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

Content Provider	Semantic Scholar
Author	Bergeron, Nicolas
Copyright Year	2006
Abstract	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.
File Format	PDF HTM / HTML
Alternate Webpage(s)	https://webusers.imj-prg.fr/~nicolas.bergeron/Travaux_files/THETA8.pdf
Alternate Webpage(s)	https://arxiv.org/pdf/math/0612447v1.pdf
Language	English
Access Restriction	Open
Content Type	Text
Resource Type	Article