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

TL;DR

The article proves a uniform, explicit bound on the number of torsion points in the Abel–Jacobi embedding of a non-isotrivial genus $g\ge2$ curve over the function field of a curve, with bound $c(g)$ given explicitly (e.g., $16g^2+32g+124$ in general, plus refinements under reduction hypotheses). Central to the approach are Zhang’s admissible pairing, the arithmetic Hodge index theorem over function fields, and a Green-function bound on polarized metrized graphs that ties local reductions to global height data. The authors also establish a Bogomolov-type bound for the number of geometric points of small Néron–Tate height on the Jacobian, with explicit constants depending on $g$ and a small parameter $\\",epsilon$ controlling the height threshold. They deduce corollaries, including a corollary for curves not defined over the prime field, and provide a framework applicable to uniform Manin–Mumford problems over function fields with computable constants. Overall, the work advances effective, uniform control over near-torsion and near-zero-height points on curves over function fields, with potential implications for Diophantine geometry and arithmetic dynamics.

Abstract

We prove that any smooth projective geometrically connected non-isotrivial curve of genus $g\ge 2$ over a one-dimensional function field of any characteristic has at most $16g^2+32g+124$ torsion points for any Abel-Jacobi embedding of the curve into its Jacobian. The proof uses Zhang's admissible pairing on curves, the arithmetic Hodge index theorem over function fields, and the metrized graph analogue of Elkies' lower bound for the Green function. More generally, we prove an explicit Bogomolov-type result bounding the number of geometric points of small Néron-Tate height on the curve embedded into its Jacobian.

Theorems & Definitions (12)

• Theorem 1.1
• Corollary 1.2
• Theorem 1.3
• Proposition 2.1
• Lemma 2.2
• proof
• Lemma 2.3
• proof
• Lemma 5.1
• proof