The image of Galois representations attached to elliptic curves with an isogeny

• Autores: Ralph Greenberg
• Localización: American journal of mathematics, ISSN 0002-9327, Vol. 134, Nº 5, 2012, págs. 1167-1196
• Idioma: inglés
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• Resumen

  • 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 ${\bf Q}$.