## Title data

Greenberg, Ralph ; Rubin, Karl ; Silverberg, Alice ; Stoll, Michael:

**On elliptic curves with an isogeny of degree 7.**

*In:* American Journal of Mathematics.
Vol. 136
(2014)
Issue 1
.
- pp. 77-109.

ISSN 0002-9327

DOI: https://doi.org/10.1353/ajm.2014.0005

## Abstract in another language

We show that if $E$ is an elliptic curve over ${\bf Q}$ with a ${\bf Q}$-rational isogeny of degree $7$, then the image of the $7$-adic Galois representation attached to $E$ is as large as allowed by the isogeny, except for the curves with complex multiplication by ${\bf Q}(\sqrt{-7})$. The analogous result with $7$ replaced by a prime $p > 7$ was proved by the first author. The present case $p = 7$ has additional interesting complications. We show that any exceptions correspond to the rational points on a certain curve of genus $12$. We then use the method of Chabauty to show that the exceptions are exactly the curves with complex multiplication. As a by-product of one of the key steps in our proof, we determine exactly when there exist elliptic curves over an arbitrary field $k$ of characteristic not $7$ with a $k$-rational isogeny of degree $7$ and a specified Galois action on the kernel of the isogeny, and we give a parametric description of such curves.

## Further data

Item Type: | Article in a journal |
---|---|

Refereed: | Yes |

Institutions of the University: | Faculties > Faculty of Mathematics, Physics und Computer Science > Department of Mathematics > Chair Mathematics II (Computer Algebra) Faculties > Faculty of Mathematics, Physics und Computer Science > Department of Mathematics > Chair Mathematics II (Computer Algebra) > Chair Mathematics II (Computer Algebra) - Univ.-Prof. Dr. Michael Stoll Faculties Faculties > Faculty of Mathematics, Physics und Computer Science Faculties > Faculty of Mathematics, Physics und Computer Science > Department of Mathematics |

Result of work at the UBT: | Yes |

DDC Subjects: | 500 Science > 510 Mathematics |

Date Deposited: | 20 Nov 2014 10:58 |

Last Modified: | 20 Nov 2014 10:58 |

URI: | https://eref.uni-bayreuth.de/id/eprint/3750 |