On good reduction of some K3 surfaces related to abelian surfaces. (English) Zbl 1361.14027

Summary: The Néron-Ogg-Shafarevič criterion for abelian varieties tells that the Galois action on the \(\ell\)-adic étale cohomology of an abelian variety over a local field determines whether the variety has good reduction or not. We prove an analogue of this criterion for a certain type of K3 surfaces closely related to abelian surfaces. We also prove its \(p\)-adic analogue. This paper includes T. Ito’s unpublished result on Kummer surfaces.


14J28 \(K3\) surfaces and Enriques surfaces
11G25 Varieties over finite and local fields
14G20 Local ground fields in algebraic geometry
Full Text: DOI arXiv Euclid


[1] W. P. Barth, K. Hulek, C. A. M. Peters and A. Van de Ven, Compact complex surfaces, Second edition, Ergeb. Math. Grenzgeb. (3) [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics] 4, Springer-Verlag, Berlin, 2004.
[2] P. Berthelot and A. Ogus, \(F\)-isocrystals and de Rham cohomology. I, Invent. Math. 72 (1983), 159-199. · Zbl 0516.14017 · doi:10.1007/BF01389319
[3] S. Bosch, W. Lütkebohmert and M. Raynaud, Néron models, Ergeb. Math. Grenzgeb. (3) [Results in Mathematics and Related Areas (3)] 21, Springer-Verlag, Berlin, 1990. · Zbl 0705.14001
[4] R. Coleman and A. Iovita, The Frobenius and monodromy operators for curves and abelian varieties, Duke Math. J. 97 (1999), 171-215. · Zbl 0962.14030 · doi:10.1215/S0012-7094-99-09708-9
[5] P. Deligne, Relèvement des surfaces \(K3\) en caractéristique nulle, prepared for publication by Luc Illusie, Lecture Notes in Math. 868, Algebraic surfaces (Orsay, 1976-78), 58-79, Springer, Berlin-New York, 1981.
[6] I. V. Dolgachev and J. Keum, Finite groups of symplectic automorphisms of \(K3\) surfaces in positive characteristic, Ann. of Math. (2) 169 (2009), 269-313. · Zbl 1187.14047 · doi:10.4007/annals.2009.169.269
[7] G. Faltings, Crystalline cohomology and \(p\)-adic Galois-representations, in Algebraic analysis, geometry, and number theory (Baltimore, MD, 1988), 25-80, Johns Hopkins Univ. Press, Baltimore, MD, 1989. · Zbl 0805.14008
[8] A. Grothendieck, Fondements de la géométrie algébrique. [Extraits du Séminaire Bourbaki, 1957-1962.], Secrétariat mathématique, Paris, 1962.
[9] T. Ito, Good reduction of Kummer surfaces, Master’s thesis, University of Tokyo, 2001. (unpublished) · Zbl 1322.81067
[10] K. Kodaira, On compact analytic surfaces. II, III, Ann. of Math. (2) 77 (1963), 563-626; ibid. 78 (1963), 1-40. · Zbl 0118.15802 · doi:10.2307/1970131
[11] V. S. Kulikov, Degenerations of \(K3\) surfaces and Enriques surfaces, Izv. Akad. Nauk SSSR Ser. Mat. 41 (1977), no. 5, 1008-1042, 1199. · Zbl 0367.14014
[12] M. Kuwata and T. Shioda, Elliptic parameters and defining equations for elliptic fibrations on a Kummer surface, in Algebraic geometry in East Asia–Hanoi 2005, 177-215, Adv. Stud. Pure Math. 50, Math. Soc. Japan, Tokyo, 2008. · Zbl 1139.14032
[13] Q. Liu, Algebraic geometry and arithmetic curves, translated from the French by Reinie Erné, Oxf. Grad. Texts Math. 6, Oxford Science Publications, Oxford University Press, Oxford, 2002.
[14] S. Ma, Decompositions of an Abelian surface and quadratic forms, Ann. Inst. Fourier (Grenoble) 61 (2011), 717-743. · Zbl 1231.14036 · doi:10.5802/aif.2627
[15] Y. Matsumoto, Good reduction criterion for K3 surfaces, preprint, 2014, available at http://arxiv.org/abs/1401.1261v1. · Zbl 1317.14089 · doi:10.1007/s00209-014-1365-8
[16] D. R. Morrison, On \(K3\) surfaces with large Picard number, Invent. Math. 75 (1984), 105-121. · Zbl 0509.14034 · doi:10.1007/BF01403093
[17] V. V. Nikulin, Finite groups of automorphisms of Kählerian \(K3\) surfaces, Trudy Moskov. Mat. Obshch. 38 (1979), 75-137. · Zbl 0433.14024
[18] I. I. Pjatecki\?-Šapiro and I. R. Šafarevič, Torelli’s theorem for algebraic surfaces of type K3, Izv. Akad. Nauk SSSR Ser. Mat. 35 (1971), 530-572.
[19] T. Saito, Weight spectral sequences and independence of \(l\), J. Inst. Math. Jussieu 2 (2003), 583-634. · Zbl 1084.14027 · doi:10.1017/S1474748003000173
[20] J.-P. Serre and J. Tate, Good reduction of abelian varieties, Ann. of Math. (2) 88 (1968), 492-517. · Zbl 0172.46101 · doi:10.2307/1970722
[21] Théorie des topos et cohomologie étale des schémas. Tome 3, Séminaire de Géométrie Algébrique du Bois-Marie 1963-1964 (SGA 4), dirigé par M. Artin, A. Grothendieck et J. L. Verdier. Avec la collaboration de P. Deligne et B. Saint-Donat, Lecture Notes in Mathematics, Vol. 305, Springer-Verlag, Berlin, 1973.
[22] T. Shioda, Kummer sandwich theorem of certain elliptic \(K3\) surfaces, Proc. Japan Acad. Ser. A Math. Sci. 82 (2006), 137-140. · Zbl 1112.14044 · doi:10.3792/pjaa.82.137
[23] T. Shioda and H. Inose, On singular \(K3\) surfaces, in Complex analysis and algebraic geometry, 119-136, Iwanami Shoten, Tokyo, 1977. · Zbl 0374.14006 · doi:10.1017/CBO9780511569197.010
[24] J. H. Silverman, Advanced topics in the arithmetic of elliptic curves, Grad. Texts in Math. 151, Springer-Verlag, New York, 1994.
[25] T. Tsuji, \(p\)-adic étale cohomology and crystalline cohomology in the semi-stable reduction case, Invent. Math. 137 (1999), 233-411. · Zbl 0945.14008 · doi:10.1007/s002220050330
[26] T. Tsuji, Semi-stable conjecture of Fontaine-Jannsen: a survey, Cohomologies \(p\)-adiques et applications arithmétiques, II, Astérisque No. 279 (2002), 323-370. · Zbl 1041.14003
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.