Year: 2026
Type: preprint
Abstract: In this paper, we propose a method to study the {\it Uniform Boundedness Conjecture} and the {\it Lang-Silverman Conjecture} for abelian varieties $A$ defined over a global field $K$; the latter is a uniform lower bound on the heights of non-torsion rational points. Our method is inspired by Vojta's proof of the Mordell Conjecture (Faltings's Theorem). Over 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 $A/K$ 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.
Source: arXiv (Cornell University)
Authors: Gao, Ziyang, Gu, Kaiyuan
Institutions: Peking University, University of California, Los Angeles
Language: English
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7204156536