### 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) |
---|---|

Pages (from-to) | 1483-1506 |

Number of pages | 24 |

Journal | Journal of Number Theory |

Volume | 132 |

Issue number | 7 |

DOIs | |

State | Published - Jul 1 2012 |

Externally published | Yes |

### Fingerprint

### Keywords

- Elliptic curves
- Iwasawa theory
- Supersingular primes

### ASJC Scopus subject areas

- Algebra and Number Theory

### Cite this

**Iwasawa theory for elliptic curves at supersingular primes : A pair of main conjectures.** / Sprung, Florian.

Research output: Contribution to journal › Article

*Journal of Number Theory*, vol. 132, no. 7, pp. 1483-1506. https://doi.org/10.1016/j.jnt.2011.11.003

}

TY - JOUR

T1 - Iwasawa theory for elliptic curves at supersingular primes

T2 - A pair of main conjectures

AU - Sprung, Florian

PY - 2012/7/1

Y1 - 2012/7/1

N2 - 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.

AB - 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.

KW - Elliptic curves

KW - Iwasawa theory

KW - Supersingular primes

UR - http://www.scopus.com/inward/record.url?scp=84857683335&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84857683335&partnerID=8YFLogxK

U2 - 10.1016/j.jnt.2011.11.003

DO - 10.1016/j.jnt.2011.11.003

M3 - Article

VL - 132

SP - 1483

EP - 1506

JO - Journal of Number Theory

JF - Journal of Number Theory

SN - 0022-314X

IS - 7

ER -