Hodge classes and Tate classes on simple abelian fourfolds

previous :: next

Hodge classes and Tate classes on simple abelian fourfolds

B. J. J. Moonen and Yu. G. Zarhin

Source: Duke Math. J. Volume 77, Number 3 (1995), 553-581.

First Page: Show

Primary Subjects: 14C30

Secondary Subjects: 11G10, 14K15

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber.

If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Links and Identifiers

Permanent link to this document: http://projecteuclid.org/euclid.dmj/1077286533
Mathematical Reviews number (MathSciNet): MR1324634
Zentralblatt MATH identifier: 0874.14034
Digital Object Identifier: doi:10.1215/S0012-7094-95-07717-5

back to Table of Contents

References

[1] Y. André, Une remarque à propos des cycles de Hodge de type CM, Seminaire de Théorie des Nombres, Paris, 1989–1990, Progr. Math., vol. 102, Birkhäuser, Boston, 1992, pp. 1–7.

Mathematical Reviews (MathSciNet): MR98f:14005

Zentralblatt MATH: 0782.14003

[2] Y. André, Pour une théorie inconditionelle des motifs, Groupe d'étude sur les Problems Diophantiennes 1991–1992 eds. D. Bertrand and M. Waldschmidt, Publ. Math. Univ. Paris VI, vol. 106, Univ. de Paris VI, Paris, 1993.

[3] F. A. Bogomolov, Sur l'algébricité des représentations $l$-adiques, C. R. Acad. Sci. Paris Sér. A-B 290 (1980), no. 15, A701–A703.

Mathematical Reviews (MathSciNet): MR81c:14025

Zentralblatt MATH: 0457.14020

[4] N. Bourbaki, Groupes et Algèbres de Lie, Hermann, Paris, 1975, Chapitres 7 et 8.

Mathematical Reviews (MathSciNet): MR453824

[5] W. Chi, On the $l$-adic representations attached to some absolutely simple abelian varieties of type $\rm II$, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 37 (1990), no. 2, 467–484.

Mathematical Reviews (MathSciNet): MR91g:11060

Zentralblatt MATH: 0741.14024

[6] W. Chi, $\ell$-adic and $\lambda$-adic representations associated to abelian varieties defined over number fields, Amer. J. Math. 114 (1992), no. 2, 315–353.

Mathematical Reviews (MathSciNet): MR93c:11038

Zentralblatt MATH: 0795.14024

Digital Object Identifier: doi:10.2307/2374706

JSTOR: links.jstor.org

[7] P. Deligne, Théorie de Hodge II, Inst. Hautes Études Sci. Publ. Math. 40 (1971), 5–57.

Mathematical Reviews (MathSciNet): MR58:16653a

Zentralblatt MATH: 0219.14007

Digital Object Identifier: doi:10.1007/BF02684692

[8] P. Deligne, Variétés de Shimura: Interprétation modulaire, et techniques de construction de modèles canoniques, Automorphic forms, representations and $L$-functions (Proc. Sympos. Pure Math., Oregon State Univ., Corvallis, Ore., 1977), Part 2, Proc. Symp. Pure Math., vol. 33, Amer. Math. Soc., Providence, 1979, pp. 247–289.

Mathematical Reviews (MathSciNet): MR81i:10032

Zentralblatt MATH: 0437.14012

[9] P. Deligne, J. S. Milne, A. Ogus, and K.-Y. Shih, Hodge cycles, motives, and Shimura varieties, Lecture Notes in Mathematics, vol. 900, Springer-Verlag, Berlin, 1982.

Mathematical Reviews (MathSciNet): MR84m:14046

Zentralblatt MATH: 0465.00010

[10] G. Faltings, Endlichkeitssätze für abelsche Varietäten über Zahlkörpern, Invent. Math. 73 (1983), no. 3, 349–366.

Mathematical Reviews (MathSciNet): MR85g:11026a

Zentralblatt MATH: 0588.14026

Digital Object Identifier: doi:10.1007/BF01388432

[11] J.-M. Fontaine, Formes différentielles et modules de Tate des variétés abéliennes sur les corps locaux, Invent. Math. 65 (1981/82), no. 3, 379–409.

Mathematical Reviews (MathSciNet): MR83k:14023

Zentralblatt MATH: 0502.14015

Digital Object Identifier: doi:10.1007/BF01396625

[12] J. Giraud, Modules des variétés abéliennes et variétés de Shimura, Variétés de Shimura et fonctions $L$ eds. L. Breen and J. P. Labesse, Publ. Math. Univ. Paris VII, vol. 6, Univ. de Paris VII, Paris, 1979.

Zentralblatt MATH: 0538.14030

[13] F. Hazama, Algebraic cycles on certain abelian varieties and powers of special surfaces, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 31 (1985), no. 3, 487–520.

Mathematical Reviews (MathSciNet): MR86k:14004

Zentralblatt MATH: 0591.14006

[14] F. Hazama, Algebraic cycles on nonsimple abelian varieties, Duke Math. J. 58 (1989), no. 1, 31–37.

Mathematical Reviews (MathSciNet): MR91g:14005

Zentralblatt MATH: 0697.14028

Digital Object Identifier: doi:10.1215/S0012-7094-89-05803-1

Project Euclid: euclid.dmj/1077307371

[15] R. E. Kottwitz, Points on some Shimura varieties over finite fields, J. Amer. Math. Soc. 5 (1992), no. 2, 373–444.

Mathematical Reviews (MathSciNet): MR93a:11053

Zentralblatt MATH: 0796.14014

Digital Object Identifier: doi:10.2307/2152772

JSTOR: links.jstor.org

[16] H. Lange and Ch. Birkenhake, Complex Abelian Varieties, Grundlehren Math. Wiss., vol. 302, Springer-Verlag, Berlin, 1992.

Mathematical Reviews (MathSciNet): MR94j:14001

Zentralblatt MATH: 0779.14012

[17] H. W. Lenstra, Jr. and Yu. G. Zarhin, The Tate conjecture for almost ordinary abelian varieties over finite fields, Advances in Number Theory: Proceedings of the Third Conference of the CNTA (Kingston, ON, 1991) eds. F. Gouvêa and N. Yui, Oxford Sci. Publ., Clarendon Press, Oxford, 1993, pp. 179–194.

Mathematical Reviews (MathSciNet): MR97c:11067

Zentralblatt MATH: 0817.14022

[18] D. Mumford, Abelian Varieties, second ed., Oxford University Press, Oxford, 1974.

Zentralblatt MATH: 0326.14012

Mathematical Reviews (MathSciNet): MR282985

[19] D. Mumford, A note of Shimura's paper “Discontinuous groups and abelian varieties”, Math. Ann. 181 (1969), 345–351.

Mathematical Reviews (MathSciNet): MR40:1400

Zentralblatt MATH: 0169.23301

Digital Object Identifier: doi:10.1007/BF01350672

[20] V. K. Murty, Exceptional Hodge classes on certain abelian varieties, Math. Ann. 268 (1984), no. 2, 197–206.

Mathematical Reviews (MathSciNet): MR85m:14063

Zentralblatt MATH: 0521.14004

Digital Object Identifier: doi:10.1007/BF01456085

[21] A. L. Onishchik and E. B. Vinberg, Lie Groups and Algebraic Groups, Springer Series in Soviet Mathematics, Springer-Verlag, Berlin, 1990.

Mathematical Reviews (MathSciNet): MR91g:22001

Zentralblatt MATH: 0722.22004

[22] F. Oort, Endomorphism algebras of abelian varieties, Algebraic Geometry and Commutative Algebra II ed. H. Hijikata, et al., Kinokuniya Company, Ltd., Tokyo, 1988, pp. 469–502.

Mathematical Reviews (MathSciNet): MR90j:11049

Zentralblatt MATH: 0697.14029

[23] I. I. Piatetskii-Shapiro, Interrelations between the Tate and Hodge conjectures for abelian varieties, Math. USSR-Sb. 14 (1971), 615–625.

Zentralblatt MATH: 0239.14019

[24] I. I. Piatetskii-Shapiro and I. R. Shafarevich, The arithmetic of $K3$ surfaces, Trudy Mat. Inst. Steklov. 132 (1973), 44–54, In English: Proc. Steklov Inst. Math. 132 (1975), 45–57.

[25] H. Pohlmann, Algebraic cycles on abelian varieties of complex multiplication type, Ann. of Math. (2) 88 (1968), 161–180.

Mathematical Reviews (MathSciNet): MR37:4080

Zentralblatt MATH: 0201.23201

Digital Object Identifier: doi:10.2307/1970570

JSTOR: links.jstor.org

[26] K. Ribet, Galois action on division points of abelian varieties with real multiplications, Amer. J. Math. 98 (1976), no. 3, 751–804.

Mathematical Reviews (MathSciNet): MR56:15660

Zentralblatt MATH: 0348.14022

Digital Object Identifier: doi:10.2307/2373815

JSTOR: links.jstor.org

[27] K. Ribet, Division fields of abelian varieties with complex multiplication, Mem. Soc. Math. France (N.S.) 2 (1980), 75–94.

Mathematical Reviews (MathSciNet): MR83e:14029a

Zentralblatt MATH: 0452.14009

[28] K. Ribet, Hodge classes on certain types of abelian varieties, Amer. J. Math. 105 (1983), no. 2, 523–538.

Mathematical Reviews (MathSciNet): MR85a:14030

Zentralblatt MATH: 0586.14003

Digital Object Identifier: doi:10.2307/2374267

JSTOR: links.jstor.org

[29] C. Schoen, Hodge classes on self-products of a variety with an automorphism, Compositio Math. 65 (1988), no. 1, 3–32.

Mathematical Reviews (MathSciNet): MR89c:14013

Zentralblatt MATH: 0663.14006

[30] J.-P. Serre, Sur les groupes de Galois attachés aux groupes $p$-divisibles, Proceedings of a Conference on Local Fields (Driebergen, 1966) ed. T. A. Springer, Springer, Berlin, 1967, pp. 118–131.

Mathematical Reviews (MathSciNet): MR39:4166

Zentralblatt MATH: 0189.02901

[31] J.-P. Serre, Représentations $\ell$-adiques, Algebraic Number Theory (Kyoto Internat. Sympos., Res. Inst. Math. Sci., Univ. Kyoto, Kyoto, 1976), University of Kyoto, Kyoto, 1977, pp. 177–193.

Mathematical Reviews (MathSciNet): MR57:16310

Zentralblatt MATH: 0406.14015

[32] J.-P. Serre, Groupes algébriques associés aux modules de Hodge-Tate, Journées de Géométrie Algébrique de Rennes. (Rennes, 1978), Vol. III, Astérisque, vol. 65, Soc. Math. France, Paris, 1979, pp. 155–187.

Mathematical Reviews (MathSciNet): MR81j:14027

Zentralblatt MATH: 0446.20028

[33] J.-P. Serre, 1 January 1981, Letter to K. Ribet.

[34] G. Shimura, On analytic families of polarized abelian varieties and automorphic functions, Ann. of Math. (2) 78 (1963), 149–192.

Mathematical Reviews (MathSciNet): MR27:5934

Zentralblatt MATH: 0142.05402

Digital Object Identifier: doi:10.2307/1970507

JSTOR: links.jstor.org

[35] G. Shimura, Introduction to the Arithmetic Theory of Automorphic Functions, vol. 11, Publ. Math. Soc. Japan, Princeton University Press, Princeton, 1971.

Mathematical Reviews (MathSciNet): MR47:3318

Zentralblatt MATH: 0221.10029

[36] G. Shioda, Algebraic cycles on abelian varieties of Fermat type, Math. Ann. 258 (1981), no. 1, 65–80.

Mathematical Reviews (MathSciNet): MR83f:14036

Zentralblatt MATH: 0515.14005

Digital Object Identifier: doi:10.1007/BF01450347

[37] S. G. Tankeev, On algebraic cycles on abelian varieties II, Math USSR-Izv. 14 (1980), 383–394.

Zentralblatt MATH: 0429.14014

Mathematical Reviews (MathSciNet): MR534601

[38] S. G. Tankeev, On algebraic cycles on surfaces and abelian varieties, Math USSR-Izv 18 (1982), 349–380.

Zentralblatt MATH: 0551.14010

Mathematical Reviews (MathSciNet): MR616226

[39] J. Tate, Algebraic cycles and poles of zeta functions, Arithmetical Algebraic Geometry (Proc. Conf. Purdue Univ., 1963), Harper and Row, New York, 1965, pp. 93–110.

Mathematical Reviews (MathSciNet): MR37:1371

Zentralblatt MATH: 0213.22804

[40] A. Weil, Abelian varieties and the Hodge ring, Scientific Works: Collected Papers, Vol. 3, Springer-Verlag, New York, 1980, corrected 2nd printing, pp. 421–429.

[41] Yu. G. Zarhin, Abelian varieties, $\ell$-adic representations and Lie algebras: Rank independence on $\ell$, Invent. Math. 55 (1979), no. 2, 165–176.

Mathematical Reviews (MathSciNet): MR81j:14024

Zentralblatt MATH: 0424.14015

Digital Object Identifier: doi:10.1007/BF01390088

[42] Yu. G. Zarhin, Weights of simple Lie algebras in the cohomology of algebraic varieties, Math. USSR-Izv. 24 (1985), 245–281.

Zentralblatt MATH: 0579.14019

[43] Yu. G. Zarhin, Linear semisimple Lie algebras containing an operator with small number of eigenvalues, Arch. Math. (Basel) 46 (1986), no. 6, 522–532.

Mathematical Reviews (MathSciNet): MR88a:17016

Zentralblatt MATH: 0592.17007

Digital Object Identifier: doi:10.1007/BF01195020

[44] Yu. G. Zarhin, Torsion of abelian varieties over $\rm GL(2)$-extensions of number fields, Math. Ann. 284 (1989), no. 4, 631–646.

Mathematical Reviews (MathSciNet): MR90i:11061

Zentralblatt MATH: 0661.14032

Digital Object Identifier: doi:10.1007/BF01443356

[45] Yu. G. Zarhin, Abelian varieties having a reduction of $K3$ type, Duke Math. J. 65 (1992), no. 3, 511–527.

Mathematical Reviews (MathSciNet): MR93a:14042

Zentralblatt MATH: 0774.14039

Digital Object Identifier: doi:10.1215/S0012-7094-92-06520-3

Project Euclid: euclid.dmj/1077295269

previous :: next

Duke Mathematical Journal

Turn MathJax Off
What is MathJax?