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

TL;DR

This paper establishes a uniform upper bound on the number of torsion points on certain algebraic curves over function fields, extending Manin-Mumford type results with explicit bounds using advanced arithmetic geometry tools.

Contribution

It provides the first explicit uniform bound for torsion points on non-isotrivial curves over function fields, combining techniques from Zhang's pairing, the arithmetic Hodge index theorem, and metrized graph theory.

Findings

Bound of 16g^2+32g+124 torsion points for curves of genus g≥2

Extension of Manin-Mumford theorem to function fields with explicit bounds

Application of advanced arithmetic geometry methods to torsion point problems

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 N\'eron-Tate height on the curve embedded into its Jacobian.