Uniform boundedness of small points on abelian varieties over function fields

TL;DR

This paper establishes uniform bounds on torsion points and Néron-Tate heights for abelian varieties over function fields, confirming the Lang-Silverman conjecture in characteristic zero.

Contribution

It proves uniform boundedness of torsion points and a lower height bound for abelian varieties over function fields, extending key conjectures to this setting.

Findings

Uniform bound on torsion points depending on genus and gonality

Lower bound on Néron-Tate height related to Faltings height

Confirmation of the Lang-Silverman conjecture over function fields

Abstract

Let $k$ be a field of characteristic $0$ and let $K = k(B)$ be the function field of a geometrically irreducible projective curve $B$ over $k$. Let $A/K$ be a $g$-dimensional abelian variety with $\mathrm{Tr}_{K/k}(A) = 0$. We prove that any $K$-rational torsion point $x$ of $A$ has order uniformly bounded in terms of $g$ and the gonality of $B$. We also prove a uniform lower bound on the N\'{e}ron-Tate height $\widehat{h}_{A,L}(x)$ in terms of the stable Faltings height $h_{\mathrm{Fal}}(A)$ for any $K$-rational point $x$ whose forward orbit is Zariski dense, proving the Lang-Silverman conjecture over function fields of characteristic $0$.