José M. Muñoz Porras

Mirror Symmetry on K3 Surfaces via Fourier–Mukai Transform

Abstract:We use a relative Fourier–Mukai transform on elliptic K3 surfaces X to describe mirror symmetry. The action of this Fourier–Mukai transform on the cohomology ring of X reproduces relative T-duality and provides an infinitesimal isometry of the moduli space of algebraic structures on X which, in view of the triviality of the quantum cohomology of K3 surfaces, can be interpreted as mirror symmetry. From the mathematical viewpoint the novelty is that we exhibit another example of a Fourier–Mukai transform on K3 surfaces, whose properties are closely related to the geometry of the relative Jacobian of X.

Convolutional Goppa codes

In this correspondence, we define convolutional Goppa codes over algebraic curves and construct their corresponding dual codes. Examples over the projective line and over elliptic curves are described, obtaining in particular some maximum-distance separable (MDS) convolutional codes.

GENERALIZED RECIPROCITY LAWS

The aim of this paper is to give an abstract formulation of the classical reciprocity laws for function fields that could be generalized to the case of arbitrary (non-commutative) reductive groups as a first step to finding explicit non-commutative reciprocity laws. The main tool in this paper is the theory of determinant bundles over adelic Sato Grassmannians and the existence of a Krichever map for rank n vector bundles.

The Geometry of Riemann Surfaces and Abelian Varieties

Mobius transformations of the circle form a maximal convergence group by A. Basmajian and M. Zeinalian On the polar linear system of a foliation by curves in a projective space by A. Campillo and J. Olivares Birational classification of hyperelliptic real algebraic curves by F. J. Cirre The genus two Jacobians that are isomorphic to a product of elliptic curves by C. J. Earle Vanishing thetanulls and Jacobians by H. M. Farkas Theta functions, geometric quantization and unitary Schottky bundles by C. Florentino, J. Mourao, and J. P. Nunes Smooth double coverings of hyperelliptic curves by Y. Fuertes and G. Gonzalez-Diez Planar families of discrete groups by J. Gilman and L. Keen Generalized addition formulae for theta functions by E. Gomez Gonzalez and C. Gonzalez-Martinez Curves with a group action and Galois covers via infinite Grassmannians by E. Gomez Gonzalez and F. J. P. Martin Some recent results on the irreducible components of the singular locus of

FOUR-LOOP VACUUM AMPLITUDES FOR THE BOSONIC STRING

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Every Convolutional Code is a Goppa Code

by V. Gonzalez-Aguilera, J. M. Munoz-Porras, and A. G. Zamora A note on the arithmetic genus of reducible plane curves by M. R. Gonzalez-Dorrego Teichmuller curves defined by characteristic origamis by F. Herrlich Lowest uniformizations of compact real surfaces by R. A. Hidalgo and B. Maskit Principal polarizations on products of elliptic curves by H. Lange An approach to a 2-dimensional Contou-Carrere symbol by F. P. Romo Prym varieties and fourfold covers II: The dihedral case by S. Recillas and R. E. Rodriguez Examples for Veech groups of origamis by G. Schmithusen Genus 2 translation surfaces with an order 4 automorphism by R. Silhol The Pfaffian structure defining a Prym theta divisor by R. Smith and R. Varley.

Coverings with prescribed ramification and Virasoro groups

A proof of the conjecture of Belavin, Knizhnik and Morozov concerning the Polyakov measure of the bosonic string in genus 4 is offered. Its consequences for the universal moduli space approach to string theory are discussed.

An Explicit Reciprocity Law Associated to Some Finite Coverings of Algebraic Curves

Convolutional Goppa codes (CGC) were defined in Appl. Algebra Eng. Comm. Comput., vol. 15, pp. 51-61, 2004 and IEEE Trans. Inf. Theory, vol. 52, 340-344, 2006. In this paper, we prove that every convolutional code is a CGC defined over a smooth curve over \BBF q(z) and we give an explicit construction of convolutional codes as CGC over the projective line \BBP \BBF q(z)1. We characterize which convolutional codes are defined by a complete linear system over curves of genus 0, 1, and over hyperelliptic curves. We apply these results to provide detailed constructions of some linear block codes as Goppa codes.

Characterizations of Jacobians of curves with automorphisms

In this paper we study coverings with prescribed ramification from the point of view of the Sato Grassmannian and of the algebro-geometric theory of solitons. We show that the moduli space of such coverings, which is a Hurwitz scheme, is a subscheme of the Grassmannian. We give its equations and show that there is a Virasoro group which uniformizes it. We also characterize when a curve is a covering in terms of bilinear identities. Acknowledgements. This work is partially supported by the research contracts MTM2006-07618, MTM2009-11393 and SA112A07. Nakanishi Printing Co., Ltd. Shimotachiuri-Ogawa-Higashi, Kamikyoku, Kyoto 602-8048, Japan e-mail: infor@nacos.com Nakanishi Printing Co., Ltd. Shimotachiuri-Ogawa-Higashi, Kamikyoku, Kyoto 602-8048, Japan e-mail: infor@nacos.com 1991 Mathematics Subject Classification(s). N/A 2000 Mathematics Subject Classification(s). 14H10 (Primary) 35Q53, 58B99, 37K10 (Secondary). Additional information Communicated by N/A Received Date Additional information(s) Revised Date Revised Dates ∗Departamento de Matematicas, Universidad de Salamanca, Plaza de la Merced 1-4. 37008 Salamanca. Spain. Tel: +34 923294460. Fax: +34 923294583. E-mail: jmp@usal.es ∗∗Departamento de Matematicas, Universidad de Salamanca, Plaza de la Merced 1-4. 37008 Salamanca. Spain. Tel: +34 923294460. Fax: +34 923294583. E-mail: fplaza@usal.es 2 J. M. Munoz Porras and F. J. Plaza Martin

Survey on the Schottky Problem

We provide a new reciprocity law associated with finite coverings of algebraic curves. Moreover, we give explicit examples of this new reciprocity law that are not trivial consequences of the Weil reciprocity law over the base curve.