# Elliptic curves in honeycomb form

@article{Chan2012EllipticCI, title={Elliptic curves in honeycomb form}, author={Melody Chan and Bernd Sturmfels}, journal={arXiv: Algebraic Geometry}, year={2012} }

A plane cubic curve, defined over a field with valuation, is in honeycomb form if its tropicalization exhibits the standard hexagonal cycle. We explicitly compute such representations from a given j-invariant with negative valuation, we give an analytic characterization of elliptic curves in honeycomb form, and we offer a detailed analysis of the tropical group law on such a curve.

## 16 Citations

Theta Characteristics of Tropical K 4 -Curves

- Mathematics
- 2017

A K4-curve is a smooth proper curve X of genus 3 over a field with valuation whose Berkovich skeleton Γ is a complete graph on four vertices. The curve X has 28 effective theta characteristics—the 28…

Computing Unit Groups of Curves

- Mathematics, Computer ScienceJ. Symb. Comput.
- 2021

This work presents practical algorithms for computing unit groups of smooth curves of low genus, rooted in divisor theory, based on interpolation in the case of rational curves and on methods from algebraic number theory in the cases of elliptic curves.

Tropicalization of Canonical Curves: the Planar Case.

- Mathematics
- 2019

We study a topological version of the tropical lifting problem for canonical curves. This leads us to a tropical analogue of the notion of graph curves that we refer to as tropical graph curves. We…

Nonarchimedean geometry, tropicalization, and metrics on curves

- Mathematics
- 2011

We develop a number of general techniques for comparing analytifications and tropicalizations of algebraic varieties. Our basic results include a projection formula for tropical multiplicities and a…

Constructing smooth and fully faithful tropicalizations for Mumford curves

- MathematicsSelecta Mathematica
- 2020

The tropicalization of an algebraic variety X is a combinatorial shadow of X, which is sensitive to a closed embedding of X into a toric variety. Given a good embedding, the tropicalization can…

Real rank geometry of ternary forms

- Mathematics
- 2016

We study real ternary forms whose real rank equals the generic complex rank, and we characterize the semialgebraic set of sums of powers representations with that rank. Complete results are obtained…

How to Repair Tropicalizations of Plane Curves Using Modifications

- Mathematics, Computer ScienceExp. Math.
- 2016

The purpose of this article is to advertise tropical modifications as a tool to locally repair bad embeddings of plane curves, allowing the re-embedded tropical curve to better reflect the geometry of the input curve.

Tropical Graph Curves

- Mathematics
- 2016

We study a tropical analogue of the notion of graph curves. Given a connected $3$-regular graph $G$, we define a notion of tropical graph curve associated to $G$ and show their existence when $G$ is…

Faithful tropicalizations of elliptic curves using minimal models and inflection points

- MathematicsArnold Mathematical Journal
- 2019

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…

Computations and Moduli Spaces for Non-archimedean Varieties

- Mathematics
- 2014

Tropical geometry and non-archimedean analytic geometry study algebraic varieties over a field K with a non-archimedean valuation. One of the major goals is to classify varietiess over K by intrinsic…

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