The Hodge conjecture and arithmetic quotients of complex balls

TL;DR

This paper proves that Hodge classes are algebraic in most degrees for certain Shimura varieties related to complex balls, using automorphic representation classification, with results conditional on trace formula stabilization.

Contribution

It extends the verification of the Hodge conjecture to degrees outside a middle range for Shimura varieties associated with unitary groups, utilizing recent automorphic classification results.

Findings

Hodge classes are algebraic outside the middle degree range

Results extend to Shimura varieties of unitary groups of any signature

Conditional on the stabilization of the trace formula for certain groups

Abstract

Let $S$ be a closed Shimura variety uniformized by the complex $n$-ball. The Hodge conjecture predicts that every Hodge class in $H^{2k} (S, \Q)$, $k=0, \ldots, n$, 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 $\GL (N) \rtimes \langle \theta \rangle$ 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.