Tate cycles on some unitary Shimura varieties mod $p$

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 $G\left(U\left(1,n-1\right)\times U\left(n-1,1\right)\right)$ 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.