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

## TL;DR

This paper introduces an invariant for elliptic curves with a p-isogeny over p-adic fields and relates it to the geometric structure of the curves' models, especially in the case of additive, potentially supersingular reduction.

## Contribution

It establishes a link between the invariant bla_{/K} and the number of geometric components in the special fibers of minimal models for certain elliptic curves.

## Key findings

- bla_{/K} is determined by the number of geometric components.
- The result applies specifically to unramified extensions of _p and curves with additive, potentially supersingular reduction.
- Provides a geometric interpretation of the invariant bla_{/K}.

## 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 $E^\prime$.