The Tropical Manin–Mumford Conjecture

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 that 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 that is a sharp upper bound for |$d$| such that the higher-degree Manin–Mumford property holds.