Tropical Split Jacobians of genus 2 and optimal covers

TL;DR

This paper investigates tropical split Jacobians of genus 2 curves, focusing on their decomposition into elliptic curves via optimal coverings, and provides an algorithmic framework for understanding their non-uniqueness.

Contribution

It introduces the concept of tropical split Jacobians for genus 2 curves and proposes an algorithmic method to compute their canonical decompositions.

Findings

Split Jacobians are not unique, complicating their analysis.

Optimal coverings provide a canonical and algorithmic way to resolve indeterminacy.

The work connects tropical geometry with classical algebraic concepts through constructive methods.

Abstract

We explore connections between the category of tropical abelian varieties (tav), $\mathbb{T}\mathcal{A}$, and the the category of tropical curves, $\mathbb{T}\mathcal{C}$, first in a broader context and then specifically by studying the phenomenon of tropical split Jacobians. Jacobians of genus $2$ curves are two-dimensional tav and as such more complicated than their one-dimensional cousins. Whenever $\Gamma$, however, is a covering of an elliptic curve, it so happens that $Jac(\Gamma)$ splits into simpler objects, the direct sum of elliptic curves. This relation is pathological in essentially two ways, the splitting of $Jac(\Gamma)$ is not unique, and it is a priori not clear how to compute it. Similar to algebraic geometry, optimal coverings offer a remedy for both: They resolve indeterminacy as they provide us with a canonical choice. They resolve indeterminability as they provide us with an algorithmic approach. Our methods build on theory developed by Len, Mikhalkin, R\"ohrle, Ulirsch, Zakharov, Zharkov and many more. In accordance with this heritage, we want to put forth tropical geometry as a setting in which we can see abstraction at work. This means pairing "abstract machinery" with a constructive/algorithmic approach.