Related papers: \'Equation de Pell-Abel et applications

Let $C$ be a hyperelliptic curve defined over $\mathbb{Q}$, whose Weierstrass points are defined over extensions of $\mathbb{Q}$ of degree at most three, and at least one of them is rational. Generalizing a result of R. Soleng (in the case…

We study the Betti map of a particular (but relevant) section of the family of Jacobians of hyperelliptic curves using the polynomial Pell equation $A^2-DB^2=1$, with $A,B,D\in \mathbb C[t]$ and certain ramified covers ${\mathbb P}^1\to…

Let $\mathcal{M}_{g,2}$ be the moduli space of curves of genus $g$ with a level-2 structure. We prove here that there is always a non hyperelliptic element in the intersection of four thetanull divisors in $\mathcal{M}_{6,2}$. We prove also…

In this paper we study bielliptic curves of genus 3 defined over an algebraically closed field $k$ and the intersection of the moduli space $\M_3^b$ of such curves with the hyperelliptic moduli $\H_3$. Such intersection $\S$ is an…

For the superelliptic curves of the form $$ (x+1) \cdots(x+i-1)(x+i+1)\cdots (x+k)=y^\ell$$ with $x,y \in \mathbb{Q}$, $y\neq 0$, $k \geq 3$, $1\leq i\leq k$, $\ell \geq 2,$ a prime, Das, Laishram, Saradha, and Edis showed that the…

We show variants of the genus inequality for the irreducible components of the special fiber of an arithmetic curve over a henselian discrete valuation ring of residue characteristic zero that take into account the non-existence of…

We study the local invariants that a meromorphic $k$-differential on a Riemann surface of genus $g\geq0$ can have. These local invariants are the orders of zeros and poles, and the $k$-residues at the poles. We show that for a given pattern…

We consider a family of genus $g$ hyperelliptic curves as double ramified coverings over the Riemann sphere with the set of branch points of the form $\{0, \infty, x_1, \dots, x_g, u_1, \dots, u_g\}$. The branch point at infinity $P_\infty$…

The purpose of this paper is to study hyperelliptic curves with extra involutions. The locus $\L_g$ of such genus $g$ hyperelliptic curves is a $g$-dimensional subvariety of the moduli space of hyperelliptic curves $\H_g$. We discover a…

It follows from the Grothendieck-Ogg-Shafarevich formula that the rank of an abelian variety (with trivial trace) defined over the function field of a curve is bounded by a quantity which depends on the genus of the base curve and on bad…

Let $k$ be a finite field and $L$ be the function field of a curve $C/k$ of genus $g\geq 1$. In the first part of this note, we show that the number of separable $S$-integral points on a constant elliptic curve $E/L$ is bounded solely in…

Consider a hyperelliptic curve of genus $2$ over a field $K$ of characteristic zero. After extending $K$ we can view it as a marked curve with its $6$ Weierstrass points. We classify the structure of the potentially stable reduction of such…

This paper deals with singularities of genus 2 curves on a general (d_1,d_2)-polarized abelian surface (S,L). In analogy with Chen's results concerning rational curves on K3 surfaces [Ch1,Ch2], it is natural to ask whether all such curves…

Let $K$ be a number field, $n>4$ an integer, $f(x)$ an irreducible polynomial over $K$ of degree $n$, whose Galois group is either the full symmetric group $S_n$ or the alternating group $A_n$. Suppose $C:y^2=f(x)$ is the corresponding…

This paper is devoted to the study of the $\ell$-adic representations of the absolute Galois group $G$ of ${\mathbb Q}_p$, $p\geq 5$, associated to an elliptic curve over ${\mathbb Q}_p$, as $\ell$ runs through the set of all prime numbers…

We investigate a class of odd (ramification) coverings $C \to \mathbb{P}^1$ where $C$ is hyperelliptic, its Weierstrass points maps to one fixed point of $\mathbb{P}^1$ and the covering map makes the hyperelliptic involution of $C$ commute…

We present an index calculus algorithm with double large prime variation which lends itself well to a rigorous analysis. Using this algorithm we prove that for fixed genus $g \geq 2$, the discrete logarithm problem in degree 0 class groups…

A problem of current interest, also motivated by applications to Coding theory, is to find explicit equations for \textit{maximal} curves, that are projective, geometrically irreducible, non-singular curves defined over a finite field…

We propose interconnections between some problems of PDE, geometry, algebra, calculus and physics. Uniqueness of a solution of the Dirichlet problem and of some other boundary value problems for the string equation inside an arbitrary…

We construct families of smooth, proper, algebraic curves in characteristic 0, of arbitrary genus g, together with g elements in the kernel of the tame symbol. We show that those elements are in general independent by a limit calculation of…