Tropical split Jacobians of curves of genus 2 II

TL;DR

This paper investigates the structure of tropical split Jacobians of genus 2 curves, focusing on their fundamental components and how they can be reconstructed from elliptic curves and finite subgroups.

Contribution

It provides a detailed analysis of the building blocks of tropical split Jacobians, offering new insights into their composition and formation.

Findings

Characterization of elliptic curve pairs and finite subgroups forming split Jacobians

Analysis of how to reassemble Jacobians from basic components

Clarification of the structure of tropical split Jacobians in genus 2

Abstract

This paper is the second in a series of two papers which study the phenomenon of tropical split Jacobians. The first paper is a contemplative study, embedded in the broader context of exploring connections between the category of tropical abelian varieties (tav), $\mathbb{T}\mathcal{A}$, and the category of tropical curves, $\mathbb{T}\mathcal{C}$. Tropical split Jacobians take on different forms depending on whether we look at them in $\mathbb{T}\mathcal{A}$ or $\mathbb{T}\mathcal{C}$: They appear either as 2 dimensional tavs that decompose into a product of two elliptic curves, or as a pair of optimal coverings. [11] examines both and then focuses on how optimal covers give rise to split Jacobians. This paper takes a different approach. Instead of looking at the phenomenon as a whole, we analyze its building blocks, a pair of elliptic curves together with a finite subgroup of their product, and how to reassemble them into a Jacobian.