On a two-parameter family of tropical Edwards curves

TL;DR

This paper introduces a two-parameter family of tropical Edwards curves, providing explicit formulas for their tropicalizations and demonstrating how to recover entire tropical curves using ultradiscretization and algebraic methods.

Contribution

It presents a new family of tropical Edwards curves with explicit coordinate formulas and connects their tropicalization to Bruhat-Tits trees using ultradiscretization and algebraic techniques.

Findings

Explicit formulas for tropical Edwards curves are derived.

The tropical curves are obtained as quotients of Bruhat-Tits trees.

The approach links ultradiscretization with algebraic methods for tropicalization.

Abstract

In this paper, a certain two-parameter family of plane-embeddings of Edwards elliptic curve $E_a: x^2+y^2=a^2(1+x^2y^2)$ is introduced to provide explicitly computed tropical curves corresponding to degeneration in $a\to 1$. Applying the theta uniformization of $E_a$ with the method of ultradiscretization by Kajiwara-Kaneko-Nobe-Tsuda, we give a formula for the coordinate functions that traces the cycle part of the tropical elliptic curve. We also illustrate how one can recover the whole part of the tropical curve as a quotient of the Bruhat-Tits tree after Speyer's algebraic approach in smooth cases.