Extending Kato’s result to elliptic curves with $p$-isogenies

Let $E$ be an elliptic curve without complex multiplication defined over $\QQ$ and let $p$ be an odd prime number at which $E$ has good and ordinary reduction. Kato has proved in~\cite{kato} half of the main conjecture for $E$ under the condition that the representation $\rho_p\colon G_{\QQ}\rTo \Aut(\Tp E)$ of the absolute Galois group of $\QQ$ attached to the Tate module $\Tp E$ is surjective. We prove here that the result still holds if $E$ admits an isogeny of degree $p$. As a by-product, we show that the $p$-adic $L$-functions attached to an elliptic curve with good ordinary reduction at $p$ is always an integral series.

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Published 1 January 2006