Real Hyperelliptic Curves Research Articles

Divisors on real curves

Let X be a smooth projective curve over R. In the first part, we study eective divi- sors on X with totally real or totally complex support. We give some numerical conditions for a linear system to contain such a divisor. In the second part, we describe the special linear sys- tems on a real hyperelliptic curve and prove a new Cliord inequality for such curves. Finally, we study the existence of complete linear systems of small degrees and dimension r on a real curve.

Group law on the neutral component of the Jacobian of a real hyperelliptic curve having many real components

Let C be a real algebraic curve of genus g with at least g real components B 1,…, B g . We give an embedding of C into a blow up in one point of the projective plane. It allows us to describe geometrically the neutral real component Pic o ( C) o of the Jacobian of C thanks to an isomorphism with the product B 1×⋯× B g . This induces an explicit geometric description of Pic o ( C) o in the projective plane, where the group law is given by intersection with curves of genus g.

Effective approach to least deviation problems

A hierarchy of extremal polynomials described in terms of real hyperelliptic curves of genus is constructed. These polynomials depend on integer-valued and continuous parameters. The classical Chebyshëv polynomials are obtained for and the Zolotarëv polynomials for .

Two-torsion in the Jacobian of hyperelliptic curves over finite fields

We determine the exact dimension of the $$ {\bf{F}}_2 $$ -vector space of $$ {\bf{F}}_q $$ -rational 2-torsion points in the Jacobian of a hyperelliptic curve over $$ {\bf{F}}_q $$ (q odd) in terms of the degrees of the rational factors of its discriminant, and relate this to genus theory for the corresponding function field. As a corollary, we prove the existence of a point of order > 2 in the Jacobian of certain real hyperelliptic curves.

Geodesics, periods, and equations of real hyperelliptic curves

In this paper we start a new approach to the uniformization problem of Riemann surfaces and algebraic curves by means of computational procedures. The following question is studied: Given a compact Riemann surface S described as the quotient of the Poincaré upper half-plane divided by the action of a Fuchsian group, find explicitly the polynomial describing S as an algebraic curve (in some normal form). The explicit computation given in this paper is based on the numerical computation of conformal capacities of hyperbolic domains. These capacities yield the period matrices of S in terms of the Fenchel-Nielsen coordinates, and from there one gets to the polynomial via theta-characteristics. The paper also contains a list of worked-out examples and a list of examples–new in the literature–where the polynomial for the curve, as a function of the corresponding Fuchsian group, is given in closed form.