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Inspired by [Pan21], we define and study a functor which associates to equivariant sheaves on P 1 automorphic sheaves on modular curves., Inspiré par [Pan21], on définit et étudie un foncteur qui associe à des faisceaux équivariants sur P 1 des faisceaux automorphes sur les courbes modulaires.

Inspired by [Pan21], we define and study a functor which associates to equivariant sheaves on $\mathbb{P}^1$ automorphic sheaves on modular curves., Comment: in French

We develop local cohomology techniques to study the finite slope part of the coherent cohomology of Shimura varieties. The local cohomology groups we consider are a generalization of overconvergent modular forms, and they are defined by using a stratification on the Shimura variety obtained from the Bruhat stratification on a flag variety via the Hodge-Tate period map. We construct a spectral sequence from the local cohomologies to the classical cohomology and use it to obtain classicality and vanishing results. We also develop a theory of p-adic families and construct eigenvarieties. As an application, we prove some new properties of Galois representations arising from certain non-regular algebraic cuspidal automorphic representations.

We construct Hida and Coleman theories for the degree 0 and 1 cohomology of automorphic line bundles on the modular curve and we define a p-adic duality pairing between the theories in degree 0 and 1.

The reduction of Siegel varieties modulo a prime number p is stratified by the multiplicative rank of the p-divisible group of the universal abelian variety. For r\geq 0 the maximal multiplicative subgroup of the restriction of the p-torsion group of the universal abelian variety to the r-th stratum lifts canonically to the tube of this stratum and defines a partial canonical subgroup of rank r. We prove that this subgroup extends in a finite flat way on some strict neighborhood of the tube. On the ordinary stratum and on its neighborhood, we recover the usual canonical subgroup considered by Abbes and Mokrane, and Andreatta and Gasbarri. ----- La reduction des varietes de Siegel modulo un nombre premier p est stratifiee par le rang multiplicatif du groupe p-divisible de la variete abelienne universelle. Pour r\geq 0, le sous-groupe multiplicatif maximal de la restriction du groupe de p-torsion de la variete abelienne universelle a la r-ieme strate se releve canoniquement sur le tube de cette strate et definit un sous-groupe canonique partiel de rang r. Nous montrons qu'il existe un voisinage strict du tube sur lequel ce sous-groupe s'etend de maniere finie et plate. Sur la strate ordinaire et au voisinage de celle-ci, on retrouve le sous-groupe canonique usuel etudie par Abbes et Mokrane d'une part, Andreatta et Gasbarri d'autre part.

We consider the problem of defining an action of Hecke operators on the coherent cohomology of certain integral models of Shimura varieties. We formulate a general conjecture describing which Hecke operators should act integrally and solve the conjecture in certain cases. As a consequence, we obtain p-adic estimates of Satake parameters of certain nonregular self-dual automorphic representations of $\mathrm {GL}_n$ .

We investigate the p-adic properties of higher coherent cohomology of au-tomorphic vector bundles of singular weight on the Siegel threefolds. As an application we prove a modular lifting theorem for certain abelian surfaces defined over the rationals.

Nous démontrons un théorème de relévement modulaire pour des représentations galoisiennes p-adiques de dimension 2, non-ramifiées en p, des corps totalement réels peu ramifiés en p.