  We consider the unramified Iwasawa module $X (F_\infty)$ of a certain
$\mathbb{Z}_p$-extension $F_\infty/F_0$ generated by division points of an
elliptic curve with complex multiplication. This $\mathbb{Z}_p$-extension has
properties similar to those of the cyclotomic $\mathbb{Z}_p$-extension of a
real abelian field, however, it is already known that $X (F_\infty)$ can be
infinite. That is, an analog of Greenberg's conjecture for this
$\mathbb{Z}_p$-extension fails. In this paper, we mainly consider analogs of
weak forms of Greenberg's conjecture.
