The group law on a tropical elliptic curve

TL;DR

This paper defines a group law on smooth plane tropical cubic curves, demonstrating that the resulting group structure is isomorphic to the circle group $S^1$, thus extending classical elliptic curve theory to tropical geometry.

Contribution

It introduces a novel group law on tropical elliptic curves and proves its isomorphism to the circle group, bridging classical and tropical algebraic geometry.

Findings

The group law on tropical elliptic curves is well-defined.

The tropical elliptic curve's group is isomorphic to $S^1$.

This work extends classical elliptic curve properties to tropical geometry.

Abstract

In analogy with the classical group law on a plane cubic curve, we define a group law on a smooth plane tropical cubic curve. We show that the resulting group is isomorphic to $S^1$.