Anticyclotomic Iwasawa theory of CM elliptic curves
Annales de l'Institut Fourier, Volume 56 (2006) no. 4, p. 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 : https://doi.org/10.5802/aif.2206
Classification:  11G05,  11R23,  11G16
Keywords: Ellipic curves, Iwasawa theory, main conjecture, anticyclotomic, p-adic L-function
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     author = {Agboola, Adebisi and Howard, Benjamin},
     title = {Anticyclotomic Iwasawa theory of CM elliptic curves},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {56},
     number = {4},
     year = {2006},
     pages = {1001-1048},
     doi = {10.5802/aif.2206},
     zbl = {1168.11023},
     mrnumber = {2266884},
     language = {en},
     url = {https://aif.centre-mersenne.org/item/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. https://aif.centre-mersenne.org/item/AIF_2006__56_4_1001_0/

[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, Springer, Berlin (Lecture Notes in Math.) Tome 1068 (1984), pp. 17-22 | MR 756080 | Zbl 0546.14029

[3] Coates, J. Infinite descent on elliptic curves with complex multiplication, Arithmetic and Geometry, Vol. I, Birkhäuser Boston, Boston, MA (Progr. Math.) Tome 35 (1983), pp. 107-137 | MR 717591 | Zbl 0541.14026

[4] Greenberg, Ralph On the structure of certain Galois groups, Invent. Math., Tome 47 (1978) no. 1, pp. 85-99 | Article | MR 504453 | Zbl 0403.12004

[5] Greenberg, Ralph On the Birch and Swinnerton-Dyer conjecture, Invent. Math., Tome 72 (1983) no. 2, pp. 241-265 | Article | MR 700770 | Zbl 0546.14015

[6] Gross, Benedict H.; Zagier, Don B. Heegner points and derivatives of L-series, Invent. Math., Tome 84 (1986) no. 2, pp. 225-320 | Article | MR 833192 | Zbl 0608.14019

[7] Howard, Benjamin The Iwasawa theoretic Gross-Zagier theorem, Compos. Math., Tome 141 (2005) no. 4, pp. 811-846 | Article | MR 2148200 | Zbl 02211027

[8] Lang, Serge Algebraic number theory, Springer-Verlag, New York, Graduate Texts in Mathematics, Tome 110 (1994) | MR 1282723 | Zbl 0811.11001

[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 447187 | Zbl 0359.12015

[10] Mazur, B. Modular curves and arithmetic, Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Warsaw, 1983), PWN, Warsaw (1984), pp. 185-211 | MR 804682 | Zbl 0597.14023

[11] Mazur, B.; Tate, J. Canonical height pairings via biextensions, Arithmetic and geometry, Vol. I, Birkhäuser Boston, Boston, MA (Progr. Math.) Tome 35 (1983), pp. 195-237 | MR 717595 | Zbl 0574.14036

[12] Mazur, Barry Rational points of abelian varieties with values in towers of number fields, Invent. Math., Tome 18 (1972), pp. 183-266 | Article | MR 444670 | Zbl 0245.14015

[13] Mazur, Barry; Rubin, Karl Elliptic curves and class field theory, Proceedings of the International Congress of Mathematicians, Vol. II (Beijing, 2002), Higher Ed. Press, Beijing (2002), pp. 185-195 | MR 1957032 | Zbl 1036.11023

[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) | MR 2046609 | Zbl 1142.11339 | Zbl 02028845

[15] Mazur, Barry; Rubin, Karl Kolyvagin systems, Mem. Amer. Math. Soc., Tome 168 (2004), pp. viii+96 | MR 2031496 | Zbl 1055.11041

[16] Perrin-Riou, Bernadette Arithmétique des courbes elliptiques et théorie d’Iwasawa, Mém. Soc. Math. France (N.S.) (1984) no. 17, pp. 130 | Numdam | MR 799673 | Zbl 0599.14020

[17] Perrin-Riou, Bernadette Fonctions L p-adiques, théorie d’Iwasawa et points de Heegner, Bull. Soc. Math. France, Tome 115 (1987) no. 4, pp. 399-456 | Numdam | MR 928018 | Zbl 0664.12010

[18] Perrin-Riou, Bernadette Théorie d’Iwasawa et hauteurs p-adiques, Invent. Math., Tome 109 (1992) no. 1, pp. 137-185 | Article | MR 1168369 | Zbl 0781.14013

[19] Rohrlich, David E. On L-functions of elliptic curves and anticyclotomic towers, Invent. Math., Tome 75 (1984), pp. 383-408 | Article | MR 735332 | Zbl 0565.14008

[20] Rubin, Karl The “main conjectures” of Iwasawa theory for imaginary quadratic fields, Invent. Math., Tome 103 (1991) no. 1, pp. 25-68 | Article | MR 1079839 | Zbl 0737.11030

[21] Rubin, Karl p-adic L-functions and rational points on elliptic curves with complex multiplication, Invent. Math., Tome 107 (1992) no. 2, pp. 323-350 | Article | MR 1144427 | Zbl 0770.11033

[22] Rubin, Karl Abelian varieties, p-adic heights and derivatives, Algebra and number theory (Essen, 1992), de Gruyter, Berlin (1994), pp. 247-266 | MR 1285370 | Zbl 0829.11034

[23] Rubin, Karl Elliptic curves with complex multiplication and the conjecture of Birch and Swinnerton-Dyer, Arithmetic theory of elliptic curves (Cetraro, 1997), Springer, Berlin (Lecture Notes in Math.) Tome 1716 (1999), pp. 167-234 | MR 1754688 | Zbl 0991.11028

[24] Rubin, Karl Euler systems, Princeton University Press, Princeton, NJ, Annals of Mathematics Studies, Tome 147 (2000) (Hermann Weyl Lectures. The Institute for Advanced Study) | MR 1749177 | Zbl 0977.11001

[25] De Shalit, Ehud Iwasawa theory of elliptic curves with complex multiplication, Academic Press Inc., Boston, MA, Perspectives in Mathematics, Tome 3 (1987) | MR 917944 | Zbl 0674.12004

[26] Weil, A. Automorphic Forms and Dirichlet Series, Dirichlet series and automorphic forms. Lezioni fermiane., Springer (Lecture Notes in Math.) Tome 189 (1971) | Zbl 0218.10046

[27] Yager, Rodney I. On two variable p-adic L-functions, Ann. of Math. (2), Tome 115 (1982) no. 2, pp. 411-449 | Article | MR 647813 | Zbl 0496.12010