Anticyclotomic Iwasawa theory of CM elliptic curves

Agboola, Adebisi; Howard, Benjamin

Annales de l'Institut Fourier, Tome 56 (2006) no. 4, pp. 1001-1048.

Nous étudions la théorie d’Iwasawa d’une courbe elliptique $E$ à multiplication complexe, dans la ${\mathbf{Z}}_p$-extension anticyclotomique du corps de multiplication complexe (ici $p$ est un nombre premier ou $E$ a une bonne réduction ordinaire). Si la fonction $L$ complexe de $E$ a un zero à $s=1$ de multiplicité paire, la preuve de Rubin de la conjecture principale d’Iwasawa en deux variables impliquent que le dual de Pontryagin de la composante $p$-primaire du groupe de Selmer est de torsion comme module d’Iwasawa. Si la multiplicité est impaire, les travaux de Greenberg impliquent que ce module n’est pas un module de torsion. Ici nous montrons que, en cas de multiplicité impaire, le dual de Pontryagin du groupe de Selmer est un module de rang un, et nous prouvons une conjecture principale d’Iwasawa pour le sous-module de torsion.

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.

Classification : 11G05, 11R23, 11G16
Keywords: Ellipic curves, Iwasawa theory, main conjecture, anticyclotomic, $p$-adic $L$-function
Mot clés : courbes élliptiques, théorie d’Iwasawa, conjecture principale, fonction $L$ $p$-adique

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