One half log discriminant and division polynomials

Szpiro and Tucker recently proved that, under mild conditions, the valuation of the minimal discriminant of an elliptic curve with semistable reduction over a discrete valuation ring can be expressed in terms of intersections between n-torsion and 2-torsion, where n tends to infinity. The argument of Szpiro and Tucker is geometric in nature. We give a proof based on the arithmetic of division polynomials, and generalize the result to the case of hyperelliptic curves.     

Lie Algebras of Heat Operators in a Nonholonomic Frame

Mathematical Notes - We construct the Lie algebras of systems of $$2g$$ graded heat operators $$Q_0,Q_2,\dots,Q_{4g-2}$$ that determine the sigma functions $$\sigma(z,\lambda)$$ of hyperelliptic...     

The field of meromorphic functions on a sigma divisor of a hyperelliptic curve of genus 3 and applications

The field of meromorphic functions on a sigma divisor of a hyperelliptic curve of genus 3 is described in terms of the gradient of its sigma function. As an application, solutions of the corresponding families of polynomial dynamical systems in C⁴ with two polynomial integrals are constructed. These systems were introduced by Buchstaber and Mikhailov on the basis of commuting vector fields on the symmetric square of algebraic curves.     

Vanishing thetanulls on curves with involutions

The configuration of theta characteristics and vanishing thetanulls on a hyperelliptic curve is completely understood. We observe in this note that analogous results hold for the \( \sigma \)-invariant theta characteristics on any curve \( C \) with an involution \( \sigma \). As a consequence we get examples of non hyperelliptic curves with a high number of vanishing thetanulls.     

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.     

Commuting difference operators, spinor bundles and the asymptotics of orthogonal polynomials with respect to varying complex weights

The paper has three parts. In the first part we apply the theory of commuting pairs of (pseudo) difference operators to the (formal) asymptotics of orthogonal polynomials: using purely geometrical arguments we show heuristically that the asymptotics, for large degrees, of orthogonal polynomial with respect to varying weights is intimately related to certain spinor bundles on a hyperelliptic algebraic curve reproducing formulae appearing in the works of Deift et al. on the subject.In the second part we show that given an arbitrary nodal hyperelliptic curve satisfying certain conditions of admissibility we can reconstruct a sequence of polynomials orthogonal with respect to semiclassical complex varying weights supported on several curves in the complex plane. The strong asymptotics of these polynomials will be shown to be given by the spinors introduced in the first part using a Riemann-Hilbert analysis.In the third part we use Strebel theory of quadratic differentials and the procedure of welding to reconstruct arbitrary admissible hyperelliptic curves. As a result we can obtain orthogonal polynomials whose zeroes may become dense on a collection of Jordan arcs forming an arbitrary forest of trivalent loop-free trees.     

Bäcklund transformations for the nonholonomic Veselova system

We present auto and hetero Bäcklund transformations of the nonholonomic Veselova system using standard divisor arithmetic on the hyperelliptic curve of genus two. As a by-product one gets two natural integrable systems on the cotangent bundle to the unit two-dimensional sphere whose additional integrals of motion are polynomials in the momenta of fourth order.     

Continued rational fractions in hyperelliptic fields and the Mumford representation

Doklady Mathematics - A relationship between the continued fraction expansion of the quadratic irrationalities of hyperelliptic fields and the Mumford polynomials determining addition in the group...     

Elliptic Curves and Quadratic Recurrence Sequences

The explicit solution of a general three-term bilinear recurrencerelation of fourth order is constructed here in terms of theWeierstrass sigma function. The construction of the ellipticcurve associated to the Somos 4 sequence is presented as anexample. An interpretation via the theory of integrable systemsis provided, leading to a conjecture relating certain higher-orderrecurrences with hyperelliptic curves of higher genus. 2000Mathematics Subject Classification 11B37 (primary), 33E05, 37J35(secondary).     

Sur certains sous-groupes de torsion de jacobiennes de courbes hyperelliptiques de genreg ≥ 1

We construct a family of hyperelliptic curves of genusg defined over Q whose Jacobians have a rational point of order 2g(2g+1). Forl = 2g ² + 5g + 5, we construct a family of genusg hyperelliptic curves defined over Q, such that their Jacobians have a rational point of orderl orl / 2 orl / 4. We also construct a hyperelliptic curve of genusg defined over Q, which does not belong to the previous family, and whose Jacobian has a rational point of orderl.      

Systems of uniformization equations and hyperelliptic integrals

The purpose of this paper is to study systems of uniformization equations with respect to Saito free divisors which have solutions expressed in terms of hyperelliptic integrals. There are two such divisors. Both are hypersurfaces in a three-dimensional affine space defined by weighted homogeneous polynomials. One is constructed by the discriminant of a dihedral group of order 2(2n+1). The other is the discriminant of the reflection group of type H ₃. In the former case, we construct fundamental solutions by Gaussian hypergeometric functions in addition to a solution expressed by a hyperelliptic integral.     

Egorov Hydrodynamic Chains, the Chazy Equation, and SL(2,ℂ)

The general solution of the system of differential equations describing Egorov hydrodynamic chains is constructed. The solution is given in terms of the elliptic sigma function. Invariants of the sigma function are expressed as differential polynomials in a solution of the Chazy equation. The orbits of the induced action of SL(2,) and degenerating operators in the space of solutions are described.     

Factorizations of tropical and sign polynomials

In this text, we study factorizations of polynomials over the tropical hyperfield and the sign hyperfield, which we call tropical polynomials and sign polynomials, respectively. We classify all irreducible polynomials in either case. We show that tropical polynomials factor uniquely into irreducible factors, but that unique factorization fails for sign polynomials. We describe division algorithms for tropical and sign polynomials by linear terms that correspond to roots of the polynomials.     

Rational torsion of J0(N) for hyperelliptic modular curves and families of Jacobians of genus 2 and genus 3 curves with a rational point of order 5,7 or 10

We describe a way of constructing Jacobians of hyperelliptic curves of genus g ≥ 2, defined over a number field, whose Jacobians have a rational point of order of some (well chosen) integer l ≥ g + 1; the method is based on a polynomial identity. Using this approach we construct new families of genus 2 curves defined over — which contain the modular curves X₀(31) (and X₀(22) as a by-product) and X₀(29), the Jacobians of which have a rational point of order 5 and 7 respectively. We also construct a new family of hyperelliptic genus 3 curves defined over —, which contains the modular curve X₀(41), the Jacobians of which have a rational point of order 10. Finally we show that all hyperelliptic modular curves X₀(N) with N a prime number fit into the described strategy, except for N = 37 in which case we give another explanation. The authors thank the FNR (project FNR/04/MA6/11) for their support.     

The Scorza correspondence in genus 3

In this note we prove the genus 3 case of a conjecture of Farkas and Verra on the limit of the Scorza correspondence for curves with a theta-null. Specifically, we show that the limit of the Scorza correspondence for a hyperelliptic genus 3 curve C is the union of the curve ${\{x, \sigma(x) \mid x \in C\}}$ (where σ is the hyperelliptic involution), and twice the diagonal. Our proof uses the geometry of the subsystem Γ₀₀ of the linear system |2Θ|, and Riemann identities for theta constants.     

Minimum relative error approximations for 1/t

Summary We give explicit solutions to the problem of minimizing the relative error for polynomial approximations to 1/t on arbitrary finite subintervals of (0, ). We give a simple algorithm, using synthetic division, for computing practical representations of the best approximating polynomials. The resulting polynomials also minimize the absolute error in a related functional equation. We show that, for any continuous function with no zeros on the interval of interest, the geometric convergence rates for best absolute error and best relative error approximants must be equal. The approximation polynomials for 1/t are useful for finding suitably precise initial approximations in iterative methods for computing reciprocals on computers.     

Non-semistable Arakelov bound and hyperelliptic Szpiro ratio for function fields

We prove a complex function field analogue of Szpiro's conjecture for hyperelliptic curves and some applications. The cases of function fields of positive characteristic and number fields are discussed briefly.

     

Monotonicity of the zeros of orthogonal polynomials through related measures

Relation between two sequences of orthogonal polynomials, where the associated measures are related to each other by a first degree polynomial multiplication (or division), is well known. We use this relation to study the monotonicity properties of the zeros of generalized orthogonal polynomials. As examples, the Jacobi, Laguerre and Charlier polynomials are considered.     

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.     

On the Linear Complexity of the Naor–Reingold Pseudo-random Function from Elliptic Curves

We show that the elliptic curve analogue of the pseudo-random number function, introduced recently by M. Naor and O. Reingold, produces a sequence with large linear complexity. This result generalizes a similar result of F. Griffin and I. E. Shparlinski for the linear complexity of the original function of M. Naor and O. Reingold. The proof is based on some results about the distribution of subset-products in finite fields and some properties of division polynomials of elliptic curves.