Iwasawa theory for elliptic curves over imaginary quadratic fields
Journal de théorie des nombres de Bordeaux, Volume 13 (2001) no. 1, pp. 1-25.

Let E be an elliptic curve over , let K be an imaginary quadratic field, and let K be a 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 ,E p ) for i=1,2, and of the p-primary Selmer group Sel p (E/K ), viewed as discrete modules over the Iwasawa algebra of K /K.

Soit E une courbe elliptique sur , soit K un corps quadratique imaginaire, et soit K une 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 ,E p ) pour i=1,2, et de la composante p-primaire du groupe de Selmer Sel p (E/K ), considérés comme modules discrets sur l’algèbre d’Iwasawa de K /K.

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     publisher = {Universit\'e Bordeaux I},
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Bertolini, Massimo. Iwasawa theory for elliptic curves over imaginary quadratic fields. Journal de théorie des nombres de Bordeaux, Volume 13 (2001) no. 1, pp. 1-25. http://archive.numdam.org/item/JTNB_2001__13_1_1_0/

[1] M. Bertolini, Selmer groups and Heegner points in anticyclotomic Zp-extensions. Compositio Math. 99 (1995), 153-182. | Numdam | MR | Zbl

[2] M. Bertolini, An annihilator for the p-Selmer group by means of Heegner points. Atti Acc. Naz. Lincei, Classe di Sc. Fis., Mat. e Nat., Rendiconti Lincei, Mat. e Appl., Serie 9, Vol. 5, Fasc. 2 (1994), 129-140. | MR | Zbl

[3] M. Bertolini, Growth of Mordell-Weil groups in anticyclotomic towers. Symposia Mathematica, Proceedings of the Symposium in Arithmetic Geometry, Cortona 1994, E. Bombieri, et al., eds., Cambridge Univ. Press, to appear. | MR | Zbl

[4] M. Bertolini, H. Darmon, Derived heights and generalized Mazur-Tate regulators. Duke Math. Journal 76 (1994), 75-111. | MR | Zbl

[5] M. Bertolini, H. Darmon, Heegner points on Mumford-Tate curves. Inventiones Math., to appear. | Zbl

[6] J. Coates, R. Greenberg, Kummer theory for Abelian varieties over local fields. Inventiones Math. 124 (1996), 129-174. | MR | Zbl

[7] J. Coates, G. Mcconnell, Iwasawa theory of modular elliptic curves of analytic rank at most 1. J. London Math. Soc. (2) 50 (1994), 243-264. | MR | Zbl

[8] R. Greenberg, Iwasawa theory for p-adic representations. Algebraic Number Theory- in honor of K. Iwasawa, J. Coates et al., editors, Advanced Studies in Pure Mathematics, 1989, Academic Press. | MR | Zbl

[9] S. Lang, Cyclotomic fields I and II (Combined second edition,) GTM 121, 1990, Springer. | MR | Zbl

[10] B. Mazur, Rational points of Abelian Varieties with values in towers of number fields. Inventiones Math. 18 (1972), 183-266. | MR | Zbl

[11] B. Mazur, Modular Curves and Arithmetic, Proc. Int. Cong. of Math. 1983, Warszawa. | Zbl

[12] B. Mazur, Elliptic curves and towers of number fields. Unpublished manuscript.

[13] J-S. Milne, Arithmetic duality theorems. Perspective in Math., Academic Press, 1986. | MR | Zbl

[14] B. Perrin-Riou, Fonctions L p-adiques, Théorie d'Iwasawa et points de Heegner. Bull. Soc. Math. de France 115 (1987), 399-456. | Numdam | MR | Zbl

[15] B. Perrin-Riou, Théorie d'Iwasawa et hauteurs p-adiques. Inventiones Math. 109 (1992), 137-185. | MR | Zbl

[16] D. Rohrlich, On L-functions of elliptic curves and anti-cyclotomic towers. Inventiones Math. 64 (1984), 393-408. | Zbl

[17] D. Rohrlich, On L-functions of elliptic curves and cyclotomic towers. Inventiones Math. 75 (1984), 409-423. | MR | Zbl

[18] P. Schneider, Iwasawa L-functions of varieties over algebraic number fields. A first approach. Inventiones Math. 71 (1983), 251-293. | MR | Zbl

[19] P. Schneider, p-adic height pairings II. Inventiones Math. 79 (1985), 329-374. | MR | Zbl

[20] P. Schneider, Arithmetic of formal groups and applications. I. Universal norm subgroups. Inventiones Math. 87 (1987), 587-602. | MR | Zbl

[21] J.-P. Serre, Cohomologie Galoisienne. LNM 5 (cinquième édition), Springer, 1994. | MR | Zbl