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# On elliptic curves with an isogeny of degree 7

## 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 Yes Faculties > Faculty of Mathematics, Physics und Computer Science > Department of Mathematics > Chair Mathematics IIFaculties > Faculty of Mathematics, Physics und Computer Science > Department of Mathematics > Chair Mathematics II > Chair Mathematics II - Univ.-Prof. Dr. Michael StollFacultiesFaculties > Faculty of Mathematics, Physics und Computer ScienceFaculties > Faculty of Mathematics, Physics und Computer Science > Department of Mathematics Yes 500 Science > 510 Mathematics 20 Nov 2014 10:58 20 Nov 2014 10:58 https://eref.uni-bayreuth.de/id/eprint/3750