#### Vol. 11, No. 10, 2017

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Tate cycles on some unitary Shimura varieties mod $p$

### David Helm, Yichao Tian and Liang Xiao

Vol. 11 (2017), No. 10, 2213–2288
##### Abstract

Let $F$ be a real quadratic field in which a fixed prime $p$ is inert, and ${E}_{0}$ be an imaginary quadratic field in which $p$ splits; put $E={E}_{0}F$. Let $X$ be the fiber over ${\mathbb{F}}_{{p}^{2}}$ of the Shimura variety for $G\left(U\left(1,n-1\right)×U\left(n-1,1\right)\right)$ with hyperspecial level structure at $p$ for some integer $n\ge 2$. We show that under some genericity conditions the middle-dimensional Tate classes of $X$ are generated by the irreducible components of its supersingular locus. We also discuss a general conjecture regarding special cycles on the special fibers of unitary Shimura varieties, and on their relation to Newton stratification.

##### Keywords
Supersingular locus, Special fiber of Shimura varieties, Deligne–Lusztig varieties, Tate conjecture
##### Mathematical Subject Classification 2010
Primary: 11G18
Secondary: 11R39, 14C17, 14C25, 14G35