The tropical Manin-Mumford conjecture

TL;DR

This paper investigates the intersection of metric graphs with torsion points in their Jacobians, establishing finiteness and bounds under certain irrationality conditions and exploring higher-degree embeddings.

Contribution

It proves finiteness of torsion points for certain metric graphs and introduces the concept of independent girth to bound higher-degree intersections.

Findings

Number of torsion points is finite for certain metric graphs of genus ≥ 2.

Bound on torsion points is given by 3g-3 for specific graphs.

Introduces the independent girth as a key parameter for higher-degree bounds.

Abstract

In analogy with the Manin-Mumford conjecture for algebraic curves, one may ask how a metric graph under the Abel-Jacobi embedding intersects torsion points of its Jacobian. We show that the number of torsion points is finite for metric graphs of genus $g\geq 2$ which are biconnected and have edge lengths which are "sufficiently irrational" in a precise sense. Under these assumptions, the number of torsion points is bounded by $3g-3$. Next we study bounds on the number of torsion points in the image of higher-degree Abel-Jacobi embeddings, which send $d$-tuples of points to the Jacobian. This motivates the definition of the "independent girth" of a graph, a number which is a sharp upper bound for $d$ such that the higher-degree Manin-Mumford property holds.