Anticyclotomic Iwasawa theory of CM elliptic curves
Annales de l'Institut Fourier, Volume 56 (2006) no. 4, pp. 1001-1048.

We study the Iwasawa theory of a CM elliptic curve E in the anticyclotomic 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.

Nous étudions la théorie d’Iwasawa d’une courbe elliptique E à multiplication complexe, dans la 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.

DOI: 10.5802/aif.2206
Classification: 11G05,  11R23,  11G16
Keywords: Ellipic curves, Iwasawa theory, main conjecture, anticyclotomic, p-adic L-function
Agboola, Adebisi 1; Howard, Benjamin 2

1 University of California Department of Mathematics Santa Barbara, CA 93106
2 Harvard University Department of Mathematics Cambridge, MA 02138
@article{AIF_2006__56_4_1001_0,
     author = {Agboola, Adebisi and Howard, Benjamin},
     title = {Anticyclotomic {Iwasawa} theory of {CM} elliptic curves},
     journal = {Annales de l'Institut Fourier},
     pages = {1001--1048},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {56},
     number = {4},
     year = {2006},
     doi = {10.5802/aif.2206},
     mrnumber = {2266884},
     zbl = {1168.11023},
     language = {en},
     url = {http://archive.numdam.org/articles/10.5802/aif.2206/}
}
TY  - JOUR
AU  - Agboola, Adebisi
AU  - Howard, Benjamin
TI  - Anticyclotomic Iwasawa theory of CM elliptic curves
JO  - Annales de l'Institut Fourier
PY  - 2006
DA  - 2006///
SP  - 1001
EP  - 1048
VL  - 56
IS  - 4
PB  - Association des Annales de l’institut Fourier
UR  - http://archive.numdam.org/articles/10.5802/aif.2206/
UR  - https://www.ams.org/mathscinet-getitem?mr=2266884
UR  - https://zbmath.org/?q=an%3A1168.11023
UR  - https://doi.org/10.5802/aif.2206
DO  - 10.5802/aif.2206
LA  - en
ID  - AIF_2006__56_4_1001_0
ER  - 
%0 Journal Article
%A Agboola, Adebisi
%A Howard, Benjamin
%T Anticyclotomic Iwasawa theory of CM elliptic curves
%J Annales de l'Institut Fourier
%D 2006
%P 1001-1048
%V 56
%N 4
%I Association des Annales de l’institut Fourier
%U https://doi.org/10.5802/aif.2206
%R 10.5802/aif.2206
%G en
%F AIF_2006__56_4_1001_0
Agboola, Adebisi; Howard, Benjamin. Anticyclotomic Iwasawa theory of CM elliptic curves. Annales de l'Institut Fourier, Volume 56 (2006) no. 4, pp. 1001-1048. doi : 10.5802/aif.2206. http://archive.numdam.org/articles/10.5802/aif.2206/

[1] Arnold, T. Anticyclotomic main conjectures for CM modular forms (2005) (Preprint)

[2] Bertrand, D. Propriétés arithmétiques de fonctions thêta à plusieurs variables, Number theory, Noordwijkerhout 1983 (Lecture Notes in Math.), Volume 1068, Springer, Berlin, 1984, pp. 17-22 | MR | Zbl

[3] Coates, J. Infinite descent on elliptic curves with complex multiplication, Arithmetic and Geometry, Vol. I (Progr. Math.), Volume 35, Birkhäuser Boston, Boston, MA, 1983, pp. 107-137 | MR | Zbl

[4] Greenberg, Ralph On the structure of certain Galois groups, Invent. Math., Volume 47 (1978) no. 1, pp. 85-99 | DOI | MR | Zbl

[5] Greenberg, Ralph On the Birch and Swinnerton-Dyer conjecture, Invent. Math., Volume 72 (1983) no. 2, pp. 241-265 | DOI | MR | Zbl

[6] Gross, Benedict H.; Zagier, Don B. Heegner points and derivatives of L-series, Invent. Math., Volume 84 (1986) no. 2, pp. 225-320 | DOI | MR | Zbl

[7] Howard, Benjamin The Iwasawa theoretic Gross-Zagier theorem, Compos. Math., Volume 141 (2005) no. 4, pp. 811-846 | DOI | MR | Zbl

[8] Lang, Serge Algebraic number theory, Graduate Texts in Mathematics, 110, Springer-Verlag, New York, 1994 | MR | Zbl

[9] Martinet, J. Character theory and Artin L-functions, Algebraic number fields: L-functions and Galois properties (Proc. Sympos., Univ. Durham, Durham, 1975), Academic Press, London, 1977, pp. 1-87 | MR | Zbl

[10] Mazur, B. Modular curves and arithmetic, Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Warsaw, 1983) (1984), pp. 185-211 | MR | Zbl

[11] Mazur, B.; Tate, J. Canonical height pairings via biextensions, Arithmetic and geometry, Vol. I (Progr. Math.), Volume 35, Birkhäuser Boston, Boston, MA, 1983, pp. 195-237 | MR | Zbl

[12] Mazur, Barry Rational points of abelian varieties with values in towers of number fields, Invent. Math., Volume 18 (1972), pp. 183-266 | DOI | EuDML | MR | Zbl

[13] Mazur, Barry; Rubin, Karl Elliptic curves and class field theory, Proceedings of the International Congress of Mathematicians, Vol. II (Beijing, 2002) (2002), pp. 185-195 | MR | Zbl

[14] Mazur, Barry; Rubin, Karl Studying the growth of Mordell-Weil, Doc. Math. (2003) no. Extra Vol., p. 585-607 (electronic) (Kazuya Kato’s fiftieth birthday) | EuDML | MR | Zbl

[15] Mazur, Barry; Rubin, Karl Kolyvagin systems, Mem. Amer. Math. Soc., Volume 168 (2004), pp. viii+96 | MR | Zbl

[16] Perrin-Riou, Bernadette Arithmétique des courbes elliptiques et théorie d’Iwasawa, Mém. Soc. Math. France (N.S.) (1984) no. 17, pp. 130 | EuDML | Numdam | MR | Zbl

[17] Perrin-Riou, Bernadette Fonctions L p-adiques, théorie d’Iwasawa et points de Heegner, Bull. Soc. Math. France, Volume 115 (1987) no. 4, pp. 399-456 | EuDML | Numdam | MR | Zbl

[18] Perrin-Riou, Bernadette Théorie d’Iwasawa et hauteurs p-adiques, Invent. Math., Volume 109 (1992) no. 1, pp. 137-185 | DOI | EuDML | MR | Zbl

[19] Rohrlich, David E. On L-functions of elliptic curves and anticyclotomic towers, Invent. Math., Volume 75 (1984), pp. 383-408 | DOI | EuDML | MR | Zbl

[20] Rubin, Karl The “main conjectures” of Iwasawa theory for imaginary quadratic fields, Invent. Math., Volume 103 (1991) no. 1, pp. 25-68 | DOI | EuDML | MR | Zbl

[21] Rubin, Karl p-adic L-functions and rational points on elliptic curves with complex multiplication, Invent. Math., Volume 107 (1992) no. 2, pp. 323-350 | DOI | EuDML | MR | Zbl

[22] Rubin, Karl Abelian varieties, p-adic heights and derivatives, Algebra and number theory (Essen, 1992), de Gruyter, Berlin, 1994, pp. 247-266 | MR | Zbl

[23] Rubin, Karl Elliptic curves with complex multiplication and the conjecture of Birch and Swinnerton-Dyer, Arithmetic theory of elliptic curves (Cetraro, 1997) (Lecture Notes in Math.), Volume 1716, Springer, Berlin, 1999, pp. 167-234 | MR | Zbl

[24] Rubin, Karl Euler systems, Annals of Mathematics Studies, 147, Princeton University Press, Princeton, NJ, 2000 (Hermann Weyl Lectures. The Institute for Advanced Study) | MR | Zbl

[25] de Shalit, Ehud Iwasawa theory of elliptic curves with complex multiplication, Perspectives in Mathematics, 3, Academic Press Inc., Boston, MA, 1987 | MR | Zbl

[26] Weil, A. Automorphic Forms and Dirichlet Series, Dirichlet series and automorphic forms. Lezioni fermiane. (Lecture Notes in Math.), Volume 189, Springer, 1971 | Zbl

[27] Yager, Rodney I. On two variable p-adic L-functions, Ann. of Math. (2), Volume 115 (1982) no. 2, pp. 411-449 | DOI | MR | Zbl

Cited by Sources: