In this article, we formalize operations of points on an elliptic curve over GF(p). Elliptic curve cryptography [7], whose security is based on a difficulty of discrete logarithm problem of elliptic curves, is important for information security. We prove that the two operations of points: compellProjCo and addellProjCo are unary and binary operations of a point over the elliptic curve.
We study a constructive method to find an algebraic curve in the real projective plane with a (possibly singular) topological type given in advance. Our method works if the topological model T to be realized has only double singularities and gives an algebraic curve of degree 2N+2K, where N and K are the numbers of double points and connected components of T. This degree is optimal in the sense that for any choice of the numbers N and K there exist models which cannot be realized algebraically with...
In [6], orbifold G-bundles on a certain class of elliptic fibrations over a smooth complex projective curve X were related to parabolic G-bundles over X. In this continuation of [6] we define and investigate holomorphic connections on an orbifold G-bundle over an elliptic fibration.
Let X be a compact Riemann surface and associated to each point p-i of a finite subset S of X is a positive integer m-i. Fix an elliptic curve C. To this data we associate a smooth elliptic surface Z fibered over X. The group C acts on Z with X as the quotient. It is shown that the space of all vector bundles over Z equipped with a lift of the action of C is in bijective correspondence with the space of all parabolic bundles over X with parabolic structure over S and the parabolic weights at any...
Let be the moduli space of principal -bundles on a curve , and the determinant bundle on . We define an isomorphism of onto the dual of the space of -th order theta functions on the Jacobian of . This isomorphism identifies the rational map defined by the linear system with the map which associates to a quadratic bundle the theta divisor . The two components and of are mapped into the subspaces of even and odd theta functions respectively. Finally we discuss the analogous...
Let be an algebraic projective smooth and trigonal curve of genus . In this paper we define, in a way equivalent to that followed by A. Maroni in [1], an integer , called the species of , which is a birational invariant having the property that and mod(2). In section 1 we prove that for every and every , as before, there are trigonal curves of genus and species . In section 2 we define the space of moduli of trigonal curves of genus and species . We note that is irreducible...
Nous étudions les espaces analytiques rigides de dimension 1, réguliers, de genre fini sur un corps valué complet . Nous montrons qu’un tel espace admet une réduction préstable. Si est maximalement complet, se plonge dans une courbe algébrique (analytifiée). On donne aussi une caractérisation des espaces analytiques qui sont le complémentaire d’une partie compacte dans une courbe algébrique.