Hodge type theorems for arithmetic manifolds associated to orthogonal groups

Details Hodge type theorems for arithmetic manifolds associated to orthogonal groups Hodge type theorems for arithmetic manifolds associated to orthogonal groups

2011-10-13

arxiv.org/abs/1110.3049v2

Mathematics

Number Theory

81 pages

Scientific paper

We show that special cycles generate a large part of the cohomology of locally symmetric spaces associated to orthogonal groups. We prove in particular that classes of totally geodesic submanifolds generate the cohomology groups of degree $n$ of compact congruence $p$-dimensional hyperbolic manifolds "of simple type" as long as $n$ is strictly smaller than $\frac12 [\frac{p}{2}]$. We also prove that for connected Shimura varieties associated to $\OO (p,2)$ the Hodge conjecture is true for classes of degree $< 1/2 [\frac{p+1}{2}]$. The proof of our general theorem makes use of the recent endoscopic classification of automorphic representations of orthogonal groups by \cite{ArthurBook}. As such our results are conditional on the stabilization of the trace formula for the (disconnected) groups $\GL (N) \rtimes <\theta>$ and $\SO(2n) \rtimes <\theta'>$ (where $\theta$ and $\theta'$ are the outer automorphisms), see \cite[Hypothesis 3.2]{ArthurBook}. Unfortunately, at present the stabilization of the trace formula has been proved only for the case of {\it connected} groups. The extension needed is part of work in progress by the Paris-Marseille team of automorphic form researchers. For more detail, see the second paragraph of subsection \ref{org2} below.

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