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.
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 q
uaternionic 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.
Let $F$ be a totally real field in which $p$ is unramified. We study the Goren-Oort stratification of the special fibers of quaternionic Shimura varieties over a place above $p$. We show that each stratum is a $(mathbb{P}^1)^N$-bundle over other quat
ernionic Shimura varieties (for some appropriate $N$).
Let $F$ be a quadratic real field, $p$ be a rational prime inert in $F$. In this paper, we prove that an overconvergent $p$-adic Hilbert eigenform for $F$ of small slope is actually a classical Hilbert modular form.
Let $cO_K$ be a complete discrete valuation ring of residue characteristic $p>0$, and $G$ be a finite flat group scheme over $cO_K$ of order a power of $p$. We prove in this paper that the Abbes-Saito filtration of $G$ is bounded by a simple linear f
unction of the degree of $G$. Assume $cO_K$ has generic characteristic 0 and the residue field of $cO_K$ is perfect. Fargues constructed the higher level canonical subgroups for a Barsotti-Tate group $cG$ over $cO_K$ which is not too supersingular. As an application of our bound, we prove that the canonical subgroup of $cG$ of level $ngeq 2$ constructed by Fargues appears in the Abbes-Saito filtration of the $p^n$-torsion subgroup of $cG$.
Let k be an algebraically closed field of characteristic $p>0$, and $G_0$ be a Barsotti-Tate group (or $p$-divisible group) over k. We denote by $S$ the algebraic local moduli in characteristic p of $G_0$, by $G$ the universal deformation of $G_0$ ov
er $S$, and by $Usubset S$ the ordinary locus of $G$. The etale part of $G$ over $U$ gives rise to a monodromy representation $rho$ of the fundamental group of $U$ on the Tate module of $G$. Motivated by a famous theorem of Igusa, we prove in this article that $rho$ is surjective if $G_0$ is connected and HW-cyclic. This latter condition is equivalent to that Oorts $a$-number of $G_0$ equals 1, and it is satisfied by all connected one-dimensional Barsotti-Tate groups over $k$.
