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

TL;DR

This paper studies the p-adic étale cohomology of Drinfeld's tower over a p-adic field, revealing a decomposition similar to modular curves and showing differences in representation admissibility for fields other than Q_p.

Contribution

It provides a decomposition of the p-adic étale cohomology of Drinfeld's tower for F=Q_p and proves a finiteness theorem for the cohomology modulo p, extending to all F.

Findings

Decomposition of cohomology analogous to Emerton's for Q_p

Finiteness of arithmetic étale cohomology modulo p

Non-admissibility of certain representations for F ≠ Q_p

Abstract

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 $F\neq {\mathbf Q}_p$, 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 $F = {\mathbf Q}_p$.