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 $SL_{n+1}(K)$ (with respect to a fixed place of $K$), under the hypothesis that these groups have no $p'$-torsion (any arithmetic group possesses a normal subgroup of finite index without $p'$-torsion). We define the cohomology of $\Gamma $ with compact supports and values in ${\Bbb Z}[1/p]$, and we relate it to spaces of harmonic cocycles, also with compact supports (\S 3). We give a description of the locus of these supports, in particular by introducing a notion of cusp in dimension $n\geq 1$ (\S 4) and we calculate "geometrically" the Euler-Poincar\'e characteristic of this cohomology, up to torsion (\S 5).