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PreprintCC BY 4.0arXiv

Hecke Orbits on Globally Drinfeld-Uniformized Shimura Varieties Are p-adically Nowhere Dense

  • Yu Fu
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Abstract

Fix a projective unitary Shimura datum satisfying Varshavsky's global Drinfeld uniformization hypotheses at a place v| p of its reflex field E. Let 𝒮h be the base change of the Shimura variety to ℂp at a sufficiently small neat finite level. We prove that for every closed reduced algebraic subvariety Z⊂𝒮h with dim Z<dim 𝒮h, the p-adic closure of its full finite adelic Hecke orbit has empty interior in 𝒮h(ℂp). At certain compatible levels, the nowhere density theorem also holds for Shimura varieties associated with a related PEL datum having the same adjoint datum as Varshavsky's. Moreover, for every rational prime p and every integer d≥2, there is a projective unitary Shimura variety of dimension d−1 satisfying these uniformization hypotheses at a place above p for which the nowhere density theorem holds.

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Cite

BibTeXRIS
BibTeX
@misc{fu2026hecke,
  title = {{Hecke Orbits on Globally Drinfeld-Uniformized Shimura Varieties Are $p$-adically Nowhere Dense}},
  author = {Yu Fu},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.28869},
  eprint = {2610.28869},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.28869},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Hecke Orbits on Globally Drinfeld-Uniformized Shimura Varieties Are p-adically Nowhere Dense
AU  - Yu Fu
PY  - 2026
DA  - 2026/10/20
PB  - arXiv
DO  - 10.48550/arxiv.2610.28869
UR  - https://arxiv.org/abs/2610.28869
AB  - Fix a projective unitary Shimura datum satisfying Varshavsky's global Drinfeld uniformization hypotheses at a place v| p of its reflex field E. Let 𝒮h be the base change of the Shimura variety to ℂ_p at a sufficiently small neat finite level. We prove that for every closed reduced algebraic subvariety Z⊂𝒮h with dim Z<dim 𝒮h, the p-adic closure of its full finite adelic Hecke orbit has empty interior in 𝒮h(ℂ_p). At certain compatible levels, the nowhere density theorem also holds for Shimura varieties associated with a related PEL datum having the same adjoint datum as Varshavsky's. Moreover, for every rational prime p and every integer d≥2, there is a projective unitary Shimura variety of dimension d−1 satisfying these uniformization hypotheses at a place above p for which the nowhere density theorem holds.
ER  -