The vanishing of Iwasawa's \(\mu\)-invariant implies the weak Leopoldt conjecture (Q2695705)

Summary: 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.