Density of μ-ordinary Primes for K3 Surfaces
Abstract
By a result of Bogomolov and Zarhin, a K3 surface X over a number field L has ordinary reduction at a density 1 set of primes after passing to a finite extension of L. In this paper, we refine this result for non-CM K3 surfaces whose transcendental Hodge structure has endomorphism field F abelian over ℚ. We further assume a condition on the connected components of the l-adic monodromy group of X, and, when F is totally real, a parity condition on the rank of the transcendental lattice over F. Under these assumptions, we prove that the set of primes of L at which X has μ-ordinary reduction has density $1$. As a corollary, the set of primes of L at which X has ordinary reduction has density 1/[FL:L]. We include explicit examples of K3 surfaces satisfying these conditions.
Related
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BibTeX
@misc{can2026density,
title = {{Density of $μ$-ordinary Primes for K3 Surfaces}},
author = {Trung Can and Yuxin Lin},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.34093},
eprint = {2609.34093},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.34093},
}
RIS
TY - UNPB
TI - Density of μ-ordinary Primes for K3 Surfaces
AU - Trung Can
AU - Yuxin Lin
PY - 2026
DA - 2026/09/28
PB - arXiv
DO - 10.48550/arxiv.2609.34093
UR - https://arxiv.org/abs/2609.34093
AB - By a result of Bogomolov and Zarhin, a K3 surface X over a number field L has ordinary reduction at a density 1 set of primes after passing to a finite extension of L. In this paper, we refine this result for non-CM K3 surfaces whose transcendental Hodge structure has endomorphism field F abelian over ℚ. We further assume a condition on the connected components of the l-adic monodromy group of X, and, when F is totally real, a parity condition on the rank of the transcendental lattice over F. Under these assumptions, we prove that the set of primes of L at which X has μ-ordinary reduction has density $1$. As a corollary, the set of primes of L at which X has ordinary reduction has density 1/[FL:L]. We include explicit examples of K3 surfaces satisfying these conditions.
ER -