Related papers: Computing canonical heights using arithmetic inter…

We study the arithmetic of complete intersections in projective space over number fields. Our main results include arithmetic Torelli theorems and versions of the Shafarevich conjecture, as proved for curves and abelian varieties by…

We explicitly find an equation and a projective embedding of the Kummer surface associated to the Jacobian of a curve of genus 2 given by an equation of the form y^2 + h(x)y = f(x) over an arbitrary ground field as well as several maps that…

We enumerate rational curves in toric surfaces passing through points and satisfying cross-ratio constraints using tropical and combinatorial methods. Our starting point is arXiv:1509.07453, where a tropical-algebraic correspondence theorem…

A fast algorithm for counting intersections of two normal curves on a triangulated surface is proposed. It yields a convenient way for treating mapping class groups of punctured surfaces by presenting mapping classes by matrices, and the…

In this article, we present a method for computing rational points on hyperelliptic curves of genus~3 and isolated quadratic points on hyperelliptic curves of genus~2 and~3 whose Jacobians have rank~0. Our approach begins by computing the…

An approach to the calculation of local canonical morphic heights is described, motivated by the analogy between the classical height in Diophantine geometry and entropy in algebraic dynamics. We consider cases where the local morphic…

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 outline a strategy for computing intersection numbers on smooth varieties with torus actions using a residue formula of Bott. As an example, Gromov-Witten numbers of twisted cubic and elliptic quartic curves on some general complete…

We adopt a statistical point of view on the conjecture of Lang which predicts a lower bound for the canonical height of non-torsion rational points on elliptic curves defined over $\mathbb{Q}$. More specifically, we prove that among the…

In this article we give explicit formulae for a lift of the relative Frobenius morphism between elliptic curves and show how one can compute this lift in the case of ordinary reduction in odd characteristic. Our theory can also be used in…

We construct algebraic curves in abelian surfaces starting from tropical curves in real tori. We give a necessary and sufficient condition for a tropical curve in a real torus to be realizable by an algebraic curve in an abelian surface.…

Canonical differential calculus is defined for finitely generated abelian group with an involution existing consistently. Two such canonical calculi are found out. Fermionic representation for canonical calculus is defined based on…

We study the interplay between canonical heights and endomorphisms of an abelian variety $A$ over a number field $k$. In particular we show that whenever the ring of endomorphisms defined over $k$ is strictly larger than $\Z$ there will be…

Let C be a curve of genus at least 2 imbedded in a product of elliptic curves. We give an explicit upper bound for the points in the intersection of C with the union of all algebraic subgroups of a certain codimension. As a corollary we…

We introduce a new canonical height function for Jordan blocks of small eigenvalues for endomorphisms on smooth projective varieties over a number field. We prove that under an assumption on the eigenvalues of the endomorphism on the group…

By a theorem of Wahl, the canonically embedded curves which are hyperplane section of K3 surfaces are distinguished by the non-surjectivity of their Wahl map. In this paper we address the problem of distinguishing hyperplane sections of…

It is a longstanding problem to determine the precise relationship between the geodesic length spectrum of a hyperbolic manifold and its commensurability class. A well known result of Reid, for instance, shows that the geodesic length…

We enumerate plane complex algebraic curves of a given degree with one singularity of any given topological type. Our approach is to compute the homology classes of the corresponding equisingular strata in the parameter spaces of plane…

We obtain new results on the geometry of Hilbert modular varieties in positive characteristic and morphisms between them. Using these results and methods of rigid geometry, we develop a theory of canonical subgroups for abelian varieties…

We use the relations between quadrics, trace codes and algebraic curves to construct algebraic curves over finite fields with many points and to compute generalized Hamming weights of codes.