Factorization de la cohomologie \'etale p-adique de la tour de Drinfeld

For a finite extension $F$ of ${\mathbf{Q}}_p$, Drinfeld defined a tower of coverings of ${\mathbb{P}}^1\setminus {\mathbb{P}}^1(F)$ (the Drinfeld half-plane). For $F={\mathbf{Q}}_p$, 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 finitness theorem for the arithmetic 'etale cohomology modulo $p$ which is shown by first proving, via a computation of nearby cycles, that this cohomology has finite presentation. This last result holds for all $F$; for $F\mathrm{\ne }{\mathbf{Q}}_p$, it implies that the representations of ${\rm GL}_2(F)$ obtained from the cohomology of the Drinfeld tower are not admissible contrary to the case $F={\mathbf{Q}}_p$.