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 $\varphi :E\to E^{\prime }$ comes equipped with an invariant ${\alpha }_{\varphi /K}$ that measures the valuation of the leading term of the formal group homomorphism $\Phi :\hat{E}\to {\hat{E}}^{\prime }$. We prove that if $K/{\mathbb{Q}}_p$ is unramified and $E$ has additive, potentially supersingular reduction, then ${\alpha }_{\varphi /K}$ is determined by the number of distinct geometric components on the special fibers of the minimal proper regular models of $E$ and $E^{\prime }$.

6 pages