On the Z_p-ranks of tamely ramified Iwasawa modules

TL;DR

This paper derives explicit formulas for the free ranks of S-ramified Iwasawa modules over certain number fields, using p-adic logarithm independence results, advancing understanding of their structure in Iwasawa theory.

Contribution

It provides new explicit formulas for the ranks of S-ramified Iwasawa modules when S excludes p and the base field is rational or imaginary quadratic.

Findings

Explicit formulas for free ranks of Iwasawa modules.

Application of p-adic Baker's theorem to Iwasawa theory.

Results specific to rational and imaginary quadratic fields.

Abstract

For a prime number p, we denote by K the cyclotomic Z_p-extension of a number field k. For a finite set S of prime numbers, we consider the S-ramified Iwasawa module which is the Galois group of the maximal abelian pro-p-extension of K unramified outside S. This paper treats the case where S does not contain p and k is the rational number field or an imaginary quadratic field. In this case, we prove the explicit formulae for the free ranks of the S-ramified Iwasawa modules as abelian pro-p groups, by using Brumer's p-adic version of Baker's theorem on the linear independence of logarithms of algebraic numbers.