Riv. Mat. Univ. Parma, Vol. 14, No. 1, 2023

Tsuyoshi Itoh [a]

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

Pages: 153-171
Received: 7 September 2022
Accepted in revised form: 28 March 2023
Mathematics Subject Classification: 11R23, 11G05, 11G15.
Keywords: Non-cyclotomic \(\mathbb{Z}_p\)-extension, Iwasawa module, CM elliptic curve.
Authors address:
[a]: Division of Mathematics, Education Center, Faculty of Social Systems Science, Chiba Institute of Technology, 2-1-1 Shibazono, Narashino, Chiba, 275-0023, Japan

This work was partly supported by JSPS KAKENHI Grant Number JP15K04791.

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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.

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