Counting isomorphism classes of pointed hyperelliptic curves of genus 4 over finite fields with odd characteristic

This paper is devoted to computing the number of isomorphism classes of pointed hyperelliptic curves over finite fields. We deal with the genus-4 case and the finite fields are of odd characteristic. The number of isomorphism classes is computed. This number can be represented as a polynomial in q of degree 7, where q is the order of the finite field. The results have applications in the classification problems and in the hyperelliptic curve cryptosystems.     

Isomorphism classes of elliptic and hyperelliptic curves over finite fields

We give the number and representatives of isomorphism classes of hyperelliptic curves of genus g defined over finite fields , g=1,2,3. These results have applications to hyperelliptic curve cryptography.     

Isomorphism classes of hyperelliptic curves of genus 2 over finite fields with characteristic 2

In this paper we study the computation of the number of isomorphism classes of hyperelliptic curves of genus 2 over finite fields Fq with q even. We show the formula of the number of isomorphism classes, that is, for q = 2m, if 4 m, then the formula is 2q3 q2 - q; if 4 | m, then the formula is 2q3 q2 - q 8. These results can be used in the classification problems and the hyperelliptic curve cryptosystems.     

A Review on the Isomorphism Classes of Hyperelliptic Curves of Genus 2 over Finite Fields Admitting a Weierstrass Point

The reduced equations for the isomorphism classes of hyperelliptic curves of genus 2 admitting a Weierstrass point over a finite field of arbitrary characteristic, are shown and the number of such classes is included. This work picks up in a unified way a series of previous results published by several authors by using different methodologies. These classifications are of interest in designing and implementing of hyperelliptic curve cryptosystems.     

Counting hyperelliptic curves

We find a closed formula for the number hyp(g) of hyperelliptic curves of genus g over a finite field k=Fq of odd characteristic. These numbers hyp(g) are expressed as a polynomial in q with integer coefficients that depend on g and the set of divisors of q−1 and q+1. As a by-product we obtain a closed formula for the number of self-dual curves of genus g. A hyperelliptic curve is defined to be self-dual if it is k-isomorphic to its own hyperelliptic twist.     

Hyperelliptic curves of genus three over finite fields of even characteristic

In this paper we classify hyperelliptic curves of genus 3 defined over a finite field k of even characteristic. We consider rational models representing all k-isomorphy classes of curves with a given arithmetic structure for the ramification divisor and we find necessary and sufficient conditions for two models of the same type to be k-isomorphic. Also, we compute the automorphism group of each curve and an explicit formula for the total number of curves.     

Explicit bounds of polynomial coefficients and counting points on Picard curves over finite fields

In this paper, we describe an algorithm for computing the order of the Jacobian varieties of Picard curves over finite fields. This is an extension of the algorithm of Matsuo, Chao and Tsujii (MCT) [K. Matsuo, J. Chao, S. Tsujii, An improved baby step algorithm for point counting of hyperelliptic curves over finite fields, in: LNCS vol. 2369, Springer-Verlag, 2005, pp. 461–474] for hyperelliptic curves. We study the characteristic polynomials and the Jacobian varieties of algebraic curves of genus three over finite fields. Based on this, we investigate the explicit computable bounds for coefficients of the characteristic polynomial and improve a part of the baby-step giant-step of the counting points algorithm. Usefulness of the proposed method is illustrated and verified by the simple examples.     

On the number of curves of genus 2 over a finite field

In this paper we compute the number of curves of genus 2 defined over a finite field k of odd characteristic up to isomorphisms defined over k; the even characteristic case is treated in an ongoing work (G. Cardona, E. Nart, J. Pujolàs, Curves of genus 2 over field of even characteristic, 2003, submitted for publication). To this end, we first give a parametrization of all points in , the moduli variety that classifies genus 2 curves up to isomorphism, defined over an arbitrary perfect field (of zero or odd characteristic) and corresponding to curves with non-trivial reduced group of automorphisms; we also give an explicit representative defined over that field for each of these points. Then, we use cohomological methods to compute the number of k-isomorphism classes for each point in .     

The fluctuations in the number of points on a hyperelliptic curve over a finite field

The number of points on a hyperelliptic curve over a field of q elements may be expressed as q+1+S where S is a certain character sum. We study fluctuations of S as the curve varies over a large family of hyperelliptic curves of genus g. For fixed genus and growing q, Katz and Sarnak showed that is distributed as the trace of a random 2g×2g unitary symplectic matrix. When the finite field is fixed and the genus grows, we find that the limiting distribution of S is that of a sum of q independent trinomial random variables taking the values ±1 with probabilities 1/2(1+q⁻¹) and the value 0 with probability 1/(q+1). When both the genus and the finite field grow, we find that has a standard Gaussian distribution.     

The Maximum or Minimum Number of Rational Points on Genus Three Curves over Finite Fields

We show that for all finite fields F_q, there exists a curve C over F_q of genus 3 such that the number of rational points on C is within 3 of the Serre–Weil upper or lower bound. For some q, we also obtain improvements on the upper bound for the number of rational points on a genus 3 curve over F_q.with an Appendix by Jean-Pierre Serre     

Constructing pairing-friendly hyperelliptic curves using Weil restriction

A pairing-friendly curve is a curve over a finite field whose Jacobian has small embedding degree with respect to a large prime-order subgroup. In this paper we construct pairing-friendly genus 2 curves over finite fields Fq whose Jacobians are ordinary and simple, but not absolutely simple. We show that constructing such curves is equivalent to constructing elliptic curves over Fq that become pairing-friendly over a finite extension of Fq. Our main proof technique is Weil restriction of elliptic curves. We describe adaptations of the Cocks-Pinch and Brezing-Weng methods that produce genus 2 curves with the desired properties. Our examples include a parametric family of genus 2 curves whose Jacobians have the smallest recorded ρ-value for simple, non-supersingular abelian surfaces.     

The statistics of the number of rational points in the hyperelliptic ensemble over a finite field

We study the distribution of the numbers of \({F_{{q^r}}}\)-rational points of hyperelliptic curves over a finite field F_q in odd characteristic. This extends the result of Kurlberg and Rudnick [4]. We also study the distribution of the number of \({F_{{q^r}}}\)-rational points and the trace of high powers of the Frobenius class of real hyperelliptic curves over a finite field F_q in even characteristic.     

On Moduli Spaces,Equidistribution, Estimates,and Rational Points of Algebraic Curves

We consider the moduli spaces of hyperelliptic curves, Artin–Schreier coverings, and some other families of curves of this type over fields of characteristic p. By using the Postnikov method, we obtain expressions for the Kloosterman sums. The distribution of angles of the Kloosterman sums was investigated on a computer. For small prime p, we study rational points on curves y ² = f(x). We consider the problem of the accuracy of estimates of the number of rational points of hyperelliptic curves and the existence of rational points of curves of the indicated type on the moduli spaces of these curves over a prime finite field.     

Torsionfree sheaves over a nodal curve of arithmetic genus one

We classify all isomorphism classes of stable torsionfree sheaves on an irreducible nodal curve of arithmetic genus one defined over ℂ. Let X be a nodal curve of arithmetic genus one defined over ℝ, with exactly one node, such that X does not have any real points apart from the node. We classify all isomorphism classes of stable real algebraic torsionfree sheaves over X of even rank. We also classify all isomorphism classes of real algebraic torsionfree sheaves over X of rank one.     

Witt equivalence of algebraic function fields over real closed fields

We examine the conditions for two algebraic function fields over real closed fields to be Witt equivalent. We show that there are only two Witt classes of algebraic function fields with a fixed real closed field of constants: real and non-real ones. The first of them splits further into subclasses corresponding to the tame equivalence. This condition has a natural interpretation in terms of both: orderings (the associated Harrison isomorphism maps 1-pt fans onto 1-pt fans), and geometry and topology of associated real curves (the bijection of points is a homeomorphism and these two curves have the same number of semi-algebraically connected components). Finally, we derive some immediate consequences of those theorems. In particular we describe all the Witt classes of algebraic function fields of genus 0 and 1 over the fixed real closed field. Received: 16 February 2000; in final form: 7 December 2000 / Published online: 18 January 2002     

On the isomorphism classes of Legendre elliptic curves over finite fields

In this paper,the number of isomorphism classes of Legendre elliptic curves over finite field is enumerated.     

The p-rank strata of the moduli space of hyperelliptic curves

We prove results about the intersection of the p-rank strata and the boundary of the moduli space of hyperelliptic curves in characteristic p?3. This yields a strong technique that allows us to analyze the stratum of hyperelliptic curves of genus g and p-rank f. Using this, we prove that the endomorphism ring of the Jacobian of a generic hyperelliptic curve of genus g and p-rank f is isomorphic to Z if g?4. Furthermore, we prove that the Z/?-monodromy of every irreducible component of is the symplectic group Sp2g(Z/?) if g?3, and ?≠p is an odd prime (with mild hypotheses on ? when f=0). These results yield numerous applications about the generic behavior of hyperelliptic curves of given genus and p-rank over finite fields, including applications about Newton polygons, absolutely simple Jacobians, class groups and zeta functions.