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Publisher's Note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations
- Publisher
- Springer Journals
- Copyright
- Copyright © Institute for Mathematical Sciences (IMS), Stony Brook University, NY 2019
- ISSN
- 2199-6792
- eISSN
- 2199-6806
- DOI
- 10.1007/s40598-019-00121-y
- Publisher site
- See Article on Publisher Site
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\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\ne {2,3}$$\end{document} and with v(j(E))<0\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$v(j(E))<0$$\end{document} admits a tropicalization that contains a cycle of length -v(j(E))\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$-v(j(E))$$\end{document}. 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 Baker et al. (Algebraic Geom 3(1):63–105, 2016) and construct an embedding such that its tropicalization contains a cycle of length -v(j(E))\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$-v(j(E))$$\end{document}. We call this a numerically faithful tropicalization. A key difference between this approach and the approach in Baker et al. (2016) is that we do not require any of the analytic theory on Berkovich spaces such as the Poincaré–Lelong formula or (Baker et al. 2016) to establish the numerical faithfulness of this tropicalization.
Journal
Arnold Mathematical Journal – Springer Journals
Published: Dec 27, 2019