The p-adic Kummer-Leopoldt constant -- Normalized p-adic regulator
Résumé
The p-adic Kummer--Leopoldt constant kappa_K of a number
field K is (assuming the Leopoldt conjecture) the least integer c
such that for all n >> 0, any global unit
of K, which is locally a p^(n+c)th power at the p-places,
is necessarily the p^nth power of a global unit of K. This constant has been
computed by Assim & Nguyen Quang Do using Iwasawa's techniques,
after intricate studies and calculations by many authors.
We give an elementary p-adic proof and an improvement of these
results, then a class field theory interpretation of kappa_K.
We give some applications (including generalizations of Kummer's
lemma on regular pth cyclotomic fields) and a natural definition of the
normalized p-adic regulator for any K and any p≥2.
This is done without analytical computations, using only class field theory
and especially the properties of the so-called p-torsion
group T_K of Abelian p-ramification theory over K.
Origine : Fichiers produits par l'(les) auteur(s)
Loading...