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PreprintarXiv licencearXiv

Pointless K3 surfaces over finite fields

  • Asher Auel
  • Philip Engel
  • Kiran S. Kedlaya
  • Sair Shaikh
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Abstract

We prove that K3 surfaces with no rational points over the finite field with q elements exist if and only if q ≤ 11. Our approach applies linear programming techniques to improve on generic sphere packing bounds for the discriminants of even rootless definite lattices. These bounds are applied to constrain the possible zeta functions of K3 surfaces with no rational points via the Artin--Tate formula. The remaining cases are ruled out by direct analysis of primitive embeddings of particular lattices into Niemeier lattices.

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Cite

BibTeXRIS
BibTeX
@misc{auel2026pointless,
  title = {{Pointless K3 surfaces over finite fields}},
  author = {Asher Auel and Philip Engel and Kiran S. Kedlaya and Sair Shaikh},
  howpublished = {arXiv},
  year = {2026},
  month = nov,
  doi = {10.48550/arxiv.2611.08356},
  eprint = {2611.08356},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2611.08356},
}
RIS
TY  - UNPB
TI  - Pointless K3 surfaces over finite fields
AU  - Asher Auel
AU  - Philip Engel
AU  - Kiran S. Kedlaya
AU  - Sair Shaikh
PY  - 2026
DA  - 2026/11/05
PB  - arXiv
DO  - 10.48550/arxiv.2611.08356
UR  - https://arxiv.org/abs/2611.08356
AB  - We prove that K3 surfaces with no rational points over the finite field with q elements exist if and only if q ≤ 11. Our approach applies linear programming techniques to improve on generic sphere packing bounds for the discriminants of even rootless definite lattices. These bounds are applied to constrain the possible zeta functions of K3 surfaces with no rational points via the Artin--Tate formula. The remaining cases are ruled out by direct analysis of primitive embeddings of particular lattices into Niemeier lattices.
ER  -