Let be a real quadratic field in which a fixed prime is inert, and be an imaginary quadratic field in which splits; put . Let be the fiber over of the Shimura variety for with hyperspecial level structure at for some integer . We show that under some genericity conditions the middle-dimensional Tate classes of 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.
"Tate cycles on some unitary Shimura varieties mod $p$." Algebra Number Theory 11 (10) 2213 - 2288, 2017. https://doi.org/10.2140/ant.2017.11.2213