PreprintCC BY 4.0arXiv
S-integrality for families of ordinary K3 surfaces and algebraicity theorems
Abstract
We prove an S-integrality theorem for special divisors on GSpin Shimura varieties in positive characteristic. Let C be a generically ordinary curve in such a Shimura variety not contained in any special divisor. Then, for any increasing sequence of prime-to-p positive integers mi and any finite set of closed points S⊂ C, we prove that C∖ S meets the special divisor Z(mi) for all but finitely many i. The key new input is an algebraicity theorem for formal special endomorphisms which allows us to use techniques from Diophantine approximation. We prove this algebraicity theorem using a punctual monodromy theorem and a positive-characteristic analogue of the Mumford--Tate conjecture.
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BibTeX
@misc{jiang2026sintegrality,
title = {{S-integrality for families of ordinary K3 surfaces and algebraicity theorems}},
author = {Ruofan Jiang and Ananth N. Shankar},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.25703},
eprint = {2609.25703},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.25703},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - S-integrality for families of ordinary K3 surfaces and algebraicity theorems
AU - Ruofan Jiang
AU - Ananth N. Shankar
PY - 2026
DA - 2026/09/22
PB - arXiv
DO - 10.48550/arxiv.2609.25703
UR - https://arxiv.org/abs/2609.25703
AB - We prove an S-integrality theorem for special divisors on GSpin Shimura varieties in positive characteristic. Let C be a generically ordinary curve in such a Shimura variety not contained in any special divisor. Then, for any increasing sequence of prime-to-p positive integers m_i and any finite set of closed points S⊂ C, we prove that C∖ S meets the special divisor Z(m_i) for all but finitely many i. The key new input is an algebraicity theorem for formal special endomorphisms which allows us to use techniques from Diophantine approximation. We prove this algebraicity theorem using a punctual monodromy theorem and a positive-characteristic analogue of the Mumford--Tate conjecture.
ER -