Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: The Hodge conjecture and arithmetic quotients of complex balls   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.

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The categories of this paper are: gr-qc
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The categories of this paper are: patt-sol
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The categories of this paper are: solv-int
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The categories of this paper are: bayes-an
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The categories of this paper are: nlin
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The categories of this paper are: ao-sci
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The categories of this paper are: alg-geom
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The categories of this paper are: funct-an
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The categories of this paper are: chem-ph
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The categories of this paper are: adap-org
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The categories of this paper are: hep-th
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The categories of this paper are: q-fin
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The categories of this paper are: stat
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The categories of this paper are: dg-ga
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The categories of this paper are: cmp-lg
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The categories of this paper are: cond-mat