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Densities of Picard Rank Jumps for K3 Surfaces Over Global Fields

  • Robin Huang
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Abstract

Let X be an algebraic K3 surface defined over a global field K of characteristic $0$ or p ≥ 3, assumed ordinary in the case where K is a global function field. We provide an exact formula for the density of primes p of K at which the geometric Neron-Severi group of the reduction Xp has a given rank in terms of the Galois representation attached to X. As an extension, we prove that there are infinitely many reductions of X with an element of the geometric Neron-Severi group representing a fixed square class under the intersection pairing and prove results about the density of such places. In particular, we show that if X corresponds to a sufficiently generic moduli point, a density zero set of places of K satisfy this condition.

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Cite

BibTeXRIS
BibTeX
@misc{huang2026densities,
  title = {{Densities of Picard Rank Jumps for K3 Surfaces Over Global Fields}},
  author = {Robin Huang},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.06888},
  eprint = {2610.06888},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.06888},
}
RIS
TY  - UNPB
TI  - Densities of Picard Rank Jumps for K3 Surfaces Over Global Fields
AU  - Robin Huang
PY  - 2026
DA  - 2026/10/01
PB  - arXiv
DO  - 10.48550/arxiv.2610.06888
UR  - https://arxiv.org/abs/2610.06888
AB  - Let X be an algebraic K3 surface defined over a global field K of characteristic $0$ or p ≥ 3, assumed ordinary in the case where K is a global function field. We provide an exact formula for the density of primes p of K at which the geometric Neron-Severi group of the reduction X_p has a given rank in terms of the Galois representation attached to X. As an extension, we prove that there are infinitely many reductions of X with an element of the geometric Neron-Severi group representing a fixed square class under the intersection pairing and prove results about the density of such places. In particular, we show that if X corresponds to a sufficiently generic moduli point, a density zero set of places of K satisfy this condition.
ER  -