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Tropical split Jacobians of curves of genus 2 II

Lou-Jean Leila Cobigo

TL;DR

The paper addresses the problem of understanding tropical split Jacobians for genus-2 curves by deconstructing them into building blocks—two tropical elliptic curves and a finite subgroup—and reconstructing the genus-2 Jacobian from splitting data. It develops a constructive framework that yields either a genus-2 curve (or a family) with a pair of optimal degree-$d$ covers, producing a splitting isogeny with kernel isomorphic to the $d$-torsion of one elliptic factor, and distinguishes the two principal combinatorial types (theta and dumbbell) via an explicit algorithm. The global perspective situates these results in the moduli spaces $A^{tr}_2$ and $M^{tr}_2$, characterizing the locus of split Jacobians, decomposing the $d$-split locus into $\varphi(d)$ 2D fans, and connecting to tropical Torelli theory and Schottky-type problems. The work blends tropical analogue of Mumford’s criterion, adjoint morphisms, and explicit reconstruction techniques to provide both abstract (moduli) and constructive (curve-building) insights into tropical split Jacobians and their applications.

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.

Tropical split Jacobians of curves of genus 2 II

TL;DR

The paper addresses the problem of understanding tropical split Jacobians for genus-2 curves by deconstructing them into building blocks—two tropical elliptic curves and a finite subgroup—and reconstructing the genus-2 Jacobian from splitting data. It develops a constructive framework that yields either a genus-2 curve (or a family) with a pair of optimal degree- covers, producing a splitting isogeny with kernel isomorphic to the -torsion of one elliptic factor, and distinguishes the two principal combinatorial types (theta and dumbbell) via an explicit algorithm. The global perspective situates these results in the moduli spaces and , characterizing the locus of split Jacobians, decomposing the -split locus into 2D fans, and connecting to tropical Torelli theory and Schottky-type problems. The work blends tropical analogue of Mumford’s criterion, adjoint morphisms, and explicit reconstruction techniques to provide both abstract (moduli) and constructive (curve-building) insights into tropical split Jacobians and their applications.

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), , and the category of tropical curves, . Tropical split Jacobians take on different forms depending on whether we look at them in or : 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.

Paper Structure

This paper contains 9 sections, 3 theorems, 6 equations.

Table of Contents

  1. Introduction
  2. Local: Task \ref{['task_local']}
  3. Global: Task \ref{['task_global']}
  4. Context
  5. Results
  6. Plan of Paper
  7. Acknowledgements
  8. Category of tropical abelian varieties
  9. Preliminaries

Key Result

Theorem 1.1

(arXiv:2410.13459, Theorem 61 and Theorem 66) Let $\Gamma$ be a tropical curve of genus $2$. For an optimal pair $(\mathbb{T}E,\varphi)$, that is a pair consisting of a tropical elliptic curve $\mathbb{T}E$ and a tropical optimal cover $\varphi: \Gamma \rightarrow \mathbb{T}E$, there exists another whose kernel satisfies $\mathop{\mathrm{Jac}}\nolimits_d(\mathbb{T}E')\cong \ker(\phi) \cong \matho

Theorems & Definitions (4)

  • Theorem 1.1
  • Theorem 1.6
  • Theorem 1.7
  • Definition 2.1