Iwasawa theory for elliptic curves over imaginary quadratic fields

Bertolini, Massimo

Journal de théorie des nombres de Bordeaux, Tome 13 (2001) no. 1, pp. 1-25.

Soit $E$ une courbe elliptique sur $\mathbb{Q}$, soit $K$ un corps quadratique imaginaire, et soit $K_{\infty }$ une ${\mathbb{Z}}_p$-extension de $K$. Étant donné un ensemble $\Sigma$ de places de $K$ contenant les places au dessus de $p$ et les places de mauvaise réduction de $E$, nous notons $K_{\Sigma }$ l’extension maximale de $K$ non ramifiée en-dehors de $\Sigma$. Cet article est consacré à l’étude de la structure des groupes de cohomologie $H^i(K_{\Sigma }/K_{\infty },E_{p^{\infty }})$ pour $i=1,2,$ et de la composante $p$-primaire du groupe de Selmer Sel${}_{p^{\infty }}(E/K_{\infty })$, considérés comme modules discrets sur l’algèbre d’Iwasawa de $K_{\infty }/K.$

Let $E$ be an elliptic curve over $\mathbb{Q}$, let $K$ be an imaginary quadratic field, and let $K_{\infty }$ be a ${\mathbb{Z}}_p$-extension of $K$. Given a set $\Sigma$ of primes of $K$, containing the primes above $p$, and the primes of bad reduction for $E$, write $K_{\Sigma }$ for the maximal algebraic extension of $K$ which is unramified outside $\Sigma$. This paper is devoted to the study of the structure of the cohomology groups $H^i(K_{\Sigma }/K_{\infty },E_{p^{\infty }})$ for $i=1,2,$ and of the $p$-primary Selmer group Sel${}_{p^{\infty }}(E/K_{\infty })$, viewed as discrete modules over the Iwasawa algebra of $K_{\infty }/K.$

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