On tamely ramified Iwasawa modules for the cyclotomic Z_p-extension of abelian fields

TL;DR

This paper provides a formula for the Z_p-rank of the Galois group of a maximal S-ramified abelian pro-p extension over the cyclotomic Z_p-extension of an abelian field, revealing new finiteness properties.

Contribution

It introduces a formula for the Z_p-rank of certain Galois groups and proves finiteness of specific ramified extensions in the context of Iwasawa theory.

Findings

Formula for Z_p-rank of Galois groups

Finiteness of M_q(k_)/L(k_) extensions

Extension properties for real abelian fields

Abstract

Let p be an odd prime, and k_\infty the cyclotomic Z_p-extension of an abelian field k. For a finite set S of rational primes which does not include p, we will consider the maximal S-ramified abelian pro-p extension M_S(k_\infty) over k_\infty. We shall give a formula of the Z_p-rank of Gal(M_S(k_\infty)/k_\infty). In the proof of this formula, we also show that M_{q}(k_\infty)/L(k_\infty) is a finite extension for every real abelian field k and every rational prime q distinct from p, where L(k_\infty) is the maximal unramified abelian pro-p extension over k_\infty.