Related papers: First vertices for hyperelliptic curves in charact…

In this paper, we determine the reduced automorphism groups of hyperelliptic curves of a small genus in characteristic $2$, when they are of $2$-rank $0$. Such a curve is an Artin-Schreier curve defined in the form $y^2-y=f(x)$ for a…

This paper describes a class of Artin-Schreier curves, generalizing results of Van der Geer and Van der Vlugt to odd characteristic. The automorphism group of these curves contains a large extraspecial group as a subgroup. Precise knowledge…

We prove that for any number field $K$ and any fixed genus $g \geq 2$, there are infinitely many non-isomorphic hyperelliptic curves of genus $g$ over $K$ whose Jacobians have rank over $K$ equal to each of 0, 1, or 2. As an example of our…

We establish identities of Pfaffian type for the theta function associated with twice or half the period matrix of a hyperelliptic curve. They are implied by the large size asymptotic analysis of exact Pfaffian identities for expectation…

In a previous article, a universal linear algebraic model was proposed for describing homogeneous conformal geometries, such as the spherical, Euclidean, hyperbolic, Minkowski, anti-de Sitter and Galilei planes. This formalism was…

In this paper we examine curves defined over a field of characteristic 2 which are $(\ZZ/2\ZZ)^2$-covers of the projective line. In particular, we prove which 2-ranks occur for such curves of a given genus and where possible we give…

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…

In this study, the new characterizations of special curves are investigated without using the curvatures of these special curves: general helices, slant helices, Bertrand curves, Mannheim curves. The curvatures are given by the help of the…

We compute the $\delta$-invariant of a curve singularity parameterized by generic sparse polynomials. We apply this to describe topological types of generic singularities of sparse resultants and ``algebraic knot diagrams'' (i.e. generic…

The van der Geer--van der Vlugt curves form a class of Artin--Schreier coverings of the projective line over finite fields. We provide an explicit formula for their $L$-polynomials in characteristic $2$, expressed in terms of characters of…

The zeta function of a curve over a finite field may be expressed in terms of the characteristic polynomial of a unitary symplectic matrix, called the Frobenius class of the curve. We compute the expected value of the trace of the n-th…

Building on our previous work (arXiv:1405.5711), we develop the first practical algorithm for computing topological zeta functions of nilpotent groups, non-associative algebras, and modules. While we previously depended upon non-degeneracy…

Given a principally polarized abelian variety $(A,\Theta)$, we give a characterization of all elliptic curves that lie on $A$ in terms of intersection numbers of divisor classes in its N\'eron-Severi group.

Fixing $t \in \mathbb{R}$ and a finite field $\mathbb{F}_q$ of odd characteristic, we give an explicit upper bound on the proportion of genus $g$ hyperelliptic curves over $\mathbb{F}_q$ whose zeta function vanishes at $\frac{1}{2} + it$.…

In this article, we continue to study the geometry of bisections of certain rational elliptic surfaces. As an application, we give examples of Zariski N + 1-plets of degree 2N + 4 whose irreducible components are an irreducible quartic…

Inside the moduli space of curves of genus three with one marked point, we consider the locus of hyperelliptic curves with a marked Weierstrass point, and the locus of non-hyperelliptic curves with a marked hyperflex. These loci have…

Let X be an irreducible smooth projective curve, of genus at least two, defined over an algebraically closed field of characteristic different from two. If X admits a nontrivial automorphism \sigma that fixes pointwise all the order two…

This article concerns the computational complexity of a fundamental problem in number theory: counting points on curves and surfaces over finite fields. There is no subexponential-time algorithm known and it is unclear if it can be…

Let E_G be a family of hyperelliptic curves over F2^(alg cl) with general Weierstrass equation given over a very small field F. We describe in this paper an algorithm to compute the zeta function of E_g for g in a degree n extension field…

In this paper we study genus-$4$ curves obtained as double covers of elliptic curves. Firstly we shall give explicit defining equations of such curves with explicit criterion for whether it is nonsingular, and show the irreducibility of the…