Duke Mathematical Journal

Tate cycles on some quaternionic Shimura varieties mod p

Yichao Tian and Liang Xiao

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Let F be a totally real field in which a prime number p>2 is inert. We continue the study of the (generalized) Goren–Oort strata on quaternionic Shimura varieties over finite extensions of Fp. We prove that, when the dimension of the quaternionic Shimura variety is even, the Tate conjecture for the special fiber of the quaternionic Shimura variety holds for the cuspidal π-isotypical component, as long as the two unramified Satake parameters at p are not differed by a root of unity.

Article information

Duke Math. J., Volume 168, Number 9 (2019), 1551-1639.

Received: 25 May 2017
Revised: 30 October 2018
First available in Project Euclid: 12 June 2019

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 11G18: Arithmetic aspects of modular and Shimura varieties [See also 14G35]
Secondary: 14G35: Modular and Shimura varieties [See also 11F41, 11F46, 11G18] 14C25: Algebraic cycles 11F41: Automorphic forms on GL(2); Hilbert and Hilbert-Siegel modular groups and their modular and automorphic forms; Hilbert modular surfaces [See also 14J20]

special fiber of Hibert modular varieties supersingular locus Tate conjecture Goren–Oort stratification


Tian, Yichao; Xiao, Liang. Tate cycles on some quaternionic Shimura varieties mod $p$. Duke Math. J. 168 (2019), no. 9, 1551--1639. doi:10.1215/00127094-2018-0068. https://projecteuclid.org/euclid.dmj/1560326497

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