Related papers: Computing the Cassels-Tate Pairing for Genus Two J…

Building on work of Balakrishnan, Dogra, and of the first author, we provide some improvements to the explicit quadratic Chabauty method to compute rational points on genus $2$ bielliptic curves over $\mathbb{Q}$, whose Jacobians have…

We give a bound on the number of points of order two on the theta divisor of a principally polarized abelian variety A. When A is the Jacobian of a curve C the result can be applied in estimating the number of effective square roots of a…

In this paper we study genus 2 curves whose Jacobians admit a polarized (4,4)-isogeny to a product of elliptic curves. We consider base fields of characteristic different from 2 and 3, which we do not assume to be algebraically closed. We…

We construct three families of pairs of genus 2 curves over a field K, whose Jacobians are isomorphic as unpolarized abelian varieties. Each family is parameterized by an open subset of the Projective line over K. Our construction is based…

In [5], without giving a detailed proof, Yamauchi provided a formula to calculate the genus of a certain family of smooth complete intersection algebraic curves. That formula is used extensively in [1] to study the algebraic curves for…

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…

We present an algorithm solving the following problem: given two genus 2 curves over a field k with isogenous Jacobians, compute such an isogeny explicitly. This isogeny can be either an l-isogeny or, in the real multiplication case, an…

Let p be a prime and let C be a genus one curve over a number field k representing an element of order dividing p in the Shafarevich-Tate group of its Jacobian. We describe an algorithm which computes the set of D in the Shafarevich-Tate…

We show how to efficiently compute functions on jacobian varieties and their quotients. We deduce a quasi-optimal algorithm to compute $(l,l)$ isogenies between jacobians of genus two curves.

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…

In this paper we present a new method to show that a principal homogeneous space of the Jacobian of a curve of genus two is nontrivial. The idea is to exhibit a Brauer-Manin obstruction to the existence of rational points on a quotient of…

Distortion maps are a useful tool for pairing based cryptography. Compared with elliptic curves, the case of hyperelliptic curves of genus g > 1 is more complicated since the full torsion subgroup has rank 2g. In this paper we prove that…

Let $p$ be an odd prime number and be an integer coprime to $p$. We survey an algorithm for computing explicit rational representations of $(\ell,...,\ell)$-isogenies between Jacobians of hyperelliptic curves of arbitrary genus over an…

We say that an abelian variety is degenerate if its Hodge ring is not generated by divisor classes. Degeneracy leads to some interesting challenges when computing Sato-Tate groups, and there are currently few examples and techniques…

We implement two-cover descent for plane quartics over Q with all 28 bitangents rational and show that on a significant collection of test cases, it resolves the existence of rational points. We also review a classical description of the…

We determine the limiting distribution of the normalized Euler factors of an abelian surface A defined over a number field k when A is isogenous to the square of an elliptic curve defined over k with complex multiplication. As an…

We consider the identity component of the Sato-Tate group of the Jacobian of curves of the form $$C_1\colon y^2=x^{2g+2}+c, C_2\colon y^2=x^{2g+1}+cx, C_3\colon y^2=x^{2g+1} +c,$$ where $g$ is the genus of the curve and $c\in\mathbb Q^*$ is…

Let $A$ be an abelian variety defined over a number field $K$. The number of torsion points that are rational over a finite extension $L$ is bounded polynomially in terms of the degree $[L:K]$ of $L$ over $K$. Under the following three…

Suppose $X$ is a hyperelliptic curve of genus $g$ defined over an algebraically closed field $k$ of characteristic $p=2$. We prove that the de Rham cohomology of $X$ decomposes into pieces indexed by the branch points of the hyperelliptic…

Let k be a number field and X a smooth projective k-variety. In this paper, we study the information obtainable from descent via torsors under finite k-group schemes on the location of the k-rational points on X within the adelic points.…