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For a finite extension $F$ of ${\mathbf Q}_p$, Drinfeld defined a tower ofcoverings of ${\mathbb P}^1\setminus {\mathbb P}^1(F)$ (the Drinfeldhalf-plane). For $F = {\mathbf Q}_p$, we describe a decomposition of the$p$-adic geometric \'etale cohomology of this tower analogous to Emerton'sdecomposition of completed cohomology of the tower of modular curves. A crucialingredient is a finitness theorem for the arithmetic \'etale cohomology modulo$p$ which is shown by first proving, via a computation of nearby cycles, thatthis cohomology has finite presentation. This last result holds for all $F$;for $F\neq {\mathbf Q}_p$, it implies that the representations of ${\rmGL}_2(F)$ obtained from the cohomology of the Drinfeld tower are not admissiblecontrary to the case $F = {\mathbf Q}_p$.

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