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On potential density of rational points on hyperkahler varieties
Abstract
In this paper, we prove that the rational points on hyperkähler varieties in each of the four known deformation families admitting a rational Lagrangian fibration are potentially dense. Furthermore, we give several applications and construct examples of hyperkähler varieties beyond these families satisfying the potential density property. Our approach is first to prove that rational points on hyperkähler varieties admitting two rational Lagrangian fibrations with distinct associated isotropic rays are potentially dense. We then construct a projective Shafarevich--Tate twist admitting two such fibrations, together with a dominant rational map from the twist to the original variety.
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BibTeX
@misc{gong2026potential,
title = {{On potential density of rational points on hyperkahler varieties}},
author = {Ting Gong},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.00311},
eprint = {2611.00311},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.00311},
}
RIS
TY - UNPB
TI - On potential density of rational points on hyperkahler varieties
AU - Ting Gong
PY - 2026
DA - 2026/11/01
PB - arXiv
DO - 10.48550/arxiv.2611.00311
UR - https://arxiv.org/abs/2611.00311
AB - In this paper, we prove that the rational points on hyperkähler varieties in each of the four known deformation families admitting a rational Lagrangian fibration are potentially dense. Furthermore, we give several applications and construct examples of hyperkähler varieties beyond these families satisfying the potential density property. Our approach is first to prove that rational points on hyperkähler varieties admitting two rational Lagrangian fibrations with distinct associated isotropic rays are potentially dense. We then construct a projective Shafarevich--Tate twist admitting two such fibrations, together with a dominant rational map from the twist to the original variety.
ER -