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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 anelliptic curve with complex multiplication. This $\mathbb{Z}_p$-extension hasproperties similar to those of the cyclotomic $\mathbb{Z}_p$-extension of areal abelian field, however, it is already known that $X (F_\infty)$ can beinfinite. That is, an analog of Greenberg's conjecture for this$\mathbb{Z}_p$-extension fails. In this paper, we mainly consider analogs ofweak forms of Greenberg's conjecture.

On the unramified Iwasawa module of a $\mathbb{Z}_p$-extension generated by division points of a CM elliptic curve