Related papers: Doubly isogenous genus-2 curves with $D_4$-action

We prove the following result which was conjectured by Stichtenoth and Xing: let $g$ be the genus of a projective, irreducible non-singular curve over the finite field $\Bbb F_{q^2}$ and whose number of $\Bbb F_{q^2}$-rational points…

Let $\X$ be an irreducible, smooth, projective curve of genus $g \geq 2$ defined over the complex field $\C.$ Then there is a covering $\pi: \X \longrightarrow \P^1,$ where $\P^1$ denotes the projective line. The problem of expressing…

Boix, De Stefani and Vanzo have characterized ordinary/supersingular elliptic curves over $\mathbb{F}_p$ in terms of the level of the defining cubic homogenous polynomial. We extend their study to arbitrary genus, in particular we prove…

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…

A superspecial curve is a (non-singular) curve over a field of positive characteristic whose Jacobian variety is isomorphic to a product of supersingular elliptic curves over the algebraic closure. It is known that for given genus and…

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…

Consider the Jacobian of a hyperelliptic genus two curve defined over a prime field of characteristic p and with complex multiplication. In this paper we show that the p-Sylow subgroup of the Jacobian is either trivial or of order p.

A basis for the space of generalized theta functions of level one for the spin groups, parameterized by the theta characteristics (the even theta characteristcs for the odd spin groups) on a curve, is shown to be projectively flat over the…

We study the set of isomorphism classes of principal polarizations on abelian varieties of GL2-type. As applications of our results, we construct examples of curves C, C'/\Q of genus two which are nonisomorphic over \bar \Q and share…

We show that finite fields over which there is a curve of a given genus g with its Jacobian having a small exponent, are very rare. This extends a recent result of W. Duke in the case g=1. We also show when g=1 or g=2 that our bounds are…

We show that up to isomorphism there are exactly twenty pairs $(C,E)$, where $C$ is a genus-$2$ curve over ${\mathbf C}$, where $E$ is an elliptic curve over ${\mathbf C}$, and where for every integer $n>1$ there is a map of degree $n$ from…

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…

We present a construction explaining the existence of (unexpected) curves of degree $d+k$, passing through a set $Z$ of points on $\mathbb{P}^2$, and having a generic point $P$ of multiplicity $d$. The construction is based on the syzygies…

Let $K$ be an algebraically closed field of characteristic different from 2, $g$ a positive integer, $f(x)$ a degree $(2g+1)$ polynomial with coefficients in $K$ and without multiple roots, $C: y^2=f(x)$ the corresponding genus $g$…

Let G2 be the exceptional Lie group of automorphisms of the complex Cayley algebra and C be a smooth, connected, projective curve of genus at least 2. Using the map obtained from extension of structure groups, we prove explicit links…

We study genus 2 covers of relative elliptic curves over an arbitrary base in which 2 is invertible. Particular emphasis lies on the case that the covering degree is 2. We show that the data in the "basic construction" of genus 2 covers of…

Let K be an algebraically closed field of characteristic 2, and let X be a curve over K of genus g>1 and 2-rank r>0. For 2-subgroups S of the K-automorphism group Aut(X) of X, the Nakajima bound is |S| < 4g-3. For every g=2^h+1>8, we…

For a given hyperelliptic curve $C$ over a finite field with Jacobian $J_C$, we consider the hyperelliptic analogue of the congruential generator defined by $W_n=W_{n-1}+D$ for $n\geq 1$ and $D,W_0\in J_C$. We show that curves of genus 2…

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 construct six infinite series of families of pairs of curves (X,Y) of arbitrarily high genus, defined over number fields, together with an explicit isogeny from the Jacobian of X to the Jacobian of Y splitting multiplication by 2, 3, or…