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

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.