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

It was shown by Faltings and Hriljac that the N\'eron-Tate height of a point on the Jacobian of a curve can be expressed as the self-intersection of a corresponding divisor on a regular model of the curve. We make this explicit and use it…

The aim of this paper is twofold. First, we study the asymptotics of the N\'eron height pairing between degree-zero divisors on a family of degenerating compact Riemann surfaces parametrized by an algebraic curve. We show that if the…

We investigate N\'eron models of Jacobians of singular curves over strictly Henselian discretely valued fields, and their behaviour under tame base change. For a semiabelian variety, this behaviour is governed by a finite sequence of (a…

We develop a method to calculate the N\'eron-Tate height of tautological integral cycles on jacobians of curves defined over number fields. As examples we obtain closed expressions for the N\'eron-Tate height of the difference surface, the…

We use Arakelov theory to define a height on divisors of degree zero on a hyperelliptic curve over a global field, and show that this height has computably bounded difference from the N\'eron-Tate height of the corresponding point on the…

We develop a theory of $p$-adic N\'eron functions on abelian varieties, depending on various auxiliary choices, and show that the global $p$-adic height functions constructed by Mazur and Tate can be decomposed into a sum of $p$-adic…

N\'eron models of abelian varieties do not necessarily exist if the base $S$ has dimension higher than 1. We introduce a new condition, called toric additivity, on a family of smooth curves having nodal reduction over a normal crossing…

Let $A$ be an abelian variety over a discretely valued field. Edixhoven has defined a filtration on the special fiber of the N\'eron model of $A$ that measures the behaviour of the N\'eron model under tame base change. We interpret the…

Let $C$ be a genus $2$ hyperelliptic curve over a number field $K$, with a Weierstrass point $\infty$ at infinity, let $J$ be its Jacobian, let $\Theta$ be the theta divisor with respect to $\infty$, and let $p$ be any prime number. We give…

On an abelian scheme over a smooth curve over $\overline{\mathbb Q}$ a symmetric relatively ample line bundle defines a fiberwise N\'eon-Tate height. If the base curve is inside a projective space, we also have a height on its…

We show that the Jacobians of prestable curves over toroidal varieties always admit N\'eron models. These models are rarely quasi-compact or separated, but we also give a complete classification of quasi-compact separated group-models of…

Following our work in \cite{papas2022height}, we extend the height bounds established by Y. Andr\'e in his seminal research monograph \cite{andre1989g} for $1$-parameter families of abelian varieties defined over number fields. In our…

In this paper, we establish the following family version of Habegger's bounded height theorem on abelian varieties: a locally closed subvariety of an abelian scheme with Gao's $t^{\mathrm{th}}$ degeneracy locus removed, intersected with all…

We prove a new Bertini-type Theorem with explicit control of the genus, degree, height, and the field of definition of the constructed curve. As a consequence we provide a general strategy to reduce certain height and rank estimates on…

The jacobian of the universal curve over $\mathcal{M}_{g,n}$ is an abelian scheme over $\mathcal{M}_{g,n}$. Our main result is the construction of an algebraic space $\beta\colon \tilde{\mathcal{M}}_{g,n} \rightarrow…

A divisibility sequence is a sequence of integers $\{d_n\}$ such that $d_m$ divides $d_n$ if $m$ divides $n$. Results of Bugeaud, Corvaja, Zannier, among others, have shown that the gcd of two divisibility sequences corresponding to…

In this paper we study the singularities of the invariant metric of the Poincar\'e bundle over a family of abelian varieties and their duals over a base of arbitrary dimension. As an application of this study we prove the effectiveness of…

We prove a formula, which, given a principally polarized abelian variety $(A,\lambda)$ over the field of algebraic numbers, relates the stable Faltings height of $A$ with the N\'eron--Tate height of a symmetric theta divisor on $A$. Our…

We prove the Relative Manin-Mumford Conjecture for families of abelian varieties in characteristic 0. We follow the Pila-Zannier method to study special point problems, and we use the Betti map which goes back to work of Masser and Zannier…

We develop an explicit theory of Kummer varieties associated to Jacobians of hyperelliptic curves of genus 3, over any field $k$ of characteristic $\neq 2$. In particular, we provide explicit equations defining the Kummer variety $\mathcal…