## Duke Mathematical Journal

### Tate cycles on some quaternionic Shimura varieties mod $p$

#### Abstract

Let $F$ be a totally real field in which a prime number $p\gt 2$ is inert. We continue the study of the (generalized) Goren–Oort strata on quaternionic Shimura varieties over finite extensions of $\mathbb{F}_{p}$. 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 $\pi$-isotypical component, as long as the two unramified Satake parameters at $p$ are not differed by a root of unity.

#### Article information

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

Dates
Revised: 30 October 2018
First available in Project Euclid: 12 June 2019

https://projecteuclid.org/euclid.dmj/1560326497

Digital Object Identifier
doi:10.1215/00127094-2018-0068

Mathematical Reviews number (MathSciNet)
MR3961211

Zentralblatt MATH identifier
07097310

#### Citation

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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