### Harmonic cocycles and cohomology of arithmetic groups (in positive characteristic), by Marc Reversat

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).

Marc Reversat <reversat@picard.ups-tlse.fr>