Faithful tropicalizations of elliptic curves using minimal models and inflection points

TL;DR

This paper provides an elementary proof that elliptic curves over non-archimedean fields with certain properties admit tropicalizations containing a cycle of length related to the valuation of their j-invariant, using minimal models and inflection points.

Contribution

It introduces a new elementary approach to constructing faithful tropicalizations of elliptic curves without relying on Berkovich space theory.

Findings

Constructed explicit families of elliptic curves with multiplicative reduction and inflection points.

Proved the existence of tropicalizations with cycles of length tied to the j-invariant valuation.

Provided a simplified proof of known theorems in non-archimedean geometry.

Abstract

We give an elementary proof of the fact that any elliptic curve $E$ over an algebraically closed non-archimedean field $K$ with residue characteristic $\neq{2,3}$ and with $v(j(E))<0$ admits a tropicalization that contains a cycle of length $-v(j(E))$. We first define an adapted form of minimal models over non-discrete valuation rings and we recover several well-known theorems from the discrete case. Using these, we create an explicit family of marked elliptic curves $(E,P)$, where $E$ has multiplicative reduction and $P$ is an inflection point that reduces to the singular point on the reduction of $E$. We then follow the strategy as in \cite[Theorem 6.2]{BPR11} and construct an embedding such that its tropicalization contains a cycle of length $-v(j(E))$. We call this a numerically faithful tropicalization. A key difference between this approach and the approach in \cite{BPR11} is that we do not require any of the analytic theory on Berkovich spaces such as the {\it{Poincar\'{e}-Lelong formula}} or \cite[Theorem 5.25]{BPR11} to establish the numerical faithfulness of this tropicalization.