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 ≠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 [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 BPR11 is that we do not require any of the analytic theory on Berkovich spaces such as the Poincar\'e-Lelong formula or [Theorem 5.25]BPR11 to establish the numerical faithfulness of this tropicalization.