A uniform quantitative Manin-Mumford theorem for curves over function fields

Looper, Nicole; Silverman, Joseph H.; Wilms, Robert

doi:10.48550/arxiv.2101.11593

Cited by 1 publication

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“…Now consider applying Lemma 6.12 to the graph G 0 , the divisor (14), and the edge e * ∈ E(G 0 ). The lemma concludes that as a function of the edge-lengths of Γ 0 , the slope of f 0 (equivalently f ) on e * is a nonconstant ratio of polynomials.…”

Section: Let J Ymentioning

confidence: 99%

“…Katz, Rabinoff, and Zureick-Brown [11] used tropical methods to prove a uniform bound on the number of torsion points on an algebraic curve of fixed genus, which satisfy an additional technical constraint on the reduction type. Kühne [13] (in characteristic zero) and Looper, Silverman, and Wilms [14] (in positive characteristic) recently proved uniform bounds on the number of torsion points on an algebraic curve.…”

Section: Introductionmentioning

confidence: 99%

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The tropical Manin-Mumford conjecture

Richman¹

2021

Preprint

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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.

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Section: Let J Ymentioning

confidence: 99%