相关论文: Canonical local heights and Berkovich skeleta

We study analogues of Tate's conjecture on homomorphisms for abelian varieties when the ground field is finitely generated over an algebraic closure of a finite field. Our results cover the case of abelian varieties without nontrivial…

Let $\pi : E\to B$ be an elliptic surface defined over a number field $K$, where $B$ is a smooth projective curve, and let $P: B \to E$ be a section defined over $K$ with canonical height $\hat{h}_E(P)\not=0$. In this article, we show that…

We deal with locally free $\mathcal{O}_X$-modules with connection over a Berkovich curve $X$. As a main result we prove local and global decomposition theorems of such objects by the radii of convergence of their solutions. We also derive a…

In this paper we will prove that Tate conjecture of abelian varieties over finite field is equivalent to the finiteness of isomorphism classes of abelian varieties with a fixed dimension. We give a different approach with Zarhin's result.

For $\theta$ a non-algebraic point on a quasi projective variety over a number field, I prove that $\theta$ has an approximation by a series of algebraic points of bounded height and degree which is essentially best possible. Applications…

Let $X$ be a smooth projective variety defined over a number field $K$ and let $\varphi: X \to X$ a polarized endomorphism of degree $d \geq 2$. Let $\widehat{h}_{\varphi}$ be the canonical height associated to $\varphi$ on…

Given a minimal surface equipped with a generically finite map to an Abelian variety, we give an optimal bound on the canonical degree of a rational or an elliptic curve. As a corollary, we obtain the finiteness of rational and elliptic…

We obtain explicit, computable upper bounds for the Neron-Tate height of rational points on curves of genus at least two over number fields. The bounds use automorphisms acting on the Mordell-Weil lattice of the Jacobian. We prove an…

In this paper, we first study the local rings of a Berkovich analytic space from the point of view of commutative algebra. We show that those rings are excellent ; we introduce the notion of a an analytically separable extension of…

We prove that, for certain extensions of valued fields which admit a sensible theory of ramification groups, there exist canonical towers that correspond to the break-points of their Herbrand function. In particular, each of the…

We give a version of the Artin-Tate formula for surfaces over finite fields not assuming Tate's conjecture. It gives an equality between terms related to the Brauer group on the one hand and terms related to the Neron-Severi group on the…

Let $K$ be a number field and let $C/K$ be a curve of genus 2 with Jacobian variety $J$. In this paper, we study the canonical height $\hat{h} \colon J(K) \to \mathbb R$. More specifically, we consider the following two problems, which are…

In two earlier articles, we proved that, if the Hodge conjecture is true for ALL CM abelian varieties over the complex numbers, then both the Tate conjecture and the standard conjectures are true for abelian varieties over finite fields.…

Using original ideas from J.-B. Bost and S. David, we provide an explicit comparison between the Theta height and the stable Faltings height of a principally polarized abelian variety. We also give as an application an explicit upper bound…

We prove some new results on the arithmetic of abelian varieties over function fields of one variable over finitely generated (infinite) fields. Among other things, we introduce certain new natural objects `discrete Selmer groups' and…

Let $k$ be a field of characteristic $0$ and let $K = k(B)$ be the function field of a geometrically irreducible projective curve $B$ over $k$. Let $A/K$ be a $g$-dimensional abelian variety with $\mathrm{Tr}_{K/k}(A) = 0$. We prove that…

Let $S$ be a smooth irreducible curve over $\overline{\mathbb{Q}}$, and let $\mathcal{A} \to S$ be an abelian scheme with a curve $C \subset \mathcal{A}$, both defined over $\overline{\mathbb{Q}}$. In 2020, Barroero and Capuano proved that…

We study the asymptotic growth of the number of rational points of bounded height on smooth projective split toric varieties with Picard rank 2 over number fields, with respect to Arakelov height functions associated with big metrized line…

We prove a collection of results involving Colmez's periods and the Colmez Conjecture. Using Colmez's theory of periods of CM abelian varieties, we propose a definition for the height of a partial CM-type and prove that the Colmez…

We construct height functions defined stochastically on projective varieties equipped with endomorphisms, and we prove that these functions satisfy analogs of the usual properties of canonical heights. Moreover, we give a dynamical…