On a local invariant of elliptic curves with a p-isogeny

Abstract

An elliptic curve $E$ defined over a $p$-adic field $K$ with a $p$-isogeny $\phi:E\rightarrow E^\prime$ comes equipped with an invariant $\alpha_{\phi/K}$ that measures the valuation of the leading term of the formal group homomorphism $\Phi:\hat E \rightarrow \hat E^\prime$. We prove that if $K/\mathbb{Q}_p$ is unramified and $E$ has additive, potentially supersingular reduction, then $\alpha_{\phi/K}$ is determined by the number of distinct geometric components on the special fibers of the minimal proper regular models of $E$ and $E^\prime$.