On the Z_p-ranks of tamely ramified Iwasawa modules

Details On the Z_p-ranks of tamely ramified Iwasawa modules On the Z_p-ranks of tamely ramified Iwasawa modules

2011-03-21

arxiv.org/abs/1103.3916v1

Mathematics

Number Theory

Scientific paper

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.

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