Related papers: A sharp Bogomolov-type bound

We prove new cases of the Tate conjecture for abelian varieties over finite fields, extending previous results of Dupuy--Kedlaya--Zureick-Brown, Lenstra--Zarhin, Tankeev, and Zarhin. Notably, our methods allow us to prove the Tate…

Heuristics based on the Sato--Tate conjecture suggest that an abelian surface defined over a number field has infinitely many places of split reduction. We prove this result for abelian surfaces having real multiplication. Similar to…

Let A be an abelian variety defined over a number field K, the number of torsion points rational over a finite extension L is bounded polynomially in terms of the degree [L : K]. When A is isogenous to a product of simple abelian varieties…

Given a family of abelian covers of $\mathbb{P}^1$ and a prime $p$ of good reduction, by considering the associated Deligne--Mostow Shimura variety, we obtain lower bounds for the Ekedahl-Oort type, and the Newton polygon, at $p$ of the…

We give a lower bound for the essential dimension of isogenies of complex abelian varieties. The bound is sharp in many cases. In particular, the multiplication-by-$m$ map is incompressible for every $m\geq 2$, confirming a conjecture of…

We recognise that an entropy inequality akin to the main intermediate goal of recent works (Gowers, Green, Manners, Tao [3],[2]) regarding a conjecture of Marton provides a black box from which we can also through a short deduction recover…

This work is the third part of a series of papers. In the first two we consider curves and varieties in a power of an elliptic curve. Here we deal with subvarieties of an abelian variety in general. Let V be an irreducible variety of…

We show that a principally polarized abelian variety over a field $k$ is, as an abelian variety, a direct summand of a product of Jacobians of curves which contain a $k$-point if and only if the polarization and the minimal class are both…

We give upper and lower bounds on the number of points on abelian varieties over finite fields, and lower bounds specific to Jacobian varieties. We also determine exact formulas for the maximum and minimum number of points on Jacobian…

In this paper we prove a lower bound for the least number of one-periodic solutions of nondegenerate locally Hamiltonian equations on compact symplectic manifolds in terms of the Betti numbers of the Novikov homology associated to the…

Let $K$ be a number field, $A/K$ be an absolutely simple abelian variety of CM type, and $\ell$ be a prime number. We give explicit bounds on the degree over $K$ of the division fields $K(A[\ell^n])$, and when $A$ is an elliptic curve we…

We refine and generalize the results of K. E. Lauter and E. W. Howe on principal polarizations on products of abelian varieties over finite fields. Firstly, we study the reasons for the absence of an irreducible principal polarization in…

In "Mordell-Lang plus Bogomolov", we formulated a conjecture combining the Mordell-Lang conjecture (now proven) and the Bogomolov conjecture, and proved the Mordellic part when A is almost split (i.e., isogenous to the product of an abelian…

In this paper, we prove that the number $B(p,g)$ of isomorphism classes of abelian varieties over a prime field $\mathbb{F}_p$ of dimension $g$ has a lower bound $p^{\frac{1}{2} g^2 (1+o(1))}$ as $g \rightarrow \infty$. This is the first…

We provide new, improved lower bounds for the Hodge and Frobenius colevels of algebraic varieties (over $\mathbf{C}$ or over a finite field) in all cohomological degrees. These bounds are expressed in terms of the dimension of the variety…

We establish bounds on the minimal asymptotic pseudo-Anosov translation lengths on the complex of curves of orientable surfaces. In particular, for a closed surface with genus $g \geqslant 2$, we show that there are positive constants $a_1…

The Bogomolov Conjecture is a finiteness statement about algebraic points of small height on a smooth complete curve defined over a global field. We verify an effective form of the Bogomolov Conjecture for all curves of genus at most 4…

We prove that the geometric Bogomolov conjecture holds for nowhere degenerate abelian varieties of dimension $5$ with trivial trace. By this result together with our previous work, we see that the conjecture holds for an abelian variety…

We prove a sharp lower bound on the curvatures of non-flat ASD connections over the cylinder.

We show that, for an abelian variety defined over a $p$-adic field $K$ which has potential good reduction, its torsion subgroup with values in the composite field of $K$ and a certain Lubin-Tate extension over a $p$-adic field is finite.