Related papers: Equations defining Jacobians with Real Multiplicat…

We approach the Torelli problem of recostructing a curve from its Jacobian from a computational point of view. Following Dubrovin, we design a machinery to solve this problem effectively, which builds on methods in numerical algebraic…

We study the arithmetic of abelian varieties over $K=k(t)$ where $k$ is an arbitrary field. The main result relates Mordell-Weil groups of certain Jacobians over $K$ to homomorphisms of other Jacobians over $k$. Our methods also yield…

We study $2$-representation finite $\mathbb{K}$-algebras obtained from tensor products of tensor algebras of species. In earlier work we computed the higher preprojective algebra of said algebras to be given as Jacobian algebras of certain…

We formulate a solution to the Algebraic version of the Inverse Jacobi problem. Using this solution we produce explicit addition laws on any algebraic curve generalizing the law suggested by Leykin [2] in the case of (n, s) curves. This…

The two dimensional Jacobian Conjecture says that a morphism $f:\mathbb{C}[x,y]\to \mathbb{C}[x,y]$ having an invertible Jacobian, is invertible. We show that a morphism $f$ having an invertible Jacobian is invertible, in each of the…

Let A be a principally polarized abelian threefold over a perfect field k, not isomorphic to a product over the algebraic closure of k. There exists a canonical extension k' of k, of degree 1 or 2, such that A becomes isomorphic to a…

Let $K$ be a field of characteristic different from $2$, $\bar{K}$ its algebraic closure. Let $n \ge 3$ be an odd integer. Let $f(x)$ and $h(x)$ be degree $n$ polynomials with coefficients in $K$ and without repeated roots. Let us consider…

We define and study Jacobians of Hodge structures with weight greater than 1. Jacobians of weight 2 naturally come up in the context of the Brauer group and the Tate conjecture. They were previously studied in a special case by Beauville in…

We prove the following converse of Riemann's Theorem: let (A,\Theta) be an indecomposable principally polarized abelian variety whose theta divisor can be written as a sum of a curve and a codimension two subvariety \Theta=C+Y. Then C is…

We study degree 2 and 4 elliptic subcovers of hyperelliptic curves of genus 3 defined over $\mathbb C$. The family of genus 3 hyperelliptic curves which have a degree 2 cover to an elliptic curve $E$ and degree 4 covers to elliptic curves…

The Jacobian $J$ of a complete, smooth, connected curve $X$ admits a canonical divisor $\Theta$, called the Theta divisor. It is well-known that $\Theta$ is ample and, in fact, $3\Theta$ is very ample. For a general complete, integral curve…

Denote by $J_m$ the Jacobian variety of the hyperelliptic curve defined by the affine equation $y^2=x^m+1$ over $\mathbb{Q}$, where $m \geq 3$ is a fixed positive integer. We compute several interesting arithmetic invariants of $J_m$: its…

In this paper, we first show that the Jacobian Conjecture is true for non-homogeneous power linear mappings under some conditions. Secondly, we prove an equivalent statement about the Jacobian Conjecture in dimension $r\geq 1$ and give some…

In this paper, we give an explicit formula as well as a practical algorithm for computing the Cassels-Tate pairing on $\text{Sel}^{2}(J) \times \text{Sel}^{2}(J)$ where $J$ is the Jacobian variety of a genus two curve under the assumption…

This article is about polynomial maps with a certain symmetry and/or antisymmetry in their Jacobians, and whether the Jacobian Conjecture is satisfied for such maps, or whether it is sufficient to prove the Jacobian Conjecture for such…

In 2012, Zilber used model-theoretic techniques to show that a curve of high genus over an algebraically closed field is determined by its Jacobian (viewed only as an abstract group with a distinguished subset for an image of the curve). In…

Let $k$ be a number field. We investigate the Mordell-Weil ranks of Jacobian varieties $J_C$ associated with algebraic curves $C$ of genus $g \geq 1$ defined by affine equations of the form $y^s=x(ax^r+b)$, where $a, b \in k$ ($ab \neq 0$),…

We give a general framework for uniform, constant-time one-and two-dimensional scalar multiplication algorithms for elliptic curves and Jacobians of genus 2 curves that operate by projecting to the x-line or Kummer surface, where we can…

An abelian surface A over a field K has potential quaternionic multiplication if the ring End_\bar K (A) of geometric endomorphisms of A is an order in an indefinite rational division quaternion algebra. In this brief note, we study the…

Let $K$ be a number field, and let $C$ be a hyperelliptic curve over $K$ with Jacobian $J$. Suppose that $C$ is defined by an equation of the form $y^{2} = f(x)(x - \lambda)$ for some irreducible monic polynomial $f \in \mathcal{O}_{K}[x]$…