## Abstract

Text: We extend Kobayashi's formulation of Iwasawa theory for elliptic curves at supersingular primes to include the case a_{p}≠0, where a_{p} is the trace of Frobenius. To do this, we algebraically construct p-adic L-functions L_{p}^{#} and L_{p}^{≠} with the good growth properties of the classical Pollack p-adic L-functions that in fact match them exactly when a_{p}=0 and p is odd. We then generalize Kobayashi's methods to define two Selmer groups Sel^{#} and Sel^{≠} and formulate a main conjecture, stating that each characteristic ideal of the duals of these Selmer groups is generated by our p-adic L-functions L_{p}^{#} and L_{p}^{≠}. We then use results by Kato to prove a divisibility statement. Video: For a video summary of this paper, please click here or visit http://www.youtube.com/watch?v=Y7gPQsBZo6s.

Original language | English (US) |
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Pages (from-to) | 1483-1506 |

Number of pages | 24 |

Journal | Journal of Number Theory |

Volume | 132 |

Issue number | 7 |

DOIs | |

State | Published - Jul 2012 |

Externally published | Yes |

## Keywords

- Elliptic curves
- Iwasawa theory
- Supersingular primes

## ASJC Scopus subject areas

- Algebra and Number Theory