Elliptic curves in honeycomb form

TL;DR

This paper explores elliptic curves in honeycomb form, providing explicit constructions, analytic characterizations, and detailed analysis of their tropical group law, enhancing understanding of their geometric and algebraic properties.

Contribution

It introduces explicit methods to compute honeycomb form representations from a given j-invariant and analyzes the tropical group law on these curves.

Findings

Explicit computation methods for honeycomb form from j-invariant

Analytic characterization of elliptic curves in honeycomb form

Detailed analysis of the tropical group law on honeycomb curves

Abstract

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.