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

Abstract

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.

16 pages. Changes from the previous version are stated in Section 1.3