Abstract

We consider an elliptic curve $E$ defined over ${\bf Q}$ which has an isogeny of prime degree $p$ defined over ${\bf Q}$. Assuming that $E$ does not have complex multiplication and that $p > 7$, we show that the image of the Galois representation defined by the action of $G_{{\bf Q}}$ on the $p$-adic Tate module is as large as possible, given the existence of such an isogeny. Under a certain additional assumption, we also prove that result for $p = 7$. For $p = 5$, we show that the image is as large as allowed by the isogenies of $p$-power degree defined over ${\bf Q}$.

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ISSN
1080-6377
Print ISSN
0002-9327
Pages
pp. 1167-1196
Launched on MUSE
2012-09-22
Open Access
No
Archive Status