On Tate conjecture for the special fibers of some unitary Shimura varieties

Abstract

Let $F$ be a totally real field in which a fixed prime $p$ is inert, and let $E$ be a CM extension of $F$ in which $p$ splits. We fix two positive integers $r,s\in \mathbb{N}$. We investigate the Tate conjecture on the special fiber of $G(U(r,s)\times U(s,r))$-Shimura variety. We construct cycles which we conjecture to generate the Tate classes and verify our conjecture in the case of $G(U(1,s)\times U(s,1))$. We also discuss the general conjecture regarding special cycles on the special fibers of unitary Shimura varieties.

The final published version, to appear in Algebra and Number Theory

References in corpus (2)

• Cycles on Shimura varieties via geometric Satake
• Basic loci in Shimura varieties of Coxeter type

Cited by in corpus (5)

• Fully Hodge-Newton decomposable Shimura varieties
• On the Bruhat-Tits stratification of a quaternionic unitary Shimura variety
• Hodge classes and the Jacquet-Langlands correspondence
• Geometric Satake, categorical traces, and arithmetic of Shimura varieties
• Tate cycles on some quaternionic Shimura varieties mod p