-
On elliptic curves with an isogeny of degree 7
- American Journal of Mathematics
- Johns Hopkins University Press
- Volume 136, Number 1, February 2014
- pp. 77-109
- 10.1353/ajm.2014.0005
- Article
- Additional Information
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.