Related papers: Equations defining Jacobians with Real Multiplicat…

Inspired by experimental data, this paper investigates which isogeny classes of abelian varieties defined over a finite field of odd characteristic contain the Jacobian of a hyperelliptic curve. We provide a necessary condition by…

Given a sextic CM field $K$, we give an explicit method for finding all genus 3 hyperelliptic curves defined over $\mathbb{C}$ whose Jacobians are simple and have complex multiplication by the maximal order of this field, via an…

Based on the reduction of degree in polynomial mappings and some known results in algebraic geometry, by introducing the Brouwer degree, a tool from differential topology, algebraic topology and algebraic geometry, we completely prove the…

If $E$ is an elliptic curve, defined over $\mathbb{Q}$ or a number field having at least one real embedding, then Elkies proved that $E$ has supersingular reduction at infinitely many primes $p$. Baba and Granath extended this result to…

We give parametrisation of curves C of genus 2 with a maximal isotropic (ZZ/3)^2 in J[3], where J is the Jacobian variety of C, and develop the theory required to perform descent via (3,3)-isogeny. We apply this to several examples, where…

We study the inverse Jacobian problem for the case of Picard curves over $\mathbb{C}$. More precisely, we elaborate on an algorithm that, given a small period matrix $\Omega\in \mathbb{C}^{3\times 3}$ corresponding to a principally…

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…

For the hyperelliptic curve C_p with equation y^2=x(x-2p)(x-p)(x+p)(x+2p) with p a prime number, we discuss bounds for the rank of its Jacobian over Q, find many cases having 2-torsion in the associated Shafarevich-Tate group, and we…

In this article we prove lower and upper bounds for class numbers of algebraic curves defined over finite fields. These bounds turn out to be better than most of the previously known bounds obtained using combinatorics. The methods used in…

Let $p$ and $q$ be distinct primes. Consider the Shimura curve $\mathcal{X}$ associated to the indefinite quaternion algebra of discriminant $pq$ over $\mathbb{Q}$. Let $J$ be the Jacobian variety of $\mathcal{X}$, which is an abelian…

Genus 2 curves have been an object of much mathematical interest since eighteenth century and continued interest to date. They have become an important tool in many algorithms in cryptographic applications, such as factoring large numbers,…

Let $A/K$ be an abelian variety with real multiplication defined over a $p$-adic field $K$ with $p>2$. We show that $A/K$ must have either potentially good or potentially totally toric reduction. In the former case we give formulas of the…

We study deformations of rational curves and their singularities in positive characteristic. We use this to prove that if a smooth and proper surface in positive characteristic $p$ is dominated by a family of rational curves such that one…

We show that we can develop from scratch and using only classical language a theory of relative quadratic extensions of a given number field $K$ which is as explicit and easy as for the well-known case that $K$ is the field of rational…

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 consider the generalized Jacobian $\widetilde{J}$ of the modular curve $X_0(N)$ of level $N$ with respect to a reduced divisor consisting of all cusps. Supposing $N$ is square free, we explicitly determine the structure of the…

For a curve of genus at least four which is either very general or very general hyperelliptic, we classify all ways in which a power of its Jacobian can be isogenous to a product of Jacobians of curves. As an application, we show that, for…

The Jacobian algebra associated to a triangulation of a closed surface $S$ with a collection of marked points $M$ is (weakly) symmetric and tame. We show that for these algebras the Auslander-Reiten translate acts 2-periodical on objects.…

We introduce deformation theoretic methods for determining when a curve $X$ in a non-hyperelliptic jacobian $JC$ will deform with $JC$ to a non-jacobian. We apply these methods to a particular class of curves in the second symmetric power…

A smooth complex variety satisfies the Generalized Jacobian Conjecture if all its \'etale endomorphisms are proper. We study the conjecture for $\mathbb{Q}$-acyclic surfaces of negative Kodaira dimension. We show that $G$-equivariant…