The Ferrero-Washington theorem says that if $K/\mathbf{Q}$ is an abelian extension, then the cyclotomic $\mathbf{Z}_p$ extension $K^{\text{cyc}}/K$ has $\mu=0$.
In the paper "Iwasawa invariants of Elliptic Curves" by Greenberg and Vatsal, the authors give an alternative formulation of the Ferrero-Washington Theorem. Let $\psi: G_\mathbf{Q} \to \mathbf{Z}_p^{\times}$ be a character which is unramified at $p$ and odd. Let $C$ be the $G_\mathbf{Q}$-module defined by $$C = \mu_{p^{\infty}} \otimes \psi^{-1}.$$ Then $C$ is isomorphic to $\mathbf{Q}_p/\mathbf{Z}_p$ as an abelian group. One can then define a "Selmer group" $S_C(\mathbf{Q}^{\text{cyc}})$ which is a subgroup of $H^1(\mathbf{Q}^{\text{cyc}}, C)$ cut out by some local conditions. Now in page 11 of their paper, Greenberg/Vatsal say that the Ferrero-Washington Theorem implies that the $\Lambda$-module $S_C(\mathbf{Q}_{\infty})^{\wedge}$ has $\mu$-invariant zero.
My question is: why does the Ferrero-Washington theorem imply this? I cannot see the connection between the classical formulation and this new formulation. Would someone be able to sketch out the details here?
