On the non-commutative Main Conjecture for elliptic curves with complex multiplication

TL;DR

This paper proves the non-commutative Iwasawa Main Conjecture for CM-elliptic curves over , establishing the existence of the associated p-adic L-function and linking it to the Katz-measure and Rubin's work.

Contribution

It provides a proof of the non-commutative Iwasawa Main Conjecture for CM-elliptic curves, including the existence of the p-adic L-function and its relation to established measures.

Findings

Proof of the non-commutative Main Conjecture for CM-elliptic curves.

Existence of the non-commutative p-adic L-function for these curves.

Connection between the p-adic L-function and the Katz-measure, Rubin's work, and period comparisons.

Abstract

In arXiv:math/0404297 a non-commutative Iwasawa Main Conjecture for elliptic curves over $\mathbb{Q}$ has been formulated. In this note we show that it holds for all CM-elliptic curves $E$ defined over $\mathbb{Q}$. This was claimed in (loc.\ cit.) without proof, which we want to provide now assuming that the torsion conjecture holds in this case. Based on this we show firstly the existence of the (non-commutative) $p$-adic $L$-function of $E$ and secondly that the (non-commutative) Main Conjecture follows from the existence of the Katz-measure, the work of Yager and Rubin's proof of the 2-variable main conjecture. The main issues are the comparison of the involved periods and to show that the (non-commutative) $p$-adic $L$-function is defined over the conjectured in (loc.\ cit.) coefficient ring. Moreover we generalize our considerations to the case of CM-elliptic cusp forms.