The Tropical Manin–Mumford Conjecture

Abstract

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.