Tate classes on self-products of Abelian varieties over finite fields
Annales de l'Institut Fourier, Volume 72 (2022) no. 6, pp. 2339-2383.

We deal with g-dimensional abelian varieties X over finite fields. We prove that there is a universal constant (positive integer) N=N(g) that depends only on g that enjoys the following property. If a certain self-product of X carries an exotic Tate class then the self-product X 2N of X also carries an exotic Tate class. This gives a positive answer to a question of Kiran Kedlaya.

Nous étudions des variétés abéliennes X de dimension g sur des corps finis. Nous prouvons l’existence d’une constante universelle (entière positive) N=N(g), qui ne dépend que de g et a la propriété suivante : si une certaine puissance de X admet une classe de Tate exotique, la puissance X 2N de X admet une classe de Tate exotique aussi. Cela donne une réponse positive à une question de Kiran Kedlaya.

Received:
Revised:
Accepted:
Published online:
DOI: 10.5802/aif.3483
Classification: 11G10, 11G25, 14G15
Keywords: Abelian varieties, Tate classes, Finite fields
Mot clés : Variétés abéliennes, classes de Tate, corps finis

Zarhin, Yuri G. 1

1 Department of Mathematics, Pennsylvania State University, University Park, PA 16802, USA
License: CC-BY-ND 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
@article{AIF_2022__72_6_2339_0,
     author = {Zarhin, Yuri G.},
     title = {Tate classes on self-products of {Abelian} varieties over finite fields},
     journal = {Annales de l'Institut Fourier},
     pages = {2339--2383},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {72},
     number = {6},
     year = {2022},
     doi = {10.5802/aif.3483},
     language = {en},
     url = {https://aif.centre-mersenne.org/articles/10.5802/aif.3483/}
}
TY  - JOUR
AU  - Zarhin, Yuri G.
TI  - Tate classes on self-products of Abelian varieties over finite fields
JO  - Annales de l'Institut Fourier
PY  - 2022
SP  - 2339
EP  - 2383
VL  - 72
IS  - 6
PB  - Association des Annales de l’institut Fourier
UR  - https://aif.centre-mersenne.org/articles/10.5802/aif.3483/
DO  - 10.5802/aif.3483
LA  - en
ID  - AIF_2022__72_6_2339_0
ER  - 
%0 Journal Article
%A Zarhin, Yuri G.
%T Tate classes on self-products of Abelian varieties over finite fields
%J Annales de l'Institut Fourier
%D 2022
%P 2339-2383
%V 72
%N 6
%I Association des Annales de l’institut Fourier
%U https://aif.centre-mersenne.org/articles/10.5802/aif.3483/
%R 10.5802/aif.3483
%G en
%F AIF_2022__72_6_2339_0
Zarhin, Yuri G. Tate classes on self-products of Abelian varieties over finite fields. Annales de l'Institut Fourier, Volume 72 (2022) no. 6, pp. 2339-2383. doi : 10.5802/aif.3483. https://aif.centre-mersenne.org/articles/10.5802/aif.3483/

[1] Berkovich, Vladimir G. The Brauer group of abelian varieties, Funct. Anal. Appl., Volume 6 (1973), pp. 180-184 | Zbl

[2] Bourbaki, Nicolas Elements of mathematics. Algebra I, Chapters 1-3, Springer, 1989 | Zbl

[3] Cox, David A.; Little, John B.; Schenk, Henry K. Toric varieties, Graduate Studies in Mathematics, 124, American Mathematical Society, 2011 | Zbl

[4] Katz, Nicholas M. Review of l-adic cohomology, Motives, part 1. Proceedings of the summer research conference on motives, (Seattle, WA, 1991) (Jannsen, U.; Kleiman, S.; Serre, J.-P., eds.) (Proceedings of Symposia in Pure Mathematics), Volume 55, American Mathematical Society, 1994, pp. 21-30 | Zbl

[5] Kleiman, Steven L. Algebraic cycles and the Weil conjecture, Dix Exposés sur la Cohomologie des Schémas (Grothendieck, A.; Kuiper, N. H., eds.) (Advanced Studies in Pure Mathematics), Volume 3, North-Holland, 1968, pp. 359-386 | Zbl

[6] Lenstra, Herndrik W. Jr.; Zarhin, Yuriĭ G. The Tate conjecture for almost ordinary abelian varieties over finite fields, Advances in Number Theory (CNTA 91 Conference Proceedings, Kingston, ON, 1991) (Gouvea, Fernando Q.; Yui, N., eds.), Oxford University Press, 1993, pp. 179-194 | Zbl

[7] Milne, James S. Abelian Varieties, version 2.0, 2008 (https://www.jmilne.org/math/CourseNotes/av.html)

[8] Mumford, David Abelian varieties, Tata Institute of Fundamental Research Studies in Mathematics, 5, Oxford University Press, 1974 | Zbl

[9] Oort, Frans Abelian varieties over finite fields, Higher-dimensional varieties over finite fields (Kaledin, D.; Tschinkel, Yu., eds.) (NATO Science for Peace and Security Series D: Information and Communication Security), Volume 16, IOS Press, 2008, pp. 123-188 | Zbl

[10] Serre, Jean-Pierre Abelian -adic representations and elliptic curves, Advanced Book Classics, Addison-Wesley Publishing Group, 1989 | Zbl

[11] Serre, Jean-Pierre Lettres á Ken Ribet du 1/1/1981 et du 29/1/1981, Œuvres, IV, 1985–1998, Springer, 2000, pp. 1-20 | Zbl

[12] Silverberg, Alice; Zarhin, Yuriĭ G. Connectedness results for -adic representations associated to abelian varieties, Compos. Math., Volume 97 (1995) no. 1-2, pp. 273-284 | Zbl

[13] Tate, J. T. Conjectures on algebraic cycles in -adic cohomology, Motives, part 1. Proceedings of the summer research conference on motives, (Seattle, WA, 1991) (Jannsen, Uwe; Kleiman, S.; Serre, J.-P., eds.) (Proceedings of Symposia in Pure Mathematics), Volume 55, American Mathematical Society, 1994, pp. 71-83 | DOI | Zbl

[14] Tate, John T. Algebraic cycles and poles of zeta functions, Arithmetical Algebraic Geometry (Proc. Conf. Purdue Univ., 1963) (Schilling, O. F. G., ed.), Harper & Row, 1965, pp. 93-110 | Zbl

[15] Tate, John T. Endomorphisms of abelian varieties over finite field, Invent. Math., Volume 2 (1966), pp. 134-144 | Zbl

[16] Tate, John T. Classes d’isogénie de variétés abéliennes sur un corps fini (d’après T. Honda), Séminaire Bourbaki: Vol. 1968/69. Exposés 347 - 363 (Bourbaki, N., ed.) (Lecture Notes in Mathematics), Volume 179, Springer, 1971, pp. 95-110 | Zbl

[17] Zarhin, Yuriĭ G. Abelian varieties, -adic representations and Lie algebras. Rank independence on , Invent. Math., Volume 55 (1979), pp. 165-176 | Zbl

[18] Zarhin, Yuriĭ G. Abelian varieties, -adic representations and SL 2 , Math. USSR Izv., Volume 14 (1980), pp. 275-288 | Zbl

[19] Zarhin, Yuriĭ G. Abelian varieties of K3 type and -adic representations, Algebraic Geometry and Analytic Geometry, (Tokyo,1990) (Fujiki, A.; Kato, K.; Katsura, T.; Kawamata, Y.; Miyaoka, Y., eds.), Springer, 1991, pp. 231-255 | Zbl

[20] Zarhin, Yuriĭ G. Abelian varieties of K3 type, Séminaire de Théorie des Nombres, Paris, 1990-91 (Sinnou, David, ed.) (Progress in Mathematics), Birkhäuser, 1993, pp. 263-279 | Zbl

[21] Zarhin, Yuriĭ G. The Tate conjecture for non-simple abelian varieties over finite fields, Algebra and Number Theory. Proceedings of a conference held at the Institute of Experimental Mathematics, University of Essen, Germany, December 2-4, 1992 (Frey, G.; Ritter, J., eds.), Walter de Gruyter, 1994, pp. 267-296 | Zbl

[22] Zarhin, Yuriĭ G. Eigenvalues of Frobenius Endomorphisms of Abelian varieties of low dimension, J. Pure Applied Algebra, Volume 219 (2015) no. 6, pp. 2076-2098 | Zbl

Cited by Sources: