Ordinary reduction of K3 surfaces

• Autorzy: Fedor Bogomolov , Yuri Zarhin
• Języki publikacji: EN

Abstrakty

EN

Let X be a K3 surface over a number field K. We prove that there exists a finite algebraic field extension E/K such that X has ordinary reduction at every non-archimedean place of E outside a density zero set of places.

Słowa kluczowe

EN

K3 surfaces; Ordinary reduction; ℓ-adic representations; Newton polygons

Kategorie tematyczne

• 11G35: Varieties over global fields
• 14J28: K 3 surfaces and Enriques surfaces
• 11G25: Varieties over finite and local fields
• 14G25: Global ground fields

• Wydawca: De Gruyter Open
• Czasopismo: Open Mathematics
• Rocznik: 2009
• Tom: 7
• Numer: 2
• Strony: 206-213

Daty

wydano

2009-06-01

online

2009-05-24

Twórcy

autor

Fedor Bogomolov

• Courant Institute of Mathematical Sciences, New York University, New York, USA

autor

Yuri Zarhin

• Department of Mathematics, Pennsylvania State University, University Park, USA

Bibliografia

• [1] Artin M., Supersingular K3 surfaces, Ann. Sci. École Norm. Sup., Sér. 4, 1974, 7, 543–567
• [2] Artin M., Mazur B., Formal groups arising from algebraic varieties, Ann. Sci. École Norm. Sup., Sér. 4, 1977, 10, 87–131
• [3] Berthelot P., Ogus A., Notes on crystalline cohomology, Princeton University Press, Princeton, 1978
• [4] Bogomolov F.A., Sur l’algébricité des représentations ℓ-adiques, C. R. Acad. Sci. Paris Sér. A-B, 1980, 290, A701–A703 (in French)
• [5] Bogomolov F.A., Points of finite order on abelian varieties, Izv. Akad. Nauk SSSR Ser. Mat., 1980, 44, 782–804 (in Russian), Math. USSR Izv., 1981, 17, 55–72
• [6] Deligne P., La conjecture de Weil pour les surfaces K3, Invent. Math., 1972, 15, 206–226 (in French) http://dx.doi.org/10.1007/BF01404126[Crossref]
• [7] Deligne P. (rédigé par L. Illusie), Relèvement des surfaces K3 en charactéristique nulle, In: Surfaces Algébriques, Lecture Notes in Math., Springer, 1981, 868, 58–79 (in French)
• [8] Faltings G., Endlichkeitssätze für abelsche Varietäten über Zahlkörpern, Invent. Math., 1983, 73, 349–366, Erratum, 1984, 75, 381 (in German) http://dx.doi.org/10.1007/BF01388432[Crossref]
• [9] Hindry M., Silverman J.H., Diophantine geometry, An Introduction, Graduate Texts in Mathematics 201, Springer-Verlag, New York, 2000
• [10] Joshi K., Rajan C.S., Frobenius splitting and ordinarity, Int. Math. Res. Not., 2003, 2, 109–121 http://dx.doi.org/10.1155/S1073792803112135[Crossref]
• [11] Katz N., Nilpotent connections and the monodromy theorem: applications of a result of Turrittin, Inst. Hautes Études Sci. Publ. Math., 1970, 39, 175–232 http://dx.doi.org/10.1007/BF02684688[Crossref]
• [12] Katz N., Messing W., Some consequences of the Riemann hypothesis for varieties over finite fields, Invent. Math., 1974, 23, 73–77 http://dx.doi.org/10.1007/BF01405203[Crossref]
• [13] Koch H., Number theory II (Algebraic number theory), Encyclopedia of Mathematical Sciences 62, Springer Verlag, Berlin Heidelberg, 1992
• [14] Mazur B., Frobenius and the Hodge filtration, Bull. Amer. Math. Soc., 1972, 78, 653–667 http://dx.doi.org/10.1090/S0002-9904-1972-12976-8[Crossref]
• [15] Mumford D., Abelian varieties, Second Edition, Oxford University Press, London, 1974
• [16] Noot R., Abelian varieties-Galois representation and properties of ordinary reduction, Compositio Math., 1995, 97, 161–171
• [17] Noot R., Abelian varieties with ℓ-adic Galois representation of Mumford’s type, J. Reine Angew. Math., 2000, 519, 155–169
• [18] Nygaard N., The Tate conjecture for ordinary K3 surfaces over finite fields, Invent. Math., 1983, 74, 213–237 http://dx.doi.org/10.1007/BF01394314[WoS][Crossref]
• [19] Nygaard N., Ogus A., Tate’s conjecture for K3 surfaces of finite height, Ann. of Math., 1985, 122, 461–507 http://dx.doi.org/10.2307/1971327[Crossref]
• [20] Ogus A., Hodge cycles and crystalline cohomology, In: Lecture Notes in Math., Springer, 1982, 900, 357–414 http://dx.doi.org/10.1007/978-3-540-38955-2_8[Crossref]
• [21] Piatetski-Shapiro I. I., Shafarevich I.R., Arithmetic of K3 surfaces, Trudy Mat. Inst. Steklov, 1973, 132, 44–54 (in Russian), Proc. Steklov. Math. Inst., 1975, 132, 45–57
• [22] Serre J.-P., Représentations ℓ-adiques, In: Algebraic number theory (Kyoto Internat. Sympos., Res. Inst. Math. Sci., Univ. Kyoto, Kyoto, 1976), 177–193, Japan Soc. Promotion Sci., Tokyo, 1977 (in French)
• [23] Serre J.-P., Quelques applications du théorème de densité de Chebotarev, Inst. Hautes Études Sci. Publ. Math., 1981, 54, 323–401 (in French) http://dx.doi.org/10.1007/BF02698692[Crossref]
• [24] Serre J.-P., Abelian ℓ-adic representations and elliptic curves, Second Edition, Addison-Wesley, 1989
• [25] Skorobogatov A.N., Zarhin Yu.G., A finiteness theorem for the Brauer group of abelian varieties and K3 surfaces, J. Algebraic Geom., 2008, 17, 481–502 [Crossref]
• [26] Tankeev S.G., On the weights of the ℓ-adic representation and arithmetic of Frobenius eigenvalues, Izv. Ross. Akad. Nauk Ser. Mat., 1999, 63, 185–224 (in Russian), Izv. Math., 1999, 63, 181–218
• [27] Tate J., Conjectures on algebraic cycles in ℓ-adic cohomology, Motives (Seattle, WA, 1991), 71–83, Proc. Sympos. Pure Math. 55,Part 1, Amer. Math. Soc., Providence, RI, 1994
• [28] Yu J.-D., Yui N., K3 Surfaces of finite height over finite fields, J. Math. Kyoto Univ., 2008, 48, 499–519
• [29] Zarhin Yu.G., Hodge groups of K3 surfaces, J. Reine Angew. Math., 1983, 341, 193–220
• [30] Zarhin Yu.G., Weights of simple Lie algebras in the cohomology of algebraic varieties, Izv. Akad. Nauk SSSR Ser. Mat., 1984, 48, 264–304 (in Russian), Math. USSR Izv., 1985, 24, 245–282
• [31] Zarhin Yu.G., Transcendental cycles on ordinary K3 surfaces over finite fields, Duke Math. J., 1993, 72, 65–83 http://dx.doi.org/10.1215/S0012-7094-93-07203-1[Crossref]

Identyfikatory

DOI

10.2478/s11533-009-0013-8