Related papers: On the Zilber-Pink conjecture for complex abelian …

In this paper, we formulate the geometric Bogomolov conjecture for abelian varieties, and give some partial answers to it. In fact, we insist in a main theorem that under some degeneracy condition, a closed subvariety of an abelian variety…

Recently, R\'emond stated a very general conjecture on lower bounds of a normalized height on either an abelian variety or a power of the multiplicative group. In this note, we extend a particular case of this conjecture to split…

The Bogomolov conjecture for a curve claims finiteness of algebraic points on the curve which are small with respect to the canonical height. Ullmo has established this conjecture over number fields, and Moriwaki generalized it to the…

Let $A$ be an abelian variety over a field finitely generated over $\mathbb{Q}$. We show that the finiteness of the $\ell$-primary torsion subgroup of the higher Brauer group is a sufficient criterion for the Tate conjecture to hold.…

A differential analogue of the conjecture of Reichstein, Rogalski, and Zhang in algebraic dynamics is here established: if $X$ is a projective variety over an algebraically closed field of characteristic zero which admits a global algebraic…

In this paper we survey the history of, and recent developments on, two major conjectures originating in Zilber's model-theoretic work on complex exponentiation -- Existential Closedness and Zilber-Pink. The main focus is on the modular…

We prove function field versions of the Zilber-Pink conjectures for varieties supporting a variation of Hodge structures. A form of these results for Shimura varieties in the context of unlikely intersections is the following. Let $S$ be a…

Let $A$ be a non-zero abelian variety over a field $F$ that is not algebraic over a finite field. We prove that the rational rank of the abelian group $A(F)$ is infinite when $F$ is large in the sense of Pop (also called ample). The main…

The authors previously formulated the hybrid conjecture, unifying Andr\'e-Pink-Zannier and Andr\'e-Oort conjectures, and proved it in Shimura varieties of abelian type. We study its analogue for mixed Shimura varieties, and consider the…

We prove that Grothendieck's Hodge standard conjecture holds for abelian varieties in arbitrary characteristic if the Hodge conjecture holds for complex abelian varieties of CM-type. For abelian varieties with no exotic algebraic classes,…

We prove a connexity theorem for abelian varieties in characteristic $0$: if $X$ is an abelian variety and $V\rightarrow X$ and $W\rightarrow X$ two morphisms, then, under certain hypotheses, the fiber product of $V$ and $W$ over $X$ is…

Zilber's Exponential Algebraic Closedness conjecture (also known as Zilber's Nullstellensatz) gives conditions under which a complex algebraic variety should intersect the graph of the exponential map of a semiabelian variety. We prove the…

We prove the following theorems: Theorem 1: For any E-field with cyclic kernel, in particular $\mathbb C$ or the Zilber fields, all real abelian algebraic numbers are pointwise definable. Theorem 2: For the Zilber fields, the only pointwise…

The parity conjecture predicts that the parity of the rank of an abelian variety is determined by its global root number, that is by the sign in the conjectural functional equation of its L-function. Assuming the Shafarevich-Tate…

We compute the Picard group of the universal abelian variety over the moduli stack $\mathscr A_{g,n}$ of principally polarized abelian varieties over $\mathbb{C}$ with a symplectic principal level $n$-structure. We then prove that over…

Let $\Gamma\subset \bar{\mathbb{Q}}^*$ be a finitely generated subgroup. Denote by $\Gamma_\mathrm{div}$ its division group. A recent conjecture due to R\'emond, related to the Zilber-Pink conjecture, predicts that the absolute logarithmic…

We show that Ribet sections are the only obstruction to the validity of the relative Manin-Mumford conjecture for one dimensional families of semi-abelian surfaces. Applications include special cases of the Zilber-Pink conjecture for curves…

The Zilber-Pink conjecture predicts that an algebraic curve in $\mathcal{A}_2$ has only finitely many intersections with the special curves, unless it is contained in a proper special subvariety. Under a large Galois orbits hypothesis, we…

We study a geometric analogue of the Iwasawa Main Conjecture for abelian varieties in the two following cases: constant ordinary abelian varieties over $Z_p^d$-extensions of function fields ($d\geq 1$) ramified at a finite set of places,…

The endomorphism ring End(A) of an abelian variety A is an order in a semi-simple algebra over Q. The co-index of End(A) is the index to a maximal order containing it. We show that for abelian varieties of fixed dimension over any…