THE VANISHING OF IWASAWA'S μ-INVARIANT IMPLIES THE WEAK LEOPOLDT CONJECTURE

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 infinity)/K is a Z(p)-extension such that no prime above p splits completely in K-infinity/K, then the vanishing of Iwasawa's invariant mu(K-infinity/K) implies that the weak Leopoldt Conjecture holds for K-infinity/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.