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

This paper is devoted to counting 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 even characteristics. The number of isomorphism classes is computed and the explicit formulae are given. This number can be represented as a polynomial in q of degree 7, where q is the order of the finite field. The result can be used in the classification problems and it is useful for further studies of hyperelliptic curve cryptosystems, e.g. it is of interest for research on implementing the arithmetics of curves of low genus for cryptographic purposes. It could also be of interest for point counting problems; both on moduli spaces of curves, and on finding the maximal number of points that a pointed hyperelliptic curve over a given finite field may have.     

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.     

一类超椭圆曲线的整点个数

设ｎ是大于２的工整数，Ｄ是无平方因子正整数，分别是Ｋ的理想类群和类数．对于正整数ｍ，设ｇｋ（ｍ）是Ｉｘ中阶数等于ｍ的理想类的个数．本文证明了：超椭圆曲线ｆ（ｘ，ｙ）＝Ｄｘ２－４ｙｎ＋１＝０上整数点（ｘ，ｙ）的个数不超过ｍａｘ（８，２１６４Ｐ８１ｇｋ（Ｐ）），其中ｐ是ｎ的奇素因数．     

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.     

Towers of 2-covers of hyperelliptic curves

In this article, we give a way of constructing an unramified Galois-cover of a hyperelliptic curve. The geometric Galois-group is an elementary abelian -group. The construction does not make use of the embedding of the curve in its Jacobian, and it readily displays all subcovers. We show that the cover we construct is isomorphic to the pullback along the multiplication-by- map of an embedding of the curve in its Jacobian.

We show that the constructed cover has an abundance of elliptic and hyperelliptic subcovers. This makes this cover especially suited for covering techniques employed for determining the rational points on curves. In particular the hyperelliptic subcovers give a chance for applying the method iteratively, thus creating towers of elementary abelian 2-covers of hyperelliptic curves.

As an application, we determine the rational points on the genus curve arising from the question of whether the sum of the first fourth powers can ever be a square. For this curve, a simple covering step fails, but a second step succeeds.

     

关于A.Menezes和S.Vanstone一文的注记

1987年,R.Schoof对于有限域Fq上的椭圆曲线的同构类数目得出了一个公式.1990年,A.M enezes和S.V anstone对此当q为偶数时给出了一个初等证明.本文利用Burnside引理对q为偶数时给出一个更简单的证明.     

Explicit Mumford isomorphism for hyperelliptic curves

Using an explicit version of the Mumford isomorphism on the moduli space of hyperelliptic curves we derive a closed formula for the Arakelov-Green function of a hyperelliptic Riemann surface evaluated at its Weierstrass points.     

Elliptic subcovers of hyperelliptic curves

The main result of this paper states that if C is a hyperelliptic curve of even genus over an arbitrary field K , then there is a natural bijection between the set of equivalence classes of elliptic subcovers of and the set of elliptic subgroups of its Jacobian .     

Maximal amplitudes of hyperelliptic solutions of the derivative nonlinear Schrödinger equation

A simple formula is proven for an upper bound for amplitudes of hyperelliptic (finite-gap or N-phase) solutions of the derivative nonlinear Schrödinger equation. The upper bound is sharp, viz, it is attained for some initial conditions. The method used to prove the upper bound is the same method, with necessary modifications, used to prove the corresponding bound for solutions of the focusing NLS equation (Wright OC, III. Sharp upper bound for amplitudes of hyperelliptic solutions of the focusing nonlinear Schrödinger equation. Nonlinearity. 2019;32:1929-1966).     

Schottky's form and the hyperelliptic locus

We show that Schottky's modular form, , has in every genus an irreducible divisor which contains the hyperelliptic locus. We also improve a corollary of Igusa concerning Siegel modular forms that must necessarily vanish on the hyperelliptic locus.

     

Singular pencils of quadrics and compactified Jacobians of curves

LetY be an irreducible nodal hyperelliptic curve of arithmetic genusg such that its nodes are also ramification points (char ≠2). To the curveY, we associate a family of quadratic forms which is dual to a singular pencil of quadrics in with Segre symbol [2...21...1], where the number of 2's is equal to the number of nodes. We show that the compactified Jacobian ofY is isomorphic to the spaceR of (g−1) dimensional linear subspaces of which are contained in the intersectionQ of quadrics of the pencil. We also prove that (under this isomorphism) the generalized Jacobian ofY is isomorphic to the open subset ofR consisting of the (g−1) dimensional subspaces not passing through any singular point ofQ.     

Hyperelliptic Surfaces and their Moduli

The hyperelliptic portion of the moduli space of compact Riemann surfaces of genus g2 is decomposed into a lattice of nondisjoint subvarieties corresponding precisely with the lattice of maximal g-hyperelliptic group actions (classified up to topological equivalence). The resulting stratification of the hyperelliptic moduli space exhibits regularities which depend on the parity of g and can be detected at the level of groups of order 8.     

Circulant Double Coverings of a Circulant Graph of Valency Four

Several isomorphism classes of graph coverings of a graph G have been enumerated by many authors (see [3], [8]–[15]). A covering of G is called circulant if its covering graph is circulant. Recently, the authors [4] enumerated the isomorphism classes of circulant double coverings of a certain kind, called typical, and showed that no double covering of a circulant graph of valency 3 is circulant. In this paper, the isomorphism classes of connected circulant double coverings of a circulant graph of valency 4 are enumerated. As a consequence, it is shown that no double covering of a non-circulant graph G of valency 4 can be circulant if G is vertex-transitive or G has a prime power of vertices. The first author is supported by NSF of China (No. 60473019) and by NKBRPC (2004CB318000), and the second author is supported by Com²MaC-KOSEF (R11-1999-054) in Korea.     

Z_2-商的秩为r的超椭圆纤维化曲面的斜率

设Ｓ是一般型的相对极小曲面，ｆ：Ｓ→Ｃ是亏格ｇ的超椭圆纤维化．本文中我们证明了如果 Ｓ的代数基本群的垂直部分的极大挠 ２商为，那么其斜率且等号成立仅当 Ｓ上的超椭圆对合所诱导的二次复盖的分歧除子 Ｒ仅有（ｒ＋１→，＋１）（当ｒ为偶数）型奇点，或（ｒ＋２→ｒ＋２）（当ｒ为奇数）型奇点．     

The Extended Euclidian Algorithm on Polynomials, and the Computational Efficiency of Hyperelliptic Cryptosystems

After generalising two reduction algorithms tocharacteristic 2, we analyse the average complexity of thearithmetic in hyperelliptic Jacobians over any finite field.To this purpose we determine the exact average number offield operations for computing the greatest common divisor ofpolynomials over a finite field by the extended Euclidianalgorithm.     

Order and algebra isomorphisms of spaces of regular operators

If E and F are real Banach lattices and there is an algebra and order isomorphism Φ:(E) →(F) between their respective ordered Banach algebras of regular operators then there is a linear order isomorphism U:E→F such that Φ(T) =UTU⁻¹ for all T ∈ (E).     

Hamiltonians with two degrees of freedom admitting a singlevalued general solution

Following the basic principles stated by Painleve, we first revisit the process of selecting the admissible time-independent Hamiltonians H=(p12 + p22)/2 + V(q1,q2) whose some integer power qjnj (t) of the general solution is a singlevalued function of the complex time t. In addition to the well known rational potentials V of Henon-Heiles, this selects possible cases with a trigonometric dependence of V on qj. Then, by establishing the relevant confluences, we restrict the question of the explicit integration of the seven (three "cubic" plus four "quartic") rational Henon-Heiles cases to the quartic cases. Finally, we perform the explicit integration of the quartic cases, thus proving that the seven rational cases have a meromorphic general solution explicitly given by a genus two hyperelliptic function.     

Unitals and inversive planes

We show that if U is a Buekenhout-Metz unital (with respect to a point P) in any translation plane of order q ² with kernel containing GF(q), then U has an associated 2-(q²,q+1,q) design which is the point-residual of an inversive plane, generalizing results of Wilbrink, Baker and Ebert. Further, our proof gives a natural, geometric isomorphism between the resulting inversive plane and the (egglike) inversive plane arising from the ovoid involved in the construction of the Buekenhout-Metz unital. We apply our results to investigate some parallel classes and partitions of the set of blocks of any Buekenhout-Metz unital.     

Computing discrete logarithms in high-genus hyperelliptic Jacobians in provably subexponential time

We provide a subexponential algorithm for solving the discrete logarithm problem in Jacobians of high-genus hyperelliptic curves over finite fields. Its expected running time for instances with genus and underlying finite field satisfying for a positive constant is given by

The algorithm works over any finite field, and its running time does not rely on any unproven assumptions.