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• W2079720516 abstract "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." @default.
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• W2079720516 date "2014-01-01" @default.
• W2079720516 modified "2023-10-06" @default.
• W2079720516 title "On elliptic curves with an isogeny of degree 7" @default.
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• W2079720516 doi "https://doi.org/10.1353/ajm.2014.0005" @default.
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