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We prove that any smooth projective geometrically connected non-isotrivialcurve of genus $g\ge 2$ over a one-dimensional function field of anycharacteristic has at most $16g^2+32g+124$ torsion points for any Abel--Jacobiembedding of the curve into its Jacobian. The proof uses Zhang's admissiblepairing on curves, the arithmetic Hodge index theorem over function fields, andthe metrized graph analogue of Elkies' lower bound for the Green function. Moregenerally, we prove an explicit Bogomolov-type result bounding the number ofgeometric points of small N\'eron-Tate height on the curve embedded into itsJacobian.

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