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PreprintCC BY 4.0arXiv

Uniform Height Gaps in Arbitrary Characteristics

  • Alice Lin
  • Frank Lu
  • Jit Wu Yap
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Abstract

We prove uniform height gaps for subvarieties of abelian varieties in arbitrary characteristic, extending the new gap principle of Gao-Ge-Kühne. Our proof goes through the theory of adelic curves and globally valued fields, and proves the Bogomolov conjecture for an arbitrary globally valued field. This gives a different way to obtain uniform Bogomolov-type results, differing from the approaches of Dimitrov-Gao-Habegger-Kühne and Yuan. First, we prove the Bogomolov conjecture over globally valued fields for divisors. We then reduce the case of general subvarieties to the case of divisors by an induction argument. Specializing to the case of global function fields, we obtain a new proof of the geometric Bogomolov conjecture following the strategy of Gubler and Yamaki.

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Cite

BibTeXRIS
BibTeX
@misc{lin2026uniform,
  title = {{Uniform Height Gaps in Arbitrary Characteristics}},
  author = {Alice Lin and Frank Lu and Jit Wu Yap},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.11905},
  eprint = {2610.11905},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.11905},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Uniform Height Gaps in Arbitrary Characteristics
AU  - Alice Lin
AU  - Frank Lu
AU  - Jit Wu Yap
PY  - 2026
DA  - 2026/10/08
PB  - arXiv
DO  - 10.48550/arxiv.2610.11905
UR  - https://arxiv.org/abs/2610.11905
AB  - We prove uniform height gaps for subvarieties of abelian varieties in arbitrary characteristic, extending the new gap principle of Gao-Ge-Kühne. Our proof goes through the theory of adelic curves and globally valued fields, and proves the Bogomolov conjecture for an arbitrary globally valued field. This gives a different way to obtain uniform Bogomolov-type results, differing from the approaches of Dimitrov-Gao-Habegger-Kühne and Yuan. First, we prove the Bogomolov conjecture over globally valued fields for divisors. We then reduce the case of general subvarieties to the case of divisors by an induction argument. Specializing to the case of global function fields, we obtain a new proof of the geometric Bogomolov conjecture following the strategy of Gubler and Yamaki.
ER  -