Faithful tropicalizations of elliptic curves using minimal models and inflection points

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 $\ne 2,3$ 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{Poincaré-Lelong formula}} or \cite[Theorem 5.25]{BPR11} to establish the numerical faithfulness of this tropicalization.

24 pages, 12 figures. This is a pre-print of an article published in "Arnold Mathematical Journal". The final authenticated version is available online at: https://doi.org/10.1007/s40598-019-00121-y