Produits dans la cohomologie des variétés arithmétiques : quelques calculs sur les séries thêta

Details Produits dans la cohomologie des variétés arithmétiques : quelques calculs sur les séries thêta Produits dans la cohomologie des variétés arithmétiques : quelques calculs sur les séries thêta

2006-12-15

arxiv.org/abs/math/0612447v1

Mathematics

Number Theory

Scientific paper

For abelian varieties $A$, in the most interesting cohomology theories $H^* (A)$ is the exterior algebra of $H^1(A)$. In this paper we study a weak generalization of this in the case of arithmetic manifolds associated to orthogonal or unitary groups. In this latter case recall that arithmetic manifolds associated to standard unitary groups $U(p,q)$ ($p\geq q$) over a totally real numberfield have vanishing cohomology in degree $i=1, ..., q-1$ and that, following earlier works of Kazhdan and Shimura, Borel and Wallach constructed in \cite{BorelWallach} non zero degree $q$ cohomology classes. These cohomology classes arise as theta series. After generalizing the construction of these theta series. We prove that arbitrary (up to the obvious obstructions) cup-products of these theta series and their complex conjugates virtually non vanish, i.e. ``up to Hecke translate'', in the cohomology ring.

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