Related papers: Explicit Constructions for Genus 3 Jacobians

We study the Selmer variety associated to a canonical quotient of the $\Q_p$-pro-unipotent fundamental group of a smooth projective curve of genus at least two defined over $\Q$ whose Jacobian decomposes into a product of abelian varieties…

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…

Topological properties of the jacobian curve ${\mathcal J}_{\mathcal{F},\mathcal{G}}$ of two foliations $\mathcal{F}$ and $\mathcal{G}$ are described in terms of invariants associated to the foliations. The main result gives a decomposition…

A general canonical curve X determines a finite set T(X) of hyperplanes, which is in bijective correspondence with the set of odd theta-characteristics of X. The definition of T(X) can be extended to certain singular curves, in a way that…

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,…

For the evaluation and inversion of abelian integrals we show that the image of the Abel-Jacobi map of genus less than 5 hyperelliptic curve in its Jacobian is the intersection of shifted theta divisors with specified shifts. Therefore the…

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 introduce the notion of a `canonical' splitting over Z or ZxZ for a finitely generated group G. We show that when G happens to be the fundamental group of an orientable Haken manifold M with incompressible boundary, then the…

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…

This paper is concerned with the complexity and stability of arithmetic operations in the jacobian variety of curves over the field of complex numbers, as the genus grows to infinity. We focus on modular curves. Efficient and stable…

We give a new approach to the elliptic curve discrete logarithm problem over cubic extension fields $\mathbb{F}_{q^3}$. It is based on a transfer: First an $\mathbb{F}_q$-rational $(\ell,\ell,\ell)$-isogeny from the Weil restriction of the…

We study smooth curves on abelian surfaces, especially for genus 4, when the complementary subvariety in the Jacobian is also a surface. We show that up to translation there is exactly one genus 4 hyperelliptic curve on a general (1,…

If $C:y^2=x(x-1)(x-a_1)(x-a_2)(x-a_3)$ is genus $2$ curve a natural question to ask is: Under what conditions on $a_1,a_2,a_3$ does the Jacobian $J(C)$ have real multiplication by $\mathbb{Z}[\sqrt{\Delta}]$ for some $\Delta>0$. Over a…

In this article, we present a method for computing rational points on hyperelliptic curves of genus~3 and isolated quadratic points on hyperelliptic curves of genus~2 and~3 whose Jacobians have rank~0. Our approach begins by computing the…

This thesis is devoted to the study of abelian automorphism groups of surfaces and $3$-folds of general type over complex number field $\Bbb C$. We obtain a linear bound in $K^3$ for abelian automorphism groups of $3$-folds of general type…

We present an algorithm for solving the discrete logarithm problem in Jacobians of families of plane curves whose degrees in $X$ and $Y$ are low with respect to their genera. The finite base fields $\FF_q$ are arbitrary, but their sizes…

We address the question of existence of absolutely simple abelian varieties of dimension 2 with everywhere good reduction over quadratic fields. The emphasis will be given to the construction of pairs $(K,C)$, where $K$ is a quadratic…

We survey the theory of the compactified Jacobian associated to a singular curve. We focus on describing low genus examples using the Abel map.

this paper is devoted to the study of curves of genus 3 with group of automorphisms the symmetric group S3, principally over finite fields, in view to obtain optimal curves. For instance, we prove that, over the finite fields of char. 3,…

We provide a complete enumeration of all quotients of genus $0, 1$ and $2$ of the Shimura curves $X_0^D(N)$ over $\mathbb{Q}$ by non-trivial subgroups of Atkin--Lehner involutions. For all $1270$ genus $1$ quotients $X$ with $N$ squarefree,…