Duke Mathematical Journal

Transcendental cycles on ordinary $K3$ surfaces over finite fields

Yuri G. Zarhin
Source: Duke Math. J. Volume 72, Number 1 (1993), 65-83.
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Primary Subjects: 14G15
Secondary Subjects: 14G10, 14J28
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Links and Identifiers

Permanent link to this document: http://projecteuclid.org/euclid.dmj/1077289212
Mathematical Reviews number (MathSciNet): MR1242879
Zentralblatt MATH identifier: 0819.14005
Digital Object Identifier: doi:10.1215/S0012-7094-93-07203-1

References

[1] M. Artin, Supersingular $K3$ surfaces, Ann. Sci. École Norm. Sup. (4) 7 (1974), 543–567 (1975).
Mathematical Reviews (MathSciNet): MR51:8116
Zentralblatt MATH: 0322.14014
[2] M. Artin and B. Mazur, Formal groups arising from algebraic varieties, Ann. Sci. École Norm. Sup. (4) 10 (1977), no. 1, 87–131.
Mathematical Reviews (MathSciNet): MR56:15663
Zentralblatt MATH: 0351.14023
[3] M. Artin and H. P. F. Swinnerton-Dyer, The Shafarevich-Tate conjecture for pencils of elliptic curves on $K3$ surfaces, Invent. Math. 20 (1973), 249–266.
Mathematical Reviews (MathSciNet): MR54:5240
Zentralblatt MATH: 0289.14003
Digital Object Identifier: doi:10.1007/BF01394097
[4] P. Deligne, La conjecture de Weil pour les surfaces $K3$, Invent. Math. 15 (1972), 206–226.
Mathematical Reviews (MathSciNet): MR45:5137
Zentralblatt MATH: 0219.14022
Digital Object Identifier: doi:10.1007/BF01404126
[5] P. Deligne, La conjecture de Weil. I, Inst. Hautes Études Sci. Publ. Math. (1974), no. 43, 273–307.
Mathematical Reviews (MathSciNet): MR49:5013
Zentralblatt MATH: 0287.14001
Digital Object Identifier: doi:10.1007/BF02684373
[6] E. Freitag and R. Kiehl, Étale cohomology and the Weil conjecture, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 13, Springer-Verlag, Berlin, 1988.
Mathematical Reviews (MathSciNet): MR89f:14017
Zentralblatt MATH: 0643.14012
[7] N. Katz and W. Messing, Some consequences of the Riemann hypothesis for varieties over finite fields, Invent. Math. 23 (1974), 73–77.
Mathematical Reviews (MathSciNet): MR48:11117
Zentralblatt MATH: 0275.14011
Digital Object Identifier: doi:10.1007/BF01405203
[8] N. Koblitz, $p$-adic numbers, $p$-adic analysis, and zeta-functions, Graduate Texts in Mathematics, vol. 58, Springer-Verlag, New York, 1984.
Mathematical Reviews (MathSciNet): MR86c:11086
[9] H. W. Lenstra and Y. G. Zarhin, The Tate conjecture for almost ordinary abelian varieties over finite fields, Advances in number theory (Kingston, ON, 1991), Oxford Sci. Publ., Oxford University Press, New York,Oxford, 1993, pp. 179–194.
Mathematical Reviews (MathSciNet): MR97c:11067
Zentralblatt MATH: 0817.14022
[10] B. Mazur, Eigenvalues of Frobenius acting on algebraic varieties over finite fields, Algebraic geometry (Proceedings Sympos. Pure Math., Vol. 29, Humboldt State Univ., Arcata, Calif., 1974), Amer. Math. Soc., Providence, R.I., 1975, pp. 231–261.
Mathematical Reviews (MathSciNet): MR51:8113
Zentralblatt MATH: 0306.14011
[11] D. Mumford, Abelian Varieties, second ed., Tata Inst. Fund. Res. Stud. Math., vol. 5, Oxford Univ. Press, London, 1974.
Zentralblatt MATH: 0326.14012
Mathematical Reviews (MathSciNet): MR2514037
[12] N. O. Nygaard, The Tate conjecture for ordinary $K3$ surfaces over finite fields, Invent. Math. 74 (1983), no. 2, 213–237.
Mathematical Reviews (MathSciNet): MR85h:14012
Zentralblatt MATH: 0557.14002
Digital Object Identifier: doi:10.1007/BF01394314
[13] N. Nygaard and A. Ogus, Tate's conjecture for $K3$ surfaces of finite height, Ann. of Math. (2) 122 (1985), no. 3, 461–507.
Mathematical Reviews (MathSciNet): MR87h:14014
Zentralblatt MATH: 0591.14005
Digital Object Identifier: doi:10.2307/1971327
[14] I. I. Pjateckiĭ-Šapiro and I. R. Šafarevič, The arithmetic of surfaces of type $\rm K3$, Trudy Mat. Inst. Steklov. 132 (1973), 44–54, 264, English translation in Proc. Stekov Inst. Math. 132 (1975), 45–57.
Mathematical Reviews (MathSciNet): MR49:302
[15] J.-P. Serre, Abelian $l$-adic representations and elliptic curves, Second Edition ed., Advanced Book Classics, Addison-Wesley Publishing Company Advanced Book Program, Redwood City, CA, 1989.
Mathematical Reviews (MathSciNet): MR91b:11071
Zentralblatt MATH: 0709.14002
[16] J.-P. Serre, Représentations $l$-adiques, Algebraic number theory (Kyoto Internat. Sympos., Res. Inst. Math. Sci., Univ. Kyoto, Kyoto, 1976), Japan Society for the Promotion Science, Tokyo, 1977, pp. 177–193.
Mathematical Reviews (MathSciNet): MR57:16310
Zentralblatt MATH: 0406.14015
[17] T. Shioda, The Hodge conjecture and the Tate conjecture for Fermat varieties, Proc. Japan Acad. Ser. A Math. Sci. 55 (1979), no. 3, 111–114.
Mathematical Reviews (MathSciNet): MR80e:14006
Zentralblatt MATH: 0444.14017
Digital Object Identifier: doi:10.3792/pjaa.55.111
Project Euclid: euclid.pja/1195517400
[18] S. G. Tankeev, On cycles on Abelian varieties of prime dimension over finite or number fields, Math. USSR-Izv. 22 (1984), 329–337.
Zentralblatt MATH: 0583.14002
Mathematical Reviews (MathSciNet): MR697300
[19] J. Tate, Algebraic cycles and poles of zeta functions, Arithmetical Algebraic Geometry (Proc. Conf. Purdue Univ., 1963), Harper & Row, New York, 1965, pp. 93–110.
Mathematical Reviews (MathSciNet): MR37:1371
Zentralblatt MATH: 0213.22804
[20] J. Tate, Endomorphisms of abelian varieties over finite fields, Invent. Math. 2 (1966), 134–144.
Mathematical Reviews (MathSciNet): MR34:5829
Zentralblatt MATH: 0147.20303
Digital Object Identifier: doi:10.1007/BF01404549
[21] A. Weil, Basic number theory, Die Grundlehren der mathematischen Wissenschaften, Band 144, Springer-Verlag New York, Inc., New York, 1967.
Mathematical Reviews (MathSciNet): MR38:3244
Zentralblatt MATH: 0176.33601
[22] N. Yui, Special values of zeta-functions of Fermat varieties over finite fields, Number theory (New York, 1989/1990), Springer, New York, 1991, pp. 251–275.
Mathematical Reviews (MathSciNet): MR92i:11063
Zentralblatt MATH: 0791.14009
[23] N. Yui, Arithmetic of $K3$ over finite fields, in preparation.
[24] Yu. G. Zarhin, Abelian varieties of $K3$ type and $l$-adic representations, Algebraic geometry and analytic geometry (Proceedings, Tokyo, 1990), ICM-90 Satell. Conf. Proc., Springer, Tokyo, 1991, pp. 231–255.
Mathematical Reviews (MathSciNet): MR94i:14047
Zentralblatt MATH: 0788.14039
[25] Yu. G. Zarhin, Abelian varieties of $K3$ type, Séminaire de Théorie des Nombres, Paris, 1990–91, Progr. Math., vol. 108, Birkhäuser Boston, Boston, MA, 1993, pp. 263–279.
Mathematical Reviews (MathSciNet): MR95c:14056
Zentralblatt MATH: 0827.14031
[26] Yu. G. Zarhin, The Brauer group of an Abelian variety over a finite field, Izv. Akad. Nauk SSSR Ser. Mat. 46 (1982), English translation in Math. USSR-Izv. 20 (1983), 203–234.
Zentralblatt MATH: 0505.14034
Mathematical Reviews (MathSciNet): MR651646
[27] Yu. G. Zarhin, The Tate conjecture for powers of ordinary $K3$ surfaces over finite fields, in preparation.
[28] Yu. G. Zarhin, Hodge groups of $K3$ surfaces, J. Reine Angew. Math. 341 (1983), 193–220.
Mathematical Reviews (MathSciNet): MR84g:14009
Zentralblatt MATH: 0506.14034
Digital Object Identifier: doi:10.1515/crll.1983.341.193

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