Mathematische Zeitschrift

, Volume 279, Issue 1–2, pp 241–266 | Cite as

Good reduction criterion for K3 surfaces



We prove a Néron–Ogg–Shafarevich type criterion for good reduction of K3 surfaces, which states that a K3 surface over a complete discrete valuation field has potential good reduction if its \(l\)-adic cohomology group is unramified. We also prove a \(p\)-adic version of the criterion. (These are analogues of the criteria for good reduction of abelian varieties.) The model of the surface will be in general not a scheme but an algebraic space. As a corollary of the criterion we obtain the surjectivity of the period map of K3 surfaces in positive characteristic.


K3 surfaces Good reduction Galois representations Period map  Complex multiplication 

Mathematics Subject Classification

14J28 11G25 14G20 



The author expresses his sincere gratitude to his advisor Atsushi Shiho for supporting him in many ways. The author also thanks Takuma Hayashi, Tetsushi Ito, Teruhisa Koshikawa, Keerthi Madapusi Pera, Yukiyoshi Nakkajima, Takeshi Saito, Naoya Umezaki, and Kohei Yahiro for giving him helpful comments. This work was supported by Grant-in-Aid for JSPS Fellows Grant Number 12J08397.


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© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  1. 1.Graduate School of Mathematical SciencesThe University of TokyoTokyoJapan

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