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 ≥ 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 strict upper bound for d such that the higher-degree Manin-Mumford property holds.