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

We prove that any smooth projective geometrically connected non-isotrivial curve of genus <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>g</m:mi> <m:mo>≥</m:mo> <m:mn>2</m:mn> </m:mrow> </m:math> g≥ 2 over a one-dimensional function field of any characteristic has at most <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mrow> <m:mn>16</m:mn> <m:mo>⁢</m:mo> <m:msup> <m:mi>g</m:mi> <m:mn>2</m:mn> </m:msup> </m:mrow> <m:mo>+</m:mo> <m:mrow> <m:mn>32</m:mn> <m:mo>⁢</m:mo> <m:mi>g</m:mi> </m:mrow> <m:mo>+</m:mo> <m:mn>124</m:mn> </m:mrow> </m:math> 16g²+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.