Joseph Silverman | Searcharxiv

arXiv2021

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

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.

math.NT↗

arXiv2004

A Lower Bound for the Canonical Height on Abelian Varieties over Abelian Extensions

Let A be an abelian variety defined over a number field K, and consider the canonical height function attached to a symmetric ample line bundle L on A. We prove that there is a positive lower bound C (depending on A, K, and L) for the canonical height of non-torsion points on A defined over the maximal abelian extension K^ab of K.

math.NT↗