Algebra & Number Theory

Tate cycles on some unitary Shimura varieties mod $p$

David Helm, Yichao Tian, and Liang Xiao

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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 ) × 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.

Article information

Algebra Number Theory, Volume 11, Number 10 (2017), 2213-2288.

Received: 17 November 2015
Revised: 24 August 2017
Accepted: 28 September 2017
First available in Project Euclid: 1 February 2018

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 11G18: Arithmetic aspects of modular and Shimura varieties [See also 14G35]
Secondary: 11R39: Langlands-Weil conjectures, nonabelian class field theory [See also 11Fxx, 22E55] 14C17: Intersection theory, characteristic classes, intersection multiplicities [See also 13H15] 14C25: Algebraic cycles 14G35: Modular and Shimura varieties [See also 11F41, 11F46, 11G18]

Supersingular locus Special fiber of Shimura varieties Deligne–Lusztig varieties Tate conjecture


Helm, David; Tian, Yichao; Xiao, Liang. Tate cycles on some unitary Shimura varieties mod $p$. Algebra Number Theory 11 (2017), no. 10, 2213--2288. doi:10.2140/ant.2017.11.2213.

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