Uniformity in rational torsion and small points on abelian varieties

In this paper, we propose a method to study the Uniform Boundedness Conjecture and the Lang-Silverman Conjecture, which gives a uniform lower bound on the heights of non-torsion rational points, for abelian varieties $A$ defined over a global field $K$. Our method is inspired by Vojta’s proof of the Mordell Conjecture (Faltings’s Theorem). Oer function fields of characteristic $0$, a recent breakthrough of Looper-Yap (arXiv:2603.23396) proves both conjectures with inexplicit bounds. In our paper, we give a new proof of both conjectures with explicit bounds, which also depend polynomially on the field $K$ unless $A\mathrm{/}K$ admits a factor of good reduction everywhere. We also prove explicit bounds for elliptic curves over function fields of characteristic $p>0$. Over number fields, we prove both conjectures under a suitable high-dimensional Szpiro conjecture (weaker than Hindry’s version, Conjecture 3.4 of https://webusers.imj-prg.fr/~marc.hindry/MW-size.pdf) that we propose.