Related papers: Decomposing Jacobians via Galois covers

By the Jacquet-Langlands correspondence and Faltings' isogeny theorem, it is known that the abelian variety J_0^{Dpq}(N) (the Jacobian of a Shimura curve of discriminant Dpq with Gamma_0(N) level structure) is isogenous to the pq-new…

Given an abelian variety $X$ and a point $a\in X$ we denote by $<a>$ the closure of the subgroup of $X$ generated by $a$. Let $N=2^g-1$. We denote by $\kappa: X\to \kappa(X)\subset\mathbb P^N$ the map from $X$ to its Kummer variety. We…

In this thesis, we study the Berkovich skeleton of an algebraic curve over a discretely valued field $K$. We do this using coverings $C\rightarrow{\mathbb{P}^{1}}$ of the projective line. To study these coverings, we take the Galois closure…

Given a canonical genus three curve $X=\{F=0\}$, we construct, emulating Mumford discussion for hyperelliptic curves, a set of equations for an affine open subset of the jacobian $JX$. We give explicit algorithms describing the law group in…

Given a generic degree-2 cover of a genus 1 curve D by a non hyperelliptic genus 3 curve C over a field k of characteristic different from 2, we produce an explicit genus 2 curve X such that Jac(C) is isogenous to the product of Jac(D) and…

Up to isomorphism over C, every simple principally polarized abelian variety of dimension 3 is the Jacobian of a smooth projective curve of genus 3. Furthermore, this curve is either a hyperelliptic curve or a plane quartic. Given a sextic…

In this paper we study the Galois group of the Galois cover of the composition of a $q$-cyclic \'etale cover and a cyclic $p$-gonal cover for any odd prime $p$. Furthermore, we give properties of isogenous decompositions of certain Prym and…

It is well known that the Prym variety of an \'etale cyclic covering of a hyperelliptic curve is isogenous to the product of two Jacobians. Moreover, if the degree of the covering is odd or congruent to 2 mod 4, then the canonical isogeny…

Using Galois Theory, we construct explicitly absolutely simple (principally polarized) Prym varieties that are not isomorphic to jacobians of curves even if we ignore the polarizations. Our approach is based on the previous papers…

In a previous paper, the authors proved that the Prym variety of any non-cyclic etale triple cover of a smooth curve of genus 2 is a Jacobian variety of dimension 2. This gives a map from the moduli space of such covers to the moduli space…

We explore connections between the category of tropical abelian varieties (tav), $\mathbb{T}\mathcal{A}$, and the the category of tropical curves, $\mathbb{T}\mathcal{C}$, first in a broader context and then specifically by studying the…

Let $Y\to X$ be an unramified Galois cover of curves over a perfect field $k$ of characteristic $p>0$ with $\mathrm{Gal}(Y/X)\cong\mathbb{Z}/p\mathbb{Z}$, and let $J_X$ and $J_Y$ be the Jacobians of $X$ and $Y$ respectively. We consider the…

To every double cover ramified in two points of a general trigonal curve of genus g, one can associate an \'etale double cover of a tetragonal curve of genus g+1. We show that the corresponding Prym varieties are canonically isomorphic as…

We show that any polarized abelian variety over a finite field is covered by a Jacobian whose dimension is bounded by an explicit constant. We do this by first proving an effective version of Poonen's Bertini theorem over finite fields,…

In this paper we give a unified approach in categorical setting to the problem of finding the Galois closure of a finite cover, which includes as special cases the familiar finite separable field extensions, finite unramified covers of a…

The action of a finite group $G$ on a compact Riemann surface $X$ naturally induces another action of $G$ on its Jacobian variety $\operatorname{J}(X)$. In many cases, each component of the group algebra decomposition of…

We study wildly ramified G-Galois covers $\phi:Y \to X$ branched at B (defined over an algebraically closed field of characteristic p). We show that curves Y of arbitrarily high genus occur for such covers even when G, X, B and the inertia…

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

We extend our method to compute division polynomials of Jacobians of curves over Q to curves over Q(t), in view of computing mod ell Galois representations occurring in the \'etale cohomology of surfaces over Q. Although the division…

We study an explicit $(2g-1)$-dimensional family of Jacobian varieties of dimension $\frac{d-1}2(g-1)$, arising from quotient curves of unramified cyclic coverings of prime degree $d$ of hyperelliptic curves of genus $g\ge 2$. By using a…