Local units modulo circular units

Abstract:

In his paper, Some Modules in the Theory of Cyclotomic Fields [2], Iwasawa obtained the remarkable theorem that the quotient of the $p$-adic cyclotomic units by the completion of the circular units is isomorphic to the quotient of the group ring by the Stickleberger ideal. He then used this to deduce some interesting global results, the most striking of which is an explanation of the plus part of the analytic class number formula under the assumption that the class group at the first layer is cyclic, together with a regularity assumption. In this note, we will show how with the results in our paper, Division Values in Local Fields [1], it is now possible to give a substantially simpler proof of the above theorem. We also describe, briefly, how one can obtain various global results of Iwasawa from this Theorem, which are not included in either Lang’s [3], or Washington’s [4] books.