Related papers: Tate cycles on some quaternionic Shimura varieties…

In this note we study the supersingular locus of the GU(2,2) Shimura variety modulo a prime which is unramified in the imaginary quadratic extension. The supersingular locus of this Shimura variety can be related to the basic Rapoport-Zink…

Let $k$ be a number field and let ${\mathcal{A}}$ be a ${\rm GL}_2$-type variety defined over $k$ of dimension $d$. We show that for every prime number $p$ satisfying certain conditions (see Theorem 2), if the local-global divisibility…

In this paper, we study the behavior of Ekedahl-Oort strata under natural embeddings between the good reductions modulo $p$ of GSpin Shimura varieties and Rapoport-Smithling-Zhang unitary Shimura varieties, a prototypical setting for the…

Let $\mathcal{A}_g$ be the moduli space over $\overline{\mathbb{F}}_p$ of $g$-dimensional principally polarised abelian varieties, where $p$ is a prime. We show that if $g$ is even and $p\geq 5$, then every geometric generic member in the…

We construct group-theoretical generalizations of the Hasse invariant on strata closures of the stacks $G$-Zip$^{\mu}$. Restricting to zip data of Hodge type, we obtain a group-theoretical Hasse invariant on every Ekedahl-Oort stratum…

Let $F$ be a totally real field of degree $n$ and $p$ an odd prime. We prove the $p$-part of the integral Gross--Stark conjecture for the Brumer--Stark $p$-units living in CM abelian extensions of $F$. In previous work, the first author…

We present a heuristic argument based on Honda-Tate theory against many conjectures in `unlikely intersections' over the algebraic closure of a finite field; notably, we conjecture that every abelian variety of dimension 4 is isogenous to a…

We study algebraic cycles on complex Gushel-Mukai (GM) varieties. We prove the generalised Hodge conjecture, the (motivated) Mumford-Tate conjecture, and the generalised Tate conjecture for all GM varieties. We compute all integral Chow…

Let $p$ be an odd prime. We give an unconditional proof of the equivariant Iwasawa main conjecture for totally real fields for every admissible one-dimensional $p$-adic Lie extension whose Galois group has an abelian Sylow $p$-subgroup.…

We establish a derived geometric Satake equivalence for the quaternionic general linear group GL_n(H). By applying the real-symmetric correspondence for affine Grassmannians, we obtain a derived geometric Satake equivalence for the…

Let $p$ be an odd prime and $F$ be a complete discretely valued field with residue field of characteristic $p$. For any parahoric level structure of the split even orthogonal similitude group $\operatorname{GO}_{2n}$ over $F$, we prove a…

This paper is a continuation of our paper math.AG/0006222. We study the reduction of certain PEL Shimura varieties with parahoric level structure at primes p at which the group that defines the Shimura variety ramifies. We describe "good"…

In this article, we use deformation theory of Galois representations valued in the symplectic group of degree four to prove a freeness result for the cohomology of certain quaternionic unitary Shimura variety over the universal deformation…

Let $Y$ be an abelian variety over a subfield $k \subset \mathbb{C}$ that is of finite type over $\mathbb{Q}$. We prove that if the Mumford-Tate conjecture for $Y$ is true, then also some refined integral and adelic conjectures due to Serre…

Given a projective variety X defined over a finite field, the zeta function of divisors attempts to count all irreducible, codimension one subvarieties of X, each measured by their projective degree. When the dimension of X is greater than…

Let $F$ be a totally real field in which $p$ is unramfied and let $S$ denote the integral model of the Hilbert modular variety with good reduction at $p$. Consider the usual automorphic line bundle $\mathcal{L}$ over $S$. On the generic…

Consider a Shimura variety of Hodge type admitting a smooth integral model S at an odd prime p>3. Consider its perfectoid cover S(p^\infty) and the Hodge-Tate period map introduced by A. Caraiani and P. Scholze. We compare the pull-back to…

We prove the geometric Satake equivalence for mixed Tate motives over the integral motivic cohomology spectrum. This refines previous versions of the geometric Satake equivalence for split reductive groups. Our new geometric results include…

Let $k$ be an infinite finitely generated field of characteristic $p>0$. Fix a separated scheme $X$ smooth, geometrically connected, and of finite type over $k$ and a smooth proper morphism $f:Y\rightarrow X$. The main result of this paper…

We extend the well-known Cassels-Tate dual exact sequence for abelian varieties A over global fields K in two directions: we treat the p-primary component in the function field case, where p is the characteristic of K, and we dispense with…