Computing integral points on Mordell's elliptic curves.
Computing modular degrees using -functions
We give an algorithm to compute the modular degree of an elliptic curve defined over . Our method is based on the computation of the special value at of the symmetric square of the -function attached to the elliptic curve. This method is quite efficient and easy to implement.
Computing periods of cusp forms and modular elliptic curves.
Computing special values of motivic -functions.
Computing the cardinality of CM elliptic curves using torsion points
Let be an elliptic curve having complex multiplication by a given quadratic order of an imaginary quadratic field . The field of definition of is the ring class field of the order. If the prime splits completely in , then we can reduce modulo one the factors of and get a curve defined over . The trace of the Frobenius of is known up to sign and we need a fast way to find this sign, in the context of the Elliptic Curve Primality Proving algorithm (ECPP). For this purpose, we propose...
Computing the modular degree of an elliptic curve.
Computing torsion points on curves.
Concernant la relation de distribution satisfaite par la fonction ...associé à un réseau complexe.
Congruent numbers over real number fields
It is classical that a natural number n is congruent iff the rank of ℚ -points on Eₙ: y² = x³-n²x is positive. In this paper, following Tada (2001), we consider generalised congruent numbers. We extend the above classical criterion to several infinite families of real number fields.
Conjugation of Kuga fiber varieties.
Connectedness results for -adic representations associated to abelian varieties
Conservative polynomials and yet another action of on plane trees
In this paper we study an action of the absolute Galois group on bicolored plane trees. In distinction with the similar action provided by the Grothendieck theory of “Dessins d’enfants” the action is induced by the action of on equivalence classes of conservative polynomials which are the simplest examples of postcritically finite rational functions. We establish some basic properties of the action and compare it with the Grothendieck action.
Constructing elliptic curves over finite fields using double eta-quotients
We examine a class of modular functions for whose values generate ring class fields of imaginary quadratic orders. This fact leads to a new algorithm for constructing elliptic curves with complex multiplication. The difficulties arising when the genus of is not zero are overcome by computing certain modular polynomials.Being a product of four -functions, the proposed modular functions can be viewed as a natural generalisation of the functions examined by Weber and usually employed to construct...
Construction of Ray class fields by elliptic units
From complex multiplication we know that elliptic units are contained in certain ray class fields over a quadratic imaginary number field , and Ramachandra [3] has shown that these ray class fields can even be generated by elliptic units. However the generators constructed by Ramachandra involve very complicated products of high powers of singular values of the Klein form defined below and singular values of the discriminant . It is the aim of this paper to show, that in many cases a generator...
Construction of some classes in the cohomology of Siegel modular threefolds
Constructions de polynômes génériques à groupe de Galois résoluble
On sait que les seuls sous-groupes résolubles transitifs du groupe symétrique ₅ sont isomorphes au groupe de Frobenius , au groupe diédral D₅ et au groupe cyclique C₅. Nous montrerons comment construire des extensions de degré 5 à groupe de Galois résoluble à l’aide de courbes elliptiques. Dans un premier paragraphe nous utiliserons une courbe elliptique ayant un point de 5-torsion rationnel pour les groupes D₅ et C₅. Puis, dans le paragraphe suivant, nous utiliserons une courbe elliptique ayant...
Constructions of plane curves with many points
Contre-exemples au principe de Hasse pour certains tores coflasques
Nous étudions le comportement asymptotique du nombre de variétés dans une certaine classe ne satisfaisant pas le principe de Hasse. Cette étude repose sur des résultats récemment obtenus par Colliot-Thélène [3].
Corestriction of central simple algebras and families of Mumford-type
Let be a family of Mumford-type, that is, a family of polarized complex abelian fourfolds as introduced by Mumford in [9]. This family is defined starting from a quaternion algebra over a real cubic number field and imposing a condition to the corestriction of such . In this paper, under some extra conditions on the algebra , we make this condition explicit and in this way we are able to describe the polarization and the complex structures of the fibers. Then, we look at the non simple -fibers...
Corps de définition et points rationnels
Soit un objet algébrique (par exemple une courbe ou un revêtement) défini sur et de corps des modules un corps de nombres . Il est bien connu que n’admet pas nécessairement de -modèle. En utilisant deux résultats récents dus à P. Dèbes, J.-C. Douai et M. Emsalem nous donnerons un majorant pour le degré d’un corps de définition de sur . Dans une deuxième partie, nous donnerons des conditions suffisantes sur l’ordre de Aut() pour que admette un -modèle.