Addition in Jacobians of tropical hyperelliptic curves

TL;DR

This paper establishes a connection between effective divisors and the Jacobian of tropical hyperelliptic curves, enabling explicit addition operations through tropical intersections, based on a tropical Riemann-Roch theorem.

Contribution

It introduces a surjective map from effective divisors to the Jacobian in tropical hyperelliptic curves and demonstrates how to perform addition via tropical intersections.

Findings

Surjection from effective divisors to Jacobian established

Addition in Jacobian realized through tropical curve intersections

Effective divisors form a group structure compatible with Jacobian

Abstract

We show that there exists a surjection from the set of effective divisors of degree $g$ on a tropical curve of genus $g$ to its Jacobian by using a tropical version of the Riemann-Roch theorem. We then show that the restriction of the surjection is reduced to the bijection on an appropriate subset of the set of effective divisors of degree $g$ on the curve. Thus the subset of effective divisors has the additive group structure induced from the Jacobian. We finally realize the addition in Jacobian of a tropical hyperelliptic curve of genus $g$ via the intersection with a tropical curve of degree $3g/2$ or $3(g-1)/2$.