Practice of incomplete p-ramification over a number field -- History of abelian p-ramification - Archive ouverte HAL Accéder directement au contenu
Pré-Publication, Document De Travail Année : 2019

Practice of incomplete p-ramification over a number field -- History of abelian p-ramification

Résumé

The theory of p-ramification, regarding the Galois group of the maximal pro-p-extension of a number field K, unramified outside p and ∞, is well known including numerical experiments with PARI/GP programs. The case of ``incomplete p-ramification'' (i.e., when the set S of ramified places is a strict subset of the set P of the p-places) is, on the contrary, mostly unknown in a theoretical point of view. We give, in a first part, a way to compute, for any S ≤ P, the structure of the Galois group of the maximal S-ramified abelian pro-p-extension H_(K,S) of any field K given by means of an irreducible polynomial. We publish PARI/GP programs usable without any special prerequisites. Then, in an Appendix, we recall the ``story'' of abelian S-ramification restricting ourselves to elementary aspects in order to precise much basic contributions and references, often disregarded, which may be used by specialists of other domains of number theory. Indeed, the torsion T_(K,S) of Gal(H_(K,S)/K) (even if S=P) is a fundamental obstruction in many problems. All relationships involving S-ramification, as Iwasawa's theory, Galois cohomology, p-adic L-functions, elliptic curves, algebraic geometry, would merit special developments, which is not the purpose of this text.
Fichier principal
Vignette du fichier
S-ramification theory.pdf (505.71 Ko) Télécharger le fichier
Origine : Fichiers produits par l'(les) auteur(s)
Loading...

Dates et versions

hal-02106964 , version 1 (23-04-2019)
hal-02106964 , version 2 (15-05-2019)
hal-02106964 , version 3 (25-06-2019)

Identifiants

Citer

Georges Gras. Practice of incomplete p-ramification over a number field -- History of abelian p-ramification. 2019. ⟨hal-02106964v3⟩
66 Consultations
470 Téléchargements

Altmetric

Partager

Gmail Facebook X LinkedIn More