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.