Uniformity in rational torsion and small points on abelian varieties
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.
Related
- Sparsity of rational points on torsion level covers of Hilbert modular varieties
- Rational Points on Affine Varieties over Finite Fields and Polygonal Number Sequences
- On potential density of rational points on hyperkahler varieties
- Points and their multiples on curves in powers of simple abelian varieties
- A Complex Structure on A Rational Homology Six-Sphere with Two-Torsion
BibTeX
@misc{gao2026uniformity,
title = {{Uniformity in rational torsion and small points on abelian varieties}},
author = {Ziyang Gao and Kaiyuan Gu},
howpublished = {arXiv},
year = {2026},
month = aug,
doi = {10.48550/arxiv.2608.22285},
eprint = {2608.22285},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.22285},
}
RIS
TY - UNPB
TI - Uniformity in rational torsion and small points on abelian varieties
AU - Ziyang Gao
AU - Kaiyuan Gu
PY - 2026
DA - 2026/08/31
PB - arXiv
DO - 10.48550/arxiv.2608.22285
UR - https://arxiv.org/abs/2608.22285
AB - 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.
ER -