The vanishing of Iwasawa's $\mu$-invariant implies the weak Leopoldt conjecture

Abstract

Let K denote a number field containing a primitive p -th root of unity; if p=2 , then we assume K to be totally imaginary. If K_\infty/K is a \mathbb{Z}_p -extension such that no prime above p splits completely in K_\infty/K , then the vanishing of Iwasawa's invariant \mu(K_\infty/K) implies that the weak Leopoldt Conjecture holds for K_\infty/K . This is actually known due to a result of Ueda, which appears to have been forgotten. We present an elementary proof which is based on a reflection formula from class field theory. In the second part of the article, we prove a generalisation in the context of non-commutative Iwasawa theory: we consider admissible p -adic Lie extensions of number fields, and we derive a variant for fine Selmer groups of Galois representations over admissible p -adic Lie extensions.