Block systems of a Galois group.
Bounds for discriminants and related estimates for class numbers, regulators and zeros of zeta functions : a survey of recent results
A bibliography of recent papers on lower bounds for discriminants of number fields and related topics is presented. Some of the main methods, results, and open problems are discussed.
Calcul du nombre de classes d'un corps quadratique imaginaire ou réel, d'après Shanks, Williams, McCurley, A. K. Lenstra et Schnorr
Dans cette note nous décrivons différentes méthodes utilisées en pratique pour calculer le nombre de classes d'un corps quadratique imaginaire ou réel ainsi que pour calculer le régulateur d'un corps quadratique réel. En particulier nous décrivons l'infrastructure de Shanks ainsi que la méthode sous-exponentielle de McCurley.
Calcul et rationalité de fonctions de Belyi en genre 0
L’article comporte une méthode de calcul de fonctions de Belyi “optimales”, associées à des dessins plans. Cette étude conduit à s’interroger sur la possibilité de définir une fonction de Belyi sur le corps des modules du dessin. Pour les arbres par exemple, nous montrons que c’est toujours le cas. La preuve donne une méthode pour spécifier une telle fonction. Nous donnons ensuite un exemple de dessin qui n’admet pas de fonction de Belyi sur son corps des modules. Enfin, nous étudions la question...
Calculation of the greatest common divisor of perturbed polynomials
The coefficients of the greatest common divisor of two polynomials and (GCD) can be obtained from the Sylvester subresultant matrix transformed to lower triangular form, where and deg(GCD) needs to be computed. Firstly, it is supposed that the coefficients of polynomials are given exactly. Transformations of for an arbitrary allowable are in details described and an algorithm for the calculation of the GCD is formulated. If inexact polynomials are given, then an approximate greatest...
Class numbers of ring class fields of prime conductor
Class numbers of totally real fields and applications to the Weber class number problem
The determination of the class number of totally real fields of large discriminant is known to be a difficult problem. The Minkowski bound is too large to be useful, and the root discriminant of the field can be too large to be treated by Odlyzko's discriminant bounds. We describe a new technique for determining the class number of such fields, allowing us to attack the class number problem for a large class of number fields not treatable by previously known methods. We give an application to Weber's...
CM liftings of supersingular elliptic curves
Assuming GRH, we present an algorithm which inputs a prime and outputs the set of fundamental discriminants such that the reduction map modulo a prime above from elliptic curves with CM by to supersingular elliptic curves in characteristic is surjective. In the algorithm we first determine an explicit constant so that implies that the map is necessarily surjective and then we compute explicitly the cases .
Comptage exact de discriminants d'extensions abéliennes
Le but de cet article est d’expliquer comment calculer exactement le nombre de classes d’isomorphismes d’extensions abéliennes de en degré inférieur ou égal à et de discriminant majoré par une borne donnée. On parvient par exemple à calculer le nombre de corps cubiques cycliques de discriminant inférieur ou égal à .
Computation of 2-groups of positive classes of exceptional number fields
We present an algorithm for computing the 2-group of the positive divisor classes in case the number field has exceptional dyadic places. As an application, we compute the 2-rank of the wild kernel in .
Computation of -adic heights and log convergence.
Computation of relative class numbers of imaginary abelian number fields.
Computation of the fundamental units and the regulator of a cyclic cubic function field.
Computation of the Selmer groups of certain parametrized elliptic curves
Computing all elements of given index in sextic fields with a cubic subfield
Computing all monogeneous mixed dihedral quartic extensions of a quadratic field
Let be a given real quadratic field. We give a fast algorithm for determining all dihedral quartic fields with mixed signature having power integral bases and containing as a subfield. We also determine all generators of power integral bases in . Our algorithm combines a recent result of Kable [9] with the algorithm of Gaál, Pethö and Pohst [6], [7]. To illustrate the method we performed computations for
Computing Iwasawa modules of real quadratic number fields
Corps sextiques contenant un corps cubique (III)
Corps sextiques contenant un corps quadratique (I)
Corps sextiques contenant un corps quadratique (II)