Related papers: The Andre-Oort conjecture

In this paper, applying the intersection theory of local Arthur packets, for symplectic and split odd special orthogonal groups G_n, we give the first complete proof of the enhanced Shahidi conjecture on generic representations in local…

For an abelian type Shimura variety and an odd prime $p$ of good reduction, we characterize the regularity in codimension one of Zariski closures of Ekedahl--Oort strata in terms of the Frobenius action on the root datum. We give an…

The generalized Mordell-Lang conjecture (GML) is the statement that the irreducible components of the Zariski closure of a subset of a group of finite rank inside a semi-abelian variety are translates of closed algebraic subgroups. M.…

The Oort conjecture (now a theorem of Obus-Wewers and Pop) states that if k is an algebraically closed field of characteristic p, then any cyclic branched cover of smooth projective k-curves lifts to characteristic zero. This is equivalent…

The modular case of the Andr\'e-Oort Conjecture is a theorem of Andre and Pila, having at its heart the well-known modular function j. I give an overview of two other `nonclassical' classes of modular function, namely the quasimodular (QM)…

Let $X$ be a smooth projective variety defined on a finite field $\mathbb{F}_q$. On $X$ there is a special morphism $Fr_X$, which raises coordinates to exponent $q$: $t\mapsto t^q$. The two main results in this paper are: Result 1: If…

We establish the geometric Bogomolov conjecture for semiabelian varieties over function fields. We show a closed subvariety contains Zariski dense sets of small points, if and only if, after modulo its stabilizer, it is a torsion translate…

Let $X\subset Y(1)^n$ be a subvariety defined over a number field $\mathbb F$ and let $(P_1,\ldots,P_n)\in X$ be a special point not contained in a positive-dimensional special subvariety of $X$. We show that the if a coordinate $P_i$…

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…

We investigate the analogue of the Andr\'e--Pink--Zannier conjecture in characteristic $p$. Precisely, we prove it for ordinary function field-valued points with big monodromy, in Shimura varieties of Hodge type. We also prove an algebraic…

In this note, we prove the Zilber--Pink conjecture for subvarieties of mixed Shimura varieties, which are not defined over~$\overline{\mathbb Q}$ in a strong sense. We prove similar results for general variations of mixed Hodge structure of…

We use Vaughan's variation on Vinogradov's three-primes theorem to prove Zariski-density of prime points in several infinite families of hypersurfaces, including level sets of some quadratic forms, the Permanent polynomial, and the defining…

We formulate characteristic $p$ analogues of the Mumford--Tate and the Andr\'e--Oort conjectures for ordinary mod $p$ Shimura varieties of Hodge type, and set up general frameworks for studying them. We prove the two conjectures for…

We state and prove a variant of the Andr\'e-Oort conjecture for the product of 2 modular curves in positive characteristic, assuming GRH for quadratic fields.

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.…

The purpose of this note is to wish a happy birthday to Professor Lucia Caporaso.* We prove that Conjecture H of Caporaso et. al. ([CHarM], sec. 6) together with Lang's conjecture implies the uniformity of rational points on varieties of…

We conjecture that the exceptional set in Manin's Conjecture has an explicit geometric description. Our proposal includes the rational point contributions from any generically finite map with larger geometric invariants. We prove that this…

We study the mod $p$-points of the Kisin--Pappas integral models of Shimura varieties of Hodge type with parahoric level. We show that if the group is quasi-split, then every isogeny class contains the reduction of a CM point, proving a…

In this paper we study the Oort conjecture on Shimura subvarieties contained generically in the Torelli locus in the Siegel modular variety $\mathcal{A}_g$. Using the poly-stability of Higgs bundles on curves and the slope inequality of…

We prove the Arnold conjecture for closed symplectic manifolds with $\pi_2(M)=0$ and $\cat M=\dim M$. Furthermore, we prove an analog of the Lusternik-Schnirelmann theorem for functions with ``generalized hyperbolicity'' property.