations attached to cuspidal automorphic representations of cohomological type on GL_2 over a CM field with…

This article contributes to the study of the generic part of the cohomology of Shimura varieties. Under a mild restriction of the characteristic of the coefficient field, we prove a torsion vanishing result for Shimura varieties of abelian…

We propose a conjectural theory of $p$-integral models of Shimura varieties with level structure at $p$ given by a class of normal subgroups of parahoric subgroups with abelian quotient group. The role of the theory of local models is…

This is a survey article that advertizes the idea that there should exist a theory of p-adic local analogues of Shimura varieties. Prime examples are the towers of rigid-analytic spaces defined by Rapoport-Zink spaces, and we also review…

We state a conjecture, local Langlands in families, connecting the centre of the category of smooth representations on $\mathbb{Z}[\sqrt{q}^{-1}]$-modules of a quasi-split $p$-adic group $\mathrm{G}$ (where $q$ is the cardinality of the…

Given a KHT Shimura variety provided with an action of its unramified Hecke algebra $\mathbb T$, we proved in a previous work, see also the work of Caraiani-Scholze for other PEL Shimura varieties, that its localized cohomology groups at a…

We introduce a new collection of partially global Galois cohomology classes subsuming both plectic Heegner points and mock plectic invariants. The former are recovered as localizations of plectic Heegner classes, while the latter arise as…

In a previous paper [CG], we showed how one could generalize Taylor-Wiles modularity lifting theorems [Wil95, TW95] to contexts beyond those in which the automorphic forms in question arose from the middle degree cohomology of Shimura…

We prove that the Jacquet-Langlands correspondence for cohomological automorphic forms on quaternionic Shimura varieties is realized by a Hodge class. Conditional on Kottwitz's conjecture for Shimura varieties attached to unitary similitude…

For a cuspidal automorphic representation of GL2/Q associated to a modular form, the local and global Langlands correspondences are compatible at all finite places of Q. On the p-adic Coleman-Mazur eigencurve this principle can fail (away…

Mixed-parity module emerges for instance when a de Rham Galois representation is being tensored with a square root of cyclotomic character, which produces half odd integers as the corresponding Hodge-Tate weights. We build the whole…

We introduce a method for producing congruences between Hecke eigenclasses, possibly torsion, in the coherent cohomology of automorphic vector bundles on certain good reduction Shimura varieties. The congruences are produced using some…