Harmonic cocycles and cohomology of arithmetic groups (in positive characteristic)

Abstract

Let $K$ be a global field of characteristic $p>0$. We study the cohomology of arithmetic subgroups $\Gamma$ of $S{L}_{n+1}(K)$ (with respect to a fixed place of $K$), under the hypothesis that these groups have no $p^{\prime }$-torsion (any arithmetic group possesses a normal subgroup of finite index without $p^{\prime }$-torsion). We define the cohomology of $\Gamma$ with compact supports and values in $\mathbb{Z}[1/p]$, and we relate it to spaces of harmonic cocycles, also with compact supports (§3). We give a description of the locus of these supports, in particular by introducing a notion of cusp in dimension $n\ge 1$ (§4) and we calculate "geometrically" the Euler-Poincaré characteristic of this cohomology, up to torsion (§5).