Related papers: N\'eron models and the height jump divisor

Inspired by the work of Ellenberg, Elsholtz, Hall, and Kowalski, we investigate how the property of the generic fiber of a one-parameter family of abelian varieties being geometrically simple extends to other fibers. In \cite{EEHK09}, the…

We prove a special case of the following conjecture of Zilber-Pink generalising the Manin-Mumford conjecture : let $X$ be a curve inside an Abelian variety $A$ over $\bar{\Q}$, provided $X$ is not contained in a torsion subvariety, the…

We study how Tamagawa numbers of Jacobians of hyperelliptic curves vary as one varies the base field or the curve, in the case of semistable reduction. We find that there are strong constraints on the behaviour that appears, some of which…

We apply Tate's conjecture on algebraic cycles to study the N\'eron-Severi groups of varieties fibered over a curve. This is inspired by the work of Rosen and Silverman, who carry out such an analysis to derive a formula for the rank of the…

We investigate to what extent the theory of N\'eron models of jacobians and of abel-jacobi maps extends to relative curves over base schemes of dimension greater than 1. We give a necessary and sufficient criterion for the existence of a…

Given a smooth, proper curve $C$ over a discretely valued field $k$, we equip the $k$-vector space $H^{0}(C,\omega_{C/k})$ with a canonical discrete valuation $v_{\mathrm{can}}$ which measures how canonical forms degenerate on regular…

Let $A/K$ be an abelian variety over a number field $K$. We prove in this article that a good lower bound (in terms of the degree $[K(P):K]$) for the N\'eron-Tate height of the points $P$ of infinite order modulo every strict abelian…

We identify several classes of curves $C:f=0$, for which the Hilbert vector of the Jacobian module $N(f)$ can be completely determined, namely the 3-syzygy curves, the maximal Tjurina curves and the nodal curves, having only rational…

Tropical geometry gives a bound on the ranks of divisors on curves in terms of the combinatorics of the dual graph of a degeneration. We show that for a family of examples, curves realizing this bound might only exist over certain…

A new proof is given for the explicit formulae for the non-archimedean canonical height on an elliptic curve. This arises as a direct calculation of the Haar integral in the elliptic Jensen formula.

Let $p$ be a prime number such that the modular curve $X_0(p)$ has genus at least two. We show that the only points of the reduction mod $p$ of $X_0(p)$ with image in the reduction mod $p$ of $J_0(p)$ in the cuspidal group are the two…

We present an efficient endomorphism for the Jacobian of a curve $C$ of genus 2 (hyperelliptic) for divisors having a Non disjoint support. This extends the work of Costello and Lauter in [12] who calculated explicit formulae for divisor…

Let $E/\mathbb{Q}$ be an elliptic curve which has split multiplicative reduction at a prime $p$ and whose analytic rank $r_{an}(E)$ equals one. The main goal of this article is to relate the second order derivative of the…

We introduce an algorithm to compute the rational torsion subgroup of the Jacobian of a hyperelliptic curve of genus 3 over the rationals. We apply a Magma implementation of our algorithm to a database of curves with low discriminant due to…

The divisor theory for graphs is compared to the theory of linear series on curves through the correspondence associating a curve to its dual graph. An algebro-geometric interpretation of the combinatorial rank is proposed, and proved in…

For several applications in the arithmetic of abelian varieties it is important to compute canonical heights. Following Faltings and Hriljac, we show how the canonical height on the Jacobian of a smooth projective curve can be computed…

A well-known combinatorial theorem says that a set of n non-collinear points in the plane determines at least n distinct lines. Chen and Chv\'atal conjectured that this theorem extends to metric spaces, with an appropriated definition of…

We study abelian varieties defined over function fields of curves in positive characteristic $p$, focusing on their arithmetic within the system of Artin-Schreier extensions. First, we prove that the $L$-function of such an abelian variety…

We exhibit a precise connection between N\'eron--Tate heights on smooth curves and biextension heights of limit mixed Hodge structures associated to smoothing deformations of singular quotient curves. Our approach suggests a new way to…

The Bounded Height Conjecture of Bombieri, Masser, and Zannier states that for any sufficiently generic algebraic subvariety of a semiabelian $\overline{\mathbb{Q}}$-variety $G$ there is an upper bound on the Weil height of the points…