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Adebisi Agboola; Benjamin Howard
Anticyclotomic Iwasawa theory of CM elliptic curves
(Théorie anticylotomique d’une courbe élliptique à multiplication complexe)
Annales de l'institut Fourier, 56 no. 4 (2006), p. 1001-1048, doi: 10.5802/aif.2206
Article PDF | Reviews MR 2266884 | Zbl 1168.11023
Class. Math.: 11G05, 11R23, 11G16
Keywords: Ellipic curves, Iwasawa theory, main conjecture, anticyclotomic, $p$-adic $L$-function

Résumé - 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, Rubin’s proof of the two variable main conjecture of Iwasawa theory 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 show 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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