Tropical moments of tropical Jacobians

TL;DR

This paper provides an explicit formula for the tropical moment of tropical Jacobians, linking it to potential theory and establishing a universal linear relation with other invariants, advancing understanding in tropical geometry.

Contribution

It introduces a computable formula for the tropical moment of tropical Jacobians and reveals a universal linear relation with the tau invariant and total length.

Findings

Explicit formula for tropical moment in terms of potential theory

Existence of a universal linear relation among invariants

Connection to non-archimedean analogues of classical identities

Abstract

Each metric graph has canonically associated to it a polarized real torus called its tropical Jacobian. A fundamental real-valued invariant associated to each polarized real torus is its tropical moment. We give an explicit and efficiently computable formula for the tropical moment of a tropical Jacobian in terms of potential theory on the underlying metric graph. We show that there exists a universal linear relation between the tropical moment, the tau invariant, and the total length of a metric graph. We argue that this linear relation is a non-archimedean analogue of a recent remarkable identity established by Wilms for invariants of compact Riemann surfaces. We also relate our work to the computation of heights attached to principally polarized abelian varieties.