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Class groups of abelian fields, and the main conjecture

Cornelius Greither (1992)

Annales de l'institut Fourier

This first part of this paper gives a proof of the main conjecture of Iwasawa theory for abelian base fields, including the case p = 2 , by Kolyvagin’s method of Euler systems. On the way, one obtains a general result on local units modulo circular units. This is then used to deduce theorems on the order of χ -parts of p -class groups of abelian number fields: first for relative class groups of real fields (again including the case p = 2 ). As a consequence, a generalization of the Gras conjecture is stated...

Class invariants and cyclotomic unit groups from special values of modular units

Amanda Folsom (2008)

Journal de Théorie des Nombres de Bordeaux

In this article we obtain class invariants and cyclotomic unit groups by considering specializations of modular units. We construct these modular units from functional solutions to higher order q -recurrence equations given by Selberg in his work generalizing the Rogers-Ramanujan identities. As a corollary, we provide a new proof of a result of Zagier and Gupta, originally considered by Gauss, regarding the Gauss periods. These results comprise part of the author’s 2006 Ph.D. thesis [6] in which...

Class numbers of totally real fields and applications to the Weber class number problem

John C. Miller (2014)

Acta Arithmetica

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...

Classes d'idéaux des corps abéliens et nombres de Bernoulli généralisés

Georges Gras (1977)

Annales de l'institut Fourier

Pour l premier impair, l’étude du l -groupe des classes d’idéaux des extensions abéliennes de degré premier à l se ramène à celle de groupes notés H ϕ , où ϕ parcourt un certain ensemble de caractères l -adiques irréductibles.Il est démontré, dans cet article, une généralisation des congruences de Leopoldt et Fresnel entre les fonctions L l l -adiques et les nombres de Bernoulli généralisés. Cette généralisation conduit à une amélioration de la connaissance des H ϕ  : en effet, la juxtaposition de ce résultat...

Completely normal elements in some finite abelian extensions

Ja Koo, Dong Shin (2013)

Open Mathematics

We present some completely normal elements in the maximal real subfields of cyclotomic fields over the field of rational numbers, relying on the criterion for normal element developed in [Jung H.Y., Koo J.K., Shin D.H., Normal bases of ray class fields over imaginary quadratic fields, Math. Z., 2012, 271(1–2), 109–116]. And, we further find completely normal elements in certain abelian extensions of modular function fields in terms of Siegel functions.

Currently displaying 61 – 80 of 461