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

Let F be a real quadratic field in which a fixed prime p is inert, and E0 be an imaginary quadratic field in which p splits; put E = E0F. Let X be the fiber over Fp2 of the Shimura variety for G(U(1,n 1) × U(n 1,1)) with hyperspecial level structure at p for some integer n 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.

Supersingular locus, Special fiber of Shimura varieties, Deligne–Lusztig varieties, Tate conjecture
Mathematical Subject Classification 2010
Primary: 11G18
Secondary: 11R39, 14C17, 14C25, 14G35
Received: 17 November 2015
Revised: 24 August 2017
Accepted: 28 September 2017
Published: 31 December 2017
David Helm
Department of Mathematics
Imperial College London
South Kensington Campus
United Kingdom
Yichao Tian
Mathematics Institute
University of Bonn
Liang Xiao
Department of Mathematics
University of Connecticut
Storrs, CT
United States