Tropical Split Jacobians of genus 2 and optimal covers

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.