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arXiv · 1410.2343

On Tate conjecture for the special fibers of some unitary Shimura varieties

David Helm·Yichao Tian·Liang Xiao
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Abstract

Let $F$ be a totally real field in which a fixed prime $p$ is inert, and let $E$ be a CM extension of $F$ in which $p$ splits. We fix two positive integers $r,s \in \mathbb N$. We investigate the Tate conjecture on the special fiber of $G(U(r,s) \times U(s,r))$-Shimura variety. We construct cycles which we conjecture to generate the Tate classes and verify our conjecture in the case of $G(U(1,s) \times U(s,1))$. We also discuss the general conjecture regarding special cycles on the special fibers of unitary Shimura varieties.

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David Helm, Yichao Tian, Liang Xiao. 2014-10-09. On Tate conjecture for the special fibers of some unitary Shimura varieties. https://doi.org/10.2140/ant.2017.11.2213

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