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Pointless K3 surfaces over finite fields
Abstract
We provide an algorithm for enumerating ordinary K3 surfaces over a fixed finite field based on Taelman's theorem. We use this algorithm to prove that there exist no K3 surfaces without rational points over 𝔽q for every prime power q≥ 13. For elliptic K3 surfaces, we give a geometric proof of this non-existence without computer assistance. On the other hand, we also provide explicit pointless K3 surfaces over 𝔽q for every prime power q≤ 11.
Related
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- Densities of Picard Rank Jumps for K3 Surfaces Over Global Fields
BibTeX
@misc{brandhorst2026pointless,
title = {{Pointless K3 surfaces over finite fields}},
author = {Simon Brandhorst and Yuta Takada},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.06366},
eprint = {2611.06366},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.06366},
}
RIS
TY - UNPB
TI - Pointless K3 surfaces over finite fields
AU - Simon Brandhorst
AU - Yuta Takada
PY - 2026
DA - 2026/11/04
PB - arXiv
DO - 10.48550/arxiv.2611.06366
UR - https://arxiv.org/abs/2611.06366
AB - We provide an algorithm for enumerating ordinary K3 surfaces over a fixed finite field based on Taelman's theorem. We use this algorithm to prove that there exist no K3 surfaces without rational points over 𝔽_q for every prime power q≥ 13. For elliptic K3 surfaces, we give a geometric proof of this non-existence without computer assistance. On the other hand, we also provide explicit pointless K3 surfaces over 𝔽_q for every prime power q≤ 11.
ER -