Anticyclotomic Iwasawa theory of CM elliptic curves

Adebisi Agboola[1]; Benjamin Howard[2]

  • [1] University of California Department of Mathematics Santa Barbara, CA 93106
  • [2] Harvard University Department of Mathematics Cambridge, MA 02138

Annales de l’institut Fourier (2006)

  • Volume: 56, Issue: 4, page 1001-1048
  • ISSN: 0373-0956

Abstract

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We study the Iwasawa theory of a CM elliptic curve in the anticyclotomic -extension of the CM field, where is a prime of good, ordinary reduction for . When the complex -function of vanishes to even order, Rubin’s proof of the two variable main conjecture of Iwasawa theory implies that the Pontryagin dual of the -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.

How to cite

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Agboola, Adebisi, and Howard, Benjamin. "Anticyclotomic Iwasawa theory of CM elliptic curves." Annales de l’institut Fourier 56.4 (2006): 1001-1048. <http://eudml.org/doc/10166>.

@article{Agboola2006,
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.},
affiliation = {University of California Department of Mathematics Santa Barbara, CA 93106; Harvard University Department of Mathematics Cambridge, MA 02138; Stanford University Department of Mathematics Stanford, CA 94305},
author = {Agboola, Adebisi, Howard, Benjamin},
journal = {Annales de l’institut Fourier},
keywords = {Ellipic curves; Iwasawa theory; main conjecture; anticyclotomic; $p$-adic $L$-function; ellipic curves; anticyclotomic, -adic -function},
language = {eng},
number = {4},
pages = {1001-1048},
publisher = {Association des Annales de l’institut Fourier},
title = {Anticyclotomic Iwasawa theory of CM elliptic curves},
url = {http://eudml.org/doc/10166},
volume = {56},
year = {2006},
}

TY - JOUR
AU - Agboola, Adebisi
AU - Howard, Benjamin
TI - Anticyclotomic Iwasawa theory of CM elliptic curves
JO - Annales de l’institut Fourier
PY - 2006
PB - Association des Annales de l’institut Fourier
VL - 56
IS - 4
SP - 1001
EP - 1048
AB - 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.
LA - eng
KW - Ellipic curves; Iwasawa theory; main conjecture; anticyclotomic; $p$-adic $L$-function; ellipic curves; anticyclotomic, -adic -function
UR - http://eudml.org/doc/10166
ER -

References

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