Tate cycles on some quaternionic Shimura varieties mod p

TL;DR

This paper proves the Tate conjecture for certain quaternionic Shimura varieties over finite fields, specifically when the variety's dimension is even and certain Satake parameters are not roots of unity, advancing understanding of their algebraic cycles.

Contribution

It establishes the Tate conjecture for the special fiber of quaternionic Shimura varieties under specific conditions on Satake parameters, extending previous results in the area.

Findings

Tate conjecture holds for even-dimensional quaternionic Shimura varieties in specified cases.

Goren--Oort strata analysis aids in understanding algebraic cycles on these varieties.

Conditions on Satake parameters are crucial for the conjecture's validity.

Abstract

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 $\mathbb F_p$. 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.