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Calcul du nombre de points sur une courbe elliptique dans un corps fini : aspects algorithmiques

François Morain (1995)

Journal de théorie des nombres de Bordeaux

Nous décrivons dans cet article les algorithmes nécessaires à une implantation efficace de la méthode de Schoof pour le calcul du nombre de points sur une courbe elliptique dans un corps fini. Nous tentons d’unifier pour cela les idées d’Atkin et d’Elkies. En particulier, nous décrivons le calcul d’équations pour X 0 ( ) , premier, ainsi que le calcul efficace de facteurs des polynômes de division d’une courbe elliptique.

Caractères numériques et fonctions de Macaulay.

Mireille Martin-Deschamps (2004)

Collectanea Mathematica

The postulation of Aritméticamente Cohen-Macaulay (ACM) subschemes of the projective space PkN is well known in the case of codimension 2. There are many different ways of recording this numerical information: numerical character of Gruson/Peskine, h-vector, postulation character of Martin-Deschamps/Perrin... The first aim of this paper is to show the equivalence of these notions. The second and most important aim, is to study the postulation of codimension 3 ACM subschemes of PN. We use a result...

Computing the determinantal representations of hyperbolic forms

Mao-Ting Chien, Hiroshi Nakazato (2016)

Czechoslovak Mathematical Journal

The numerical range of an n × n matrix is determined by an n degree hyperbolic ternary form. Helton-Vinnikov confirmed conversely that an n degree hyperbolic ternary form admits a symmetric determinantal representation. We determine the types of Riemann theta functions appearing in the Helton-Vinnikov formula for the real symmetric determinantal representation of hyperbolic forms for the genus g = 1 . We reformulate the Fiedler-Helton-Vinnikov formulae for the genus g = 0 , 1 , and present an elementary computation...

Counting lines on surfaces

Samuel Boissière, Alessandra Sarti (2007)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

This paper deals with surfaces with many lines. It is well-known that a cubic contains 27 of them and that the maximal number for a quartic is 64 . In higher degree the question remains open. Here we study classical and new constructions of surfaces with high number of lines. We obtain a symmetric octic with 352 lines, and give examples of surfaces of degree d containing a sequence of d ( d - 2 ) + 4 skew lines.

Counting points on elliptic curves over finite fields

René Schoof (1995)

Journal de théorie des nombres de Bordeaux

We describe three algorithms to count the number of points on an elliptic curve over a finite field. The first one is very practical when the finite field is not too large ; it is based on Shanks's baby-step-giant-step strategy. The second algorithm is very efficient when the endomorphism ring of the curve is known. It exploits the natural lattice structure of this ring. The third algorithm is based on calculations with the torsion points of the elliptic curve [18]. This deterministic polynomial...

Cutting diagram method for systems of plane curves with base points

Marcin Dumnicki (2007)

Annales Polonici Mathematici

We develop a new method of proving non-speciality of a linear system with base fat points in general position. Using this method we show that the Hirschowitz-Harbourne conjecture holds for systems with base points of equal multiplicity bounded by 42.

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