Tate cycles on some quaternionic Shimura varieties mod p

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.

58 pages; this is the published version. Some errors are corrected and the introduction section is rewritten

References in corpus (3)

• Hirzebruch-Zagier cycles and twisted triple product Selmer groups
• Algebraic cycles on Hilbert modular fourfolds and poles of L-functions
• On Tate conjecture for the special fibers of some unitary Shimura varieties

Cited by in corpus (1)

• Supersingular locus of Hilbert modular varieties, arithmetic level raising and Selmer groups