Pour deux nombres premiers distincts $\ell$ et $p$, nous étudions la ${\mathbb{Z}}_{\ell }$-cohomologie des tours de Lubin–Tate d’un corps $p$-adique. Nous prouvons tout d’abord qu’elle réalise des correspondances de type Langlands et Jacquet–Langlands pour des familles plates de representations irréductibles supercuspidales paramétrées par une ${\mathbb{Z}}_{\ell }$-algèbre $R$. Lorsque $R={\overline{\mathbb{F}}}_{\ell }$ on obtient une réalisation cohomologique de la correspondance de Langlands–Vignéras, et une nouvelle preuve de son existence. Pour $R$ une algèbre locale, on obtient des correspondances entre déformations de ${\overline{\mathbb{F}}}_{\ell }$-représentations. Par ailleurs nous obtenons, pour toutes les ${\overline{\mathbb{F}}}_{\ell }$-représentations admissibles irréductibles, une réalisation virtuelle de la correspondance de Langlands–Vignéras semi-simple et du transfert de Langlands–Jacquet, en utilisant le complexe de cohomologie et en travaillant dans une catégorie dérivée convenable.

For two primes $\ell \ne p$, we investigate the ${\mathbb{Z}}_{\ell }$-cohomology of the Lubin–Tate towers of a $p$-adic field. We prove that it realizes some version of Langlands and Jacquet–Langlands correspondences for flat families of irreducible supercuspidal representations parameterized by a ${\mathbb{Z}}_{\ell }$-algebra $R$ in a way compatible with the extension of scalars. Applied to $R={\overline{\mathbb{F}}}_{\ell }$, this gives a cohomological realization of the Langlands–Vigneras correspondence for supercuspidals and a new proof of its existence. Applied to complete local algebras, this provides bijections between deformations of matching ${\overline{\mathbb{F}}}_{\ell }$-representations. Besides, we also get a virtual realization of both the semi-simple Langlands–Vigneras correspondence and the $\ell$-modular Langlands–Jacquet transfer for all representations, by using the cohomology complex and working in a suitable Grothendieck group.

Let $S$ be a closed surface of genus at least $2$, and let $\lambda$ and $\mu$ be two laminations that fill $S$. Let $E_r^{\lambda }$ and $E_r^{\mu }$ be the right earthquakes on $\lambda$ and $\mu$, respectively. We show that the composition $E_r^{\lambda }\circ E_r^{\mu }$ has a fixed point in the Teichmüller space of $S$. Another way to state this result is that it is possible to prescribe any two measured laminations that fill a surface as the upper and lower measured bending laminations of the convex core of a globally hyperbolic anti-de Sitter (AdS) manifold. The proof uses some estimates from the geometry of those AdS manifolds.

We apply some of the ideas of Margulis’s Ph.D. dissertation to Teichmüller space. Let $X$ be a point in Teichmüller space, and let $B_R\left(X\right)$ be the ball of radius $R$ centered at $X$ (with distances measured in the Teichmüller metric). We obtain asymptotic formulas as $R$ tends to infinity for the volume of $B_R\left(X\right)$, and also for the cardinality of the intersection of $B_R\left(X\right)$ with an orbit of the mapping class group.

Given a compact PEL-type Shimura variety, a sufficiently regular weight (defined by mild and effective conditions), and a prime number $p$ unramified in the linear data and larger than an effective bound given by the weight, we show that the (Betti) cohomology with ${\mathbb{Z}}_p$-coefficients of the given weight vanishes away from the middle degree, and hence has no $p$-torsion. We do not need any other assumption (such as ones on the images of the associated Galois representations).