Examinando por Autor "Izquierdo, Milagros"
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Publicación Algebraic Curves over Finite Fields(Universidad Nacional de Educación a Distancia (España). Facultad de Ciencias, 2010-06-01) Rovi, Carmen; Izquierdo, MilagrosThis thesis surveys the issue of finding rational points on algebraic curves over finite fields. Since Goppa’s construction of algebraic geometric codes, there has been great interest in finding curves with many rational points. Here we explain the main tools for finding rational points on a curve over a finite field and provide the necessary background on ring and field theory. Four different articles are analyzed, the first of these articles gives a complete set of table showing the numbers of rational points for curves with genus up to 50. The other articles provide interesting constructions of covering curves: covers by the Hemitian curve, Kummer extensions and Artin-Schreier extensions. With these articles the great difficulty of finding explicit equations for curves with many rational points is overcome. With the method given by Arnaldo García in [6] we have been able to find examples that can be used to define the lower bounds for the corresponding entries in the tables given in http: //wins.uva.nl/˜geer, which to the time of writing this Thesis appear as ”no information available”. In fact, as the curves found are maximal, these entries no longer need a bound, they can be given by a unique entry, since the exact value of Nq(g) is now known. At the end of the thesis an outline of the construction of Goppa codes is given and the NXL and XNL codes are presented.Publicación Modular companions in planar one-dimensional equisymmetric strata(Universal Wiser, 2025) Costa González, Antonio Félix; Broughton, S. Allen; Izquierdo, MilagrosConsider, in the moduli space of Riemann surfaces of a fixed genus, the subset of surfaces with non-trivial automorphisms. Of special interest are the numerous subsets of surfaces admitting an action of a given finite group, G, acting with a specific signature. In a previous study [6], we declared two Riemann surfaces to be modular companions if they have topologically equivalent G actions, and that their G quotients are conformally equivalent orbifolds. In this article we present a geometrically-inspired measure to decide whether two modular companions are conformally equivalent (or how different), respecting the G action. Along the way, we construct a moduli space for surfaces with the specified G action and associated equivariant tilings on these surfaces. We specifically apply the ideas to planar, finite group actions whose quotient orbifold is a sphere with four cone pointsPublicación On the connectedness of the branch locus of moduli space of hyperelliptic Klein surfaces with one boundary(2015-01-01) Izquierdo, Milagros; Costa González, Antonio Félix; Porto Ferreira da Silva, Ana MaríaAbstract. In this work we prove that the hyperelliptic branch locus of ori- entable Klein surfaces of genus g with one boundary component is connected and in the case of non-orientable Klein surfaces it has g+1 2 components, if g is odd, and g+2 2 components for even g. We notice that, for non-orientable Klein surfaces with two boundary components, the hyperelliptic branch loci are connected for all genera.Publicación One dimensional equisymmetric strata in moduli space with genus 1 quotient surfaces(Springer, 2024) S. Allen Broughton; Costa González, Antonio Félix; Izquierdo, Milagros; https://orcid.org/0000-0002-9557-9566The complex orbifold structure of the moduli space of Riemann surfaces of genus g (g ≥2) produces a stratification into complex subvarieties named equisymmetric strata. Eachequisymmetric stratum is formed by the surfaces where the group ofautomorphisms acts in a topologically equivalent way. The Riemann surfaces in the equisymmetric strata of dimension one are of two structurally different types. Type 1 equisymmetric strata correspond to Riemann surfaces where the group of automorphisms produces a quotient surface of genus zero, while those of Type 2 appear when such a quotient is a surface of genus one. Type 1 equisymmetric strata have been extensively studied by the authors of the present work in a previous recent paper, we now focus on Type 2 strata. We first establish the existence of such strata and their frequency of occurrence in moduli spaces. As a main result we obtain a complete description of Type 2 strata as coverings of the sphere branched over three point (Belyi curves) and where certain isolated points (punctures) have to be eliminated. Finally, we study in detail the doubly infinite family of Type 2 strata whose automorphism groupshave order the product of two primes.Publicación Sobre Códigos Algebraico-Geométricos Basados en Curvas Ca, b(Universidad Nacional de Educación a Distancia (España). Facultad de Ciencias, 2016-10-24) Jiménez Magdaleno, Victoriano; Izquierdo, MilagrosEl objeto de este trabajo es el estudio de las curvas tipo 𝐶𝑎,𝑏 y sus aplicaciones en la teoría de códigos. Veremos cómo las curvas 𝐶𝑎,𝑏 se pueden utilizar para construir códigos MDS (maximum distance separable codes) y nos centraremos en algunas curvas 𝐶𝑎,𝑏 que poseen un grupo de automorfismos que puede determinarse.