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arXiv · 2112.00168

The tropical Manin-Mumford conjecture

David Harry Richman
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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.

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David Harry Richman. 2021-11-30. The tropical Manin-Mumford conjecture. https://doi.org/10.1093/imrn%2Frnad098

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