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Introduction To Cyclotomic Fields

We introduce the logarithmic `-class group of a number field as a cyclotomic analogue of the ordinary ideal class group and we establish the fundamental results of the arithmetic of logaritmic classes.

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Classes logarithmiques des corps de nombres

We shall show two sufficient conditions under which the Iwasawa invariants λ
 k
 and μ
 k
 of a totally real fieldk vanish for an odd primel, based on the results obtained in [1], [3] and [4]. LetKn be the composite ofk and theln-th cyclotomic extension of the fieldQ of rational numbers. LetCn be the factor group of thel-class group ofKn by a subgroup generated by ideals whose prime factors divide the principal ideal (l). Let ϕ1 be an idempotent of the group ringZl[Gal(K1/k)] defined in the below. We shall prove λ
 k
 = μ
 k
 =0 if there is a natural numbern such that e1C
 n
 vanishes, under additional conditions concerning ramifications inKn/k.

On the Iwasawa invariants of totally real number fields

https://www.rzuser.uni-heidelberg.de/~hb3/publ/pcft.pdf

Class Field Towers

Soient K un corps de nombres, de degre fini sur Q, et p un nombre premier fixe. Soient S p l’ensemble des p-places (i.e. des places au-dessus de p) de K et S un ensemble fini de places de K contenant S p . Soient K S la pro—p—extension S-ramifiee (i.e. non ramifiee en dehors de S) maximale de K, et G S = G S (K) = Gal(K S /K). L’objet essentiel de la theorie de laS-ramification, ou ratification restreinte, est l’etude du groupe de Galois G S ,dont la structure reflete les proprietes arithmetiques du corps K par rapport au nombre premier p.

Sur L’arithmétique des Corps de Nombres p-Rationnels

Sur les corps de nombres rguliers

/pdf/theorie-ell-adique-globale-du-corps-de-classes-h2ydy0q5r8.pdf

Théorie $\ell $ -adique globale du corps de classes

This note presents our implementation in the PARI/GP system of the various arithmetic invariants attached to logarithmic classes and units of number fields Our algorithms simplify and improve on works of Diaz y Diaz, Pauli, Pohst, Soriano and the second author

/pdf/the-logarithmic-class-group-package-in-pari-gp-1i6qisw4am.pdf

Jean-François Jaulent

Papers

Classes logarithmiques des corps de nombres

Sur les corps de nombres rguliers

Théorie $\ell $ -adique globale du corps de classes

The logarithmic class group package in PARI/GP

Sur l'indépendance l-adique de nombres algébriques