Tate cycles on some unitary Shimura varieties mod p

Helm, David; Tian, Yichao; Xiao, Liang

doi:10.2140/ant.2017.11.2213

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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

DOI: 10.2140/ant.2017.11.2213

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 $F_{p^{2}}$ of the Shimura variety for $G (U (1, n - 1) \times U (n - 1, 1))$ with hyperspecial level structure at $p$ for some integer $n \geq 2$ . 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

Milestones

Received: 17 November 2015

Revised: 24 August 2017

Accepted: 28 September 2017

Published: 31 December 2017

Authors

David Helm
Department of Mathematics Imperial College London South Kensington Campus London United Kingdom

Yichao Tian
Mathematics Institute University of Bonn Bonn Germany

Liang Xiao
Department of Mathematics University of Connecticut Storrs, CT United States

Vol. 11, No. 10, 2017