Iwasawa theory for elliptic curves over imaginary quadratic fields

Massimo Bertolini

doi:10.5802/jtnb.300

Massimo Bertolini

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

Abstract (VO)
Résumé

Let $E$ be an elliptic curve over $ℚ$ , let $K$ be an imaginary quadratic field, and let $K_{\infty}$ be a $ℤ_{p}$ -extension of $K$ . Given a set $Σ$ of primes of $K$ , containing the primes above $p$ , and the primes of bad reduction for $E$ , write $K_{Σ}$ for the maximal algebraic extension of $K$ which is unramified outside $Σ$ . This paper is devoted to the study of the structure of the cohomology groups $H^{i} (K_{Σ} / K_{\infty}, E_{p^{\infty}})$ 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 $K_{\infty} / K .$

Soit $E$ une courbe elliptique sur $ℚ$ , soit $K$ un corps quadratique imaginaire, et soit $K_{\infty}$ une $ℤ_{p}$ -extension de $K$ . Étant donné un ensemble $Σ$ de places de $K$ contenant les places au dessus de $p$ et les places de mauvaise réduction de $E$ , nous notons $K_{Σ}$ l’extension maximale de $K$ non ramifiée en-dehors de $Σ$ . Cet article est consacré à l’étude de la structure des groupes de cohomologie $H^{i} (K_{Σ} / K_{\infty}, E_{p^{\infty}})$ 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 $K_{\infty} / K .$

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DOI : 10.5802/jtnb.300

Massimo Bertolini. Iwasawa theory for elliptic curves over imaginary quadratic fields. Journal de théorie des nombres de Bordeaux, Tome 13 (2001) no. 1, pp. 1-25. doi: 10.5802/jtnb.300

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