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Adebisi Agboola; Benjamin Howard
Anticyclotomic Iwasawa theory of CM elliptic curves
(Théorie anticylotomique d’une courbe élliptique à multiplication complexe)
Annales de l'institut Fourier, 56 no. 4 (2006), p. 1001-1048
Article: subscription required | Reviews MR 2266884 | Zbl 1168.11023
Class. Math.: 11G05, 11R23, 11G16
Keywords: Ellipic curves, Iwasawa theory, main conjecture, anticyclotomic, $p$-adic $L$-function
[1] T. Arnold, “Anticyclotomic main conjectures for CM modular forms”, Preprint, 2005
[2] D. Bertrand, Propriétés arithmétiques de fonctions thêta à plusieurs variables, Number theory, Noordwijkerhout 1983, Springer, 1984, p. 17–22 MR 756080 | Zbl 0546.14029
[3] J. Coates, Infinite descent on elliptic curves with complex multiplication, Arithmetic and Geometry, Vol. I, Birkhäuser Boston, 1983, p. 107–137 MR 717591 | Zbl 0541.14026
[4] Ralph Greenberg, “On the structure of certain Galois groups”, Invent. Math. 47 (1978), p. 85-99
Article | MR 504453 | Zbl 0403.12004
[5] Ralph Greenberg, “On the Birch and Swinnerton-Dyer conjecture”, Invent. Math. 72 (1983), p. 241-265
Article | MR 700770 | Zbl 0546.14015
[6] Benedict H. Gross & Don B. Zagier, “Heegner points and derivatives of $L$-series”, Invent. Math. 84 (1986), p. 225-320
Article | MR 833192 | Zbl 0608.14019
[7] Benjamin Howard, “The Iwasawa theoretic Gross-Zagier theorem”, Compos. Math. 141 (2005), p. 811-846 MR 2148200 | Zbl 02211027
[8] Serge Lang, Algebraic number theory, Graduate Texts in Mathematics 110, Springer-Verlag, 1994 MR 1282723 | Zbl 0811.11001
[9] J. Martinet, Character theory and Artin $L$-functions, Algebraic number fields: $L$-functions and Galois properties (Proc. Sympos., Univ. Durham, Durham, 1975), Academic Press, 1977, p. 1–87 MR 447187 | Zbl 0359.12015
[10] B. Mazur, Modular curves and arithmetic, in Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Warsaw, 1983), PWN, 1984, p. 185-211 MR 804682 | Zbl 0597.14023
[11] B. Mazur & J. Tate, Canonical height pairings via biextensions, Arithmetic and geometry, Vol. I, Birkhäuser Boston, 1983, p. 195–237 MR 717595 | Zbl 0574.14036
[12] Barry Mazur, “Rational points of abelian varieties with values in towers of number fields”, Invent. Math. 18 (1972), p. 183-266
Article | MR 444670 | Zbl 0245.14015
[13] Barry Mazur & Karl Rubin, Elliptic curves and class field theory, in Proceedings of the International Congress of Mathematicians, Vol. II (Beijing, 2002), Higher Ed. Press, 2002, p. 185-195 MR 1957032 | Zbl 1036.11023
[14] Barry Mazur & Karl Rubin, “Studying the growth of Mordell-Weil”, Doc. Math. (2003), p. 585-607 (electronic), Kazuya Kato’s fiftieth birthday MR 2046609 | Zbl 02028845
[15] Barry Mazur & Karl Rubin, “Kolyvagin systems”, Mem. Amer. Math. Soc. 168 (2004) MR 2031496 | Zbl 1055.11041
[16] Bernadette Perrin-Riou, “Arithmétique des courbes elliptiques et théorie d’Iwasawa”, Mém. Soc. Math. France (N.S.) (1984)
Numdam | MR 799673 | Zbl 0599.14020
[17] Bernadette Perrin-Riou, “Fonctions $L$ $p$-adiques, théorie d’Iwasawa et points de Heegner”, Bull. Soc. Math. France 115 (1987), p. 399-456
Numdam | MR 928018 | Zbl 0664.12010
[18] Bernadette Perrin-Riou, “Théorie d’Iwasawa et hauteurs $p$-adiques”, Invent. Math. 109 (1992), p. 137-185
Article | MR 1168369 | Zbl 0781.14013
[19] David E. Rohrlich, “On $L$-functions of elliptic curves and anticyclotomic towers”, Invent. Math. 75 (1984), p. 383-408
Article | MR 735332 | Zbl 0565.14008
[20] Karl Rubin, “The “main conjectures” of Iwasawa theory for imaginary quadratic fields”, Invent. Math. 103 (1991), p. 25-68
Article | MR 1079839 | Zbl 0737.11030
[21] Karl Rubin, “$p$-adic $L$-functions and rational points on elliptic curves with complex multiplication”, Invent. Math. 107 (1992), p. 323-350
Article | MR 1144427 | Zbl 0770.11033
[22] Karl Rubin, Abelian varieties, $p$-adic heights and derivatives, Algebra and number theory (Essen, 1992), de Gruyter, 1994, p. 247–266 MR 1285370 | Zbl 0829.11034
[23] Karl Rubin, Elliptic curves with complex multiplication and the conjecture of Birch and Swinnerton-Dyer, Arithmetic theory of elliptic curves (Cetraro, 1997), Springer, 1999, p. 167–234 MR 1754688 | Zbl 0991.11028
[24] Karl Rubin, Euler systems, Annals of Mathematics Studies 147, Princeton University Press, 2000, Hermann Weyl Lectures. The Institute for Advanced Study MR 1749177 | Zbl 0977.11001
[25] Ehud de Shalit, Iwasawa theory of elliptic curves with complex multiplication, Perspectives in Mathematics 3, Academic Press Inc., 1987 MR 917944 | Zbl 0674.12004
[26] A. Weil, Automorphic Forms and Dirichlet Series, Dirichlet series and automorphic forms. Lezioni fermiane., Springer, 1971 Zbl 0218.10046
[27] Rodney I. Yager, “On two variable $p$-adic $L$-functions”, Ann. of Math. (2) 115 (1982), p. 411-449 MR 647813 | Zbl 0496.12010
Adebisi Agboola; Benjamin Howard
Anticyclotomic Iwasawa theory of CM elliptic curves
(Théorie anticylotomique d’une courbe élliptique à multiplication complexe)
Annales de l'institut Fourier, 56 no. 4 (2006), p. 1001-1048
Article: subscription required | Reviews MR 2266884 | Zbl 1168.11023
Class. Math.: 11G05, 11R23, 11G16
Keywords: Ellipic curves, Iwasawa theory, main conjecture, anticyclotomic, $p$-adic $L$-function
Résumé - Abstract
We study the Iwasawa theory of a CM elliptic curve $E$ in the anticyclotomic $\mathbf{Z}_p$-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.
Bibliography
[2] D. Bertrand, Propriétés arithmétiques de fonctions thêta à plusieurs variables, Number theory, Noordwijkerhout 1983, Springer, 1984, p. 17–22 MR 756080 | Zbl 0546.14029
[3] J. Coates, Infinite descent on elliptic curves with complex multiplication, Arithmetic and Geometry, Vol. I, Birkhäuser Boston, 1983, p. 107–137 MR 717591 | Zbl 0541.14026
[4] Ralph Greenberg, “On the structure of certain Galois groups”, Invent. Math. 47 (1978), p. 85-99
Article | MR 504453 | Zbl 0403.12004
[5] Ralph Greenberg, “On the Birch and Swinnerton-Dyer conjecture”, Invent. Math. 72 (1983), p. 241-265
Article | MR 700770 | Zbl 0546.14015
[6] Benedict H. Gross & Don B. Zagier, “Heegner points and derivatives of $L$-series”, Invent. Math. 84 (1986), p. 225-320
Article | MR 833192 | Zbl 0608.14019
[7] Benjamin Howard, “The Iwasawa theoretic Gross-Zagier theorem”, Compos. Math. 141 (2005), p. 811-846 MR 2148200 | Zbl 02211027
[8] Serge Lang, Algebraic number theory, Graduate Texts in Mathematics 110, Springer-Verlag, 1994 MR 1282723 | Zbl 0811.11001
[9] J. Martinet, Character theory and Artin $L$-functions, Algebraic number fields: $L$-functions and Galois properties (Proc. Sympos., Univ. Durham, Durham, 1975), Academic Press, 1977, p. 1–87 MR 447187 | Zbl 0359.12015
[10] B. Mazur, Modular curves and arithmetic, in Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Warsaw, 1983), PWN, 1984, p. 185-211 MR 804682 | Zbl 0597.14023
[11] B. Mazur & J. Tate, Canonical height pairings via biextensions, Arithmetic and geometry, Vol. I, Birkhäuser Boston, 1983, p. 195–237 MR 717595 | Zbl 0574.14036
[12] Barry Mazur, “Rational points of abelian varieties with values in towers of number fields”, Invent. Math. 18 (1972), p. 183-266
Article | MR 444670 | Zbl 0245.14015
[13] Barry Mazur & Karl Rubin, Elliptic curves and class field theory, in Proceedings of the International Congress of Mathematicians, Vol. II (Beijing, 2002), Higher Ed. Press, 2002, p. 185-195 MR 1957032 | Zbl 1036.11023
[14] Barry Mazur & Karl Rubin, “Studying the growth of Mordell-Weil”, Doc. Math. (2003), p. 585-607 (electronic), Kazuya Kato’s fiftieth birthday MR 2046609 | Zbl 02028845
[15] Barry Mazur & Karl Rubin, “Kolyvagin systems”, Mem. Amer. Math. Soc. 168 (2004) MR 2031496 | Zbl 1055.11041
[16] Bernadette Perrin-Riou, “Arithmétique des courbes elliptiques et théorie d’Iwasawa”, Mém. Soc. Math. France (N.S.) (1984)
Numdam | MR 799673 | Zbl 0599.14020
[17] Bernadette Perrin-Riou, “Fonctions $L$ $p$-adiques, théorie d’Iwasawa et points de Heegner”, Bull. Soc. Math. France 115 (1987), p. 399-456
Numdam | MR 928018 | Zbl 0664.12010
[18] Bernadette Perrin-Riou, “Théorie d’Iwasawa et hauteurs $p$-adiques”, Invent. Math. 109 (1992), p. 137-185
Article | MR 1168369 | Zbl 0781.14013
[19] David E. Rohrlich, “On $L$-functions of elliptic curves and anticyclotomic towers”, Invent. Math. 75 (1984), p. 383-408
Article | MR 735332 | Zbl 0565.14008
[20] Karl Rubin, “The “main conjectures” of Iwasawa theory for imaginary quadratic fields”, Invent. Math. 103 (1991), p. 25-68
Article | MR 1079839 | Zbl 0737.11030
[21] Karl Rubin, “$p$-adic $L$-functions and rational points on elliptic curves with complex multiplication”, Invent. Math. 107 (1992), p. 323-350
Article | MR 1144427 | Zbl 0770.11033
[22] Karl Rubin, Abelian varieties, $p$-adic heights and derivatives, Algebra and number theory (Essen, 1992), de Gruyter, 1994, p. 247–266 MR 1285370 | Zbl 0829.11034
[23] Karl Rubin, Elliptic curves with complex multiplication and the conjecture of Birch and Swinnerton-Dyer, Arithmetic theory of elliptic curves (Cetraro, 1997), Springer, 1999, p. 167–234 MR 1754688 | Zbl 0991.11028
[24] Karl Rubin, Euler systems, Annals of Mathematics Studies 147, Princeton University Press, 2000, Hermann Weyl Lectures. The Institute for Advanced Study MR 1749177 | Zbl 0977.11001
[25] Ehud de Shalit, Iwasawa theory of elliptic curves with complex multiplication, Perspectives in Mathematics 3, Academic Press Inc., 1987 MR 917944 | Zbl 0674.12004
[26] A. Weil, Automorphic Forms and Dirichlet Series, Dirichlet series and automorphic forms. Lezioni fermiane., Springer, 1971 Zbl 0218.10046
[27] Rodney I. Yager, “On two variable $p$-adic $L$-functions”, Ann. of Math. (2) 115 (1982), p. 411-449 MR 647813 | Zbl 0496.12010
© Annales de L'Institut Fourier - ISSN (électronique) : 1777-5310