The Hodge conjecture and arithmetic quotients of complex balls

Abstract

Let $S$ be a closed Shimura variety uniformized by the complex $n$-ball. The Hodge conjecture predicts that every Hodge class in $H^{2k} (S, \Q)$, $k=0,\dots ,n$, is algebraic. We show that this holds for all degree $k$ away from the neighborhood $]n/3,2n/3[$ of the middle degree. We also address the Tate conjecture and the generalized form of the Hodge conjecture and extend most of our results to Shimura varieties associated to unitary groups of any signature. The proofs make use of the recent endoscopic classification of automorphic representations of classical groups by \cite{ArthurBook,Mok}. As such our results are conditional on the stabilization of the trace formula for the (disconnected) groups $\GL (N) \rtimes \langle θ\rangle$ associated to base change. 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.

References in corpus (4)

• La formule des traces locale tordue
• Picard groups of Siegel modular threefolds and theta lifting
• Quelques conséquences des travaux d'Arthur pour le spectre et la topologie des variétés hyperboliques
• On the regularized Siegel-Weil formula (the second term identity) and non-vanishing of theta lifts from orthogonal groups

Cited by in corpus (2)

• On $(\chi ,b)$-factors of Cuspidal Automorphic Representations of Unitary Groups I
• The Relative Lie Algebra Cohomology of the Weil Representation of SO(n,1)