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.

arXiv comment16 pages. Changes from the previous version are stated in Section 1.3

References in corpus (1)

• On the $\lambda$-invariant of Selmer groups arising from certain quadratic twists of Gross curves