Produits dans la cohomologie des vari\'et\'es arithm\'etiques : quelques calculs sur les s\'eries th\^eta

For abelian varieties A, in the most interesting cohomology theories, H∗(A) is the exterior algebra of H1(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 ≥ 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 [3] non zero degree q cohomology classes. These cohomology classes arise as theta series. In this note we generalize the construction of these theta series. Applying a work of Kudla [11] we then 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. This fits inside the, partly conjectural, picture drawn in [2].