Anticyclotomic Iwasawa theory of CM elliptic curves

Abstract

We study the Iwasawa theory of a CM elliptic curve $E$ in the anticyclotomic ${\mathbf{Z}}_p$-extension of the CM field, where $p$ is a prime of good, ordinary reduction for $E$. When the complex $L$-function of $E$ vanishes to even order, the two variable main conjecture of Rubin implies that the Pontryagin dual of the $p$-power Selmer group over the anticyclotomic extension is a torsion Iwasawa module. When the order of vanishing is odd, work of Greenberg shows that it is not a torsion module. In this paper we show that in the case of odd order of vanishing the dual of the Selmer group has rank exactly one, and we prove a form of the Iwasawa main conjecture for the torsion submodule.

Final version. To appear in the Annales de L'Institut Fourier

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• On the non-vanishing of $p$-adic heights on CM abelian varieties, and the arithmetic of Katz $p$-adic $L$-functions
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• On the control theorem for fine Selmer groups and the growth of fine Tate-Shafarevich groups in ${\mathbb{Z}}_p$-extensions
• The vanishing of anticyclotomic $\mu$-invariants for non-ordinary modular forms
• Derived p-adic heights and p-adic L-functions