Anticyclotomic Iwasawa theory of CM elliptic curves

Details Anticyclotomic Iwasawa theory of CM elliptic curves Anticyclotomic Iwasawa theory of CM elliptic curves

2003-02-25

arxiv.org/abs/math/0302319v4

Ann. Inst. Fourier 56, No.4 (2006) 1001-1048

Mathematics

Number Theory

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

Scientific paper

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.

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