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

Abstract : 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.
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Contributor : Georges Gras <>
Submitted on : Friday, March 31, 2017 - 2:07:48 PM
Last modification on : Monday, April 9, 2018 - 12:20:07 PM
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  • HAL Id : hal-01444560, version 2
  • ARXIV : 1701.06857



Georges Gras. The p-adic Kummer-Leopoldt constant -- Normalized p-adic regulator. 2017. ⟨hal-01444560v2⟩



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