The p-adic Kummer-Leopoldt constant: Normalized p-adic regulator

Abstract

The p-adic Kummer-Leopoldt constant κK 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 pn+cth power at the p-places, is necessarily the pnth power of a global unit of K. This constant has been computed by Assim and 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 κK. 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 TK of Abelian p-ramification theory over K.