In this paper, we give the complete characterization of the –torsion subgroups of certain idèle–class groups associated to characteristic function fields. As an application, we answer a question which arose in the context of Tan’s approach [6] to an important particular case of a generalization of a conjecture of Gross [4] on special values of –functions.
For an algebraic number field and a prime , define the number to be the maximal number such that there exists a Galois extension of whose Galois group is a free pro--group of rank . The Leopoldt conjecture implies , ( denotes the number of complex places of ). Some examples of and with have been known so far. In this note, the invariant is studied, and among other things some examples with are given.
We give a survey of computational class field theory. We first explain how to compute ray class groups and discriminants of the corresponding ray class fields. We then explain the three main methods in use for computing an equation for the class fields themselves: Kummer theory, Stark units and complex multiplication. Using these techniques we can construct many new number fields, including fields of very small root discriminant.
We prove that for any prime p there is a constant Cₚ > 0 such that for any n > 0 and for any p-power q there is a smooth, projective, absolutely irreducible curve over of genus g ≤ Cₚqⁿ without points of degree smaller than n.
Nous construisons un analogue «tordu» de la -tour de corps de classes d’un corps de nombres ( un nombre premier) et étudions ses liens avec la théorie d’Iwasawa. Le résultat principal donne un critère du type Golod et Shafarevich pour que la tour «tordue» soit infinie.