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

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

1999-10-07

arxiv.org/abs/math/9910190v1

Mathematics

Number Theory

Scientific paper

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).

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