Related papers: $p$-adic monodromy and mod $p$ unlikely intersecti…

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…

Let $(\mathsf{G},\mathsf{X})$ be a Shimura datum of Hodge type. Let $p$ be an odd prime such that $\mathsf{G}_{\mathbb{Q}_p}$ splits after a tamely ramified extension and $p\nmid |\pi_1(\mathsf{G}^{\rm der})|$. Under some mild additional…

We prove in this paper the Ax-Lindemann-Weierstrass theorem for all mixed Shimura varieties and discuss the lower bounds for Galois orbits of special points of mixed Shimura varieties. In particular we reprove a result of Silverberg in a…

In this article we study the (cohomological) Hodge conjecture for singular varieties. We prove the conjecture for simple normal crossing varieties that can be embedded in a family where the Mumford-Tate group remains constant. We show how…

We prove a $p$-adic analogue of the Andr\'{e}-Oort conjecture for subvarieties of the universal abelian varieties containing a dense set of special points. Let $g$ and $n$ be integers with $n \geq 3$ and $p$ a prime number not dividing $n$.…

We prove the 'hybrid conjecture' which is a common generalisation of the Andre\'e-Oort conjecture and the Andr\'e-Pink-Zannier conjecture, in the case of Shimura varieties of abelian type.

In this paper, we prove the generalised Andr\'e-Pink-Zannier conjecture (an important case of the Zilber-Pink conjecture) for all Shimura varieties of abelian type. Questions of this type were first asked by Y. Andr\'e in 1989. We actually…

Let $X$ be a smooth projective variety over a finitely generated field $K$ of characteristic~$0$ and fix an embedding $K \subset \mathbb{C}$. The Mumford--Tate conjecture is a precise way of saying that certain extra structure on the…

We construct (cohomological) correspondences between mod $p$ fibers of different Shimura varieties and describe the fibers of these correspondences by studying irreducible components of affine Deligne-Lusztig varieties. In particular, in…

We prove the isogeny property for special fibres of integral canonical models of compact Shimura varieties of $A_n$, $B_n$, $C_n$, and $D_n^{\dbR}$ type. The approach used also shows that many crystalline cycles on abelian varieties over…

For each prime number $p$ and each integer $g \geqslant 5$, we construct infinitely many abelian varieties of dimension $g$ over $\overline{\mathbb{F}}_p$ satisfying the standard conjecture of Hodge type. The main tool is a recent theorem…

In this paper, we establish two main results concerning the Mumford-Tate conjecture for hyper-K\"ahler varieties. First, we prove the conjecture for the semisimplified $\ell$-adic Galois representations attached to hyper-K\"ahler varieties…

We propose a conjecture on the $p$-adic nowhere density of the Hecke orbit of subvarieties of Hodge type Shimura varieties. By investigating the monodromy of $p$-adic Galois representations associated with points on such Shimura varieties,…

Let $(G,X)$ be a Shimura pair of Hodge type such that $G$ is the Mumford--Tate group of some elements of $X$. We assume that for each simple factor $G_0$ of $G^{\ad}$ there exists a simple factor of $G_{0\dbR}$ which is compact. Let $N\Ge…

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…

We study the image of $\ell$-adic representations attached to subvarieties of Shimura varieties $Sh_K(G,X)$ that are not contained in a smaller Shimura subvariety and have no isotrivial components. We show that, for $\ell$ large enough…

We prove a general likely intersections theorem, a counterpart to the Zilber-Pink conjectures, under the assumption that the Ax-Schanuel property and some mild additional conditions are known to hold for a given category of complex quotient…

The goal of this paper is to study a $p$-adic analog of the joint of the conjectures of Andr\'e--Oort and Andr\'e--Pink. More precisely, on a product of ordinary Siegel formal moduli schemes, we study the distribution of points whose…

The main result of this paper is the existence of Galois representations associated with the mod $p$ (or mod $p^m$) cohomology of the locally symmetric spaces for $\GL_n$ over a totally real or CM field, proving conjectures of Ash and…

We define variants of PEL type of the Shimura varieties that appear in the context of the Arithmetic Gan-Gross-Prasad conjecture. We formulate for them a version of the AGGP conjecture. We also construct (global and semi-global) integral…