Factorisation de la cohomologie etalep-adique de la tour de Drinfeld

Resume For a finite extension F of , Drinfeld defined a tower of coverings of (the Drinfeld half-plane). For , we describe a decomposition of the p-adic geometric etale cohomology of this tower analogous to Emerton's decomposition of completed cohomology of the tower of modular curves. A crucial ingredient is a finiteness theorem for the arithmetic etale cohomology modulo p whose proof uses Scholze's functor, global ingredients, and a computation of nearby cycles which makes it possible to prove that this cohomology has finite presentation. This last result holds for all F; for , it implies that the representations of obtained from the cohomology of the Drinfeld tower are not admissible contrary to the case .