Related papers: Completely decomposable modular Jacobians

We present a new technique to study Jacobian variety decompositions using subgroups of the automorphism group of the curve and the corresponding intermediate covers. In particular, this new method allows us to produce many new examples of…

We give equations for 13 genus-2 curves over $\overline{\mathbb{Q}}$, with models over $\mathbb{Q}$, whose unpolarized Jacobians are isomorphic to the square of an elliptic curve with complex multiplication by a maximal order. If the…

We present three families of pairs of geometrically non-isomorphic curves whose Jacobians are isomorphic to one another as unpolarized abelian varieties. Each family is parametrized by an open subset of P^1. The first family consists of…

In this paper we give examples of smooth projective curves whose Jacobians are isogenus to a product of an arbitrarily high number of Jacobians

In this note we give explicit constructions of decomposable hyperelliptic Jacobian varieties over fields of characteristic $0$. These include hyperelliptic Jacobian varieties that are isogenous to a product of two absolutely simple…

We prove that there is only a finite number of genus 2 curves C defined over Q such that there exists a nonconstant morphism pi:X_1(N) --->C defined over Q and the jacobian of C, J(C), is a Q-factor of the new part of the jacobian of…

We construct examples of families of curves of genus 2 or 3 over Q whose Jacobians split completely and have various large rational torsion subgroups. For example, the rational points on a certain elliptic surface over P^1 of positive rank…

A curve over the field of two elements with completely decomposable Jacobian is shown to have at most six rational points and genus at most 26. The bounds are sharp. The previous upper bound for the genus was 145. We also show that a curve…

For each open subgroup $G$ of ${\rm GL}_2(\hat{\mathbb{Z}})$ containing $-I$ with full determinant, let $X_G/\mathbb{Q}$ denote the modular curve that loosely parametrizes elliptic curves whose Galois representation, which arises from the…

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…

Let $C$ be a curve of genus 2 and $\psi_1:C \lar E_1$ a map of degree $n$, from $C$ to an elliptic curve $E_1$, both curves defined over $\bC$. This map induces a degree $n$ map $\phi_1:\bP^1 \lar \bP^1$ which we call a Frey-Kani covering.…

We analyze complex multiplication for Jacobians of curves of genus 3, as well as the resulting Shimura class groups and their subgroups corresponding to Galois conjugation over the reflex field. We combine our results with numerical methods…

We produce new explicit examples of genus-2 curves over the rational numbers whose Jacobian varieties have rational torsion points of large order. In particular, we produce a family of genus-2 curves over Q whose Jacobians have a rational…

We consider families of abelian Galois coverings of the line. When the Jacobian of the general element is totally decomposable, i.e., is isogenous to a product of elliptic curves, we prove that they yield special subvarieties of $\A_g$ if…

We use methods for computing Picard numbers of reductions of K3 surfaces in order to study the decomposability of Jacobians over number fields and the variance of Mordell-Weil ranks of families of Jacobians over different ground fields. For…

For a class of non-hyperelliptic genus 3 curves C which are 2-fold coverings of elliptic curves E, we give an explicit algebraic description of all birationally non-equivalent genus 2 curves whose Jacobians are degree 2 isogeneous to the…

Let $\phi:\,X\rightarrow Y$ be a (possibly ramified) cover between two algebraic curves of positive genus. We develop tools that may identify the Prym variety of $\phi$, up to isogeny, as the Jacobian of a quotient curve $C$ in the Galois…

We investigate the Jacobian decomposition of some algebraic curves over finite fields with genus $4$, $5$ and $10$. As a corollary, explicit equations for curves that are either maximal or minimal over the finite field with $p^2$ elements…

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…

Let $K$ be a field of characteristic different from $2$, $\bar{K}$ its algebraic closure. Let $n \ge 3$ be an odd prime such that $2$ is a primitive root modulo $n$. Let $f(x)$ and $h(x)$ be degree $n$ polynomials with coefficients in $K$…