Uniform boundedness of small points on abelian varieties 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 Néron-Tate height ${\hat{h}}_{A,L}(x)$ 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 $0$.