On Frobenius traces

S. G. Tankeev

Izvestiya. Mathematics , Tome 62 (1998) no. 1, pp. 157-190.

Résumé

In this paper we discuss a certain Diophantine property of Frobenius traces associated with an Abelian variety over a number field $k$ and apply it to prove the Mumford–Tate conjecture for 4$p$-dimensional Abelian varieties $J$ over $k$, where $p$ is a prime number, $p\geqslant 17$, or (under certain weak assumptions) $\operatorname{End}^0(J\otimes\overline k)$ is an imaginary quadratic extension of $\mathbb Q$.

Bibliographie

[1] Bogomolov F. A., “Sur l'algebricité des représentations $l$-adiques”, C. R. Acad. Sci. Paris, 290 (1980), 701–703 | MR | Zbl

[2] Borovoi M. V., “Skhemy Shimury–Delinya $M_{\mathbb C}(G,h)$ i ratsionalnye klassy kogomologii tipa $(p,p)$ na abelevykh mnogoobraziyakh”, Voprosy teorii grupp i gomologicheskoi algebry, Izd-vo YarGU, Yaroslavl, 1977, 3–53 | MR

[3] Burbaki N., Gruppy i algebry Li, gl. 1–3, Mir, M., 1976 ; гл. 4–6, 1972 ; гл. 7–8, 1978 | MR | Zbl

[4] Algebraic number theory, Proc. International Conf. (Brighton, 1965), eds. Cassels J. W. S., Frohlich A., Academic Press, London; Thompson, Washington, D.C., 1967 | MR

[5] Chi W., “On the $l$-adic representations attached to simple abelian varieties of type IV”, Bull. Austral. Math. Soc., 44 (1991), 71–78 | DOI | MR | Zbl

[6] Chi W., “$l$-adic and $\lambda $-adic representations associated to abelian varieties defined over number fields”, Amer. J. Math., 114 (1992), 315–353 | DOI | MR | Zbl

[7] Deligne P., “Variétés abéliennes ordinaires sur un corps fini”, Invent. Math., 8 (1969), 238–243 | DOI | MR | Zbl

[8] Faltings G., “Endlichkeitssätze für abelsche Varietäten über Zahlkörpern”, Invent. Math., 73 (1983), 349–366 | DOI | MR | Zbl

[9] Grothendieck A., “Hodge's general conjecture is false for trivial reasons”, Topology, 8 (1969), 299–303 | DOI | MR | Zbl

[10] Hodge W. V. D., “The topological invariants of algebraic varieties”, Proc. Internat. Congr. Math., V. 1 (Cambridge, MA, 1950), Amer. Math. Soc., Providence, RI, 1952, 182–192 | MR

[11] Mumford D., “Families of abelian varieties”, Algebraic Groups and Discontinuous Subgroups, Proc. Sympos. Pure Math., 9, Amer. Math. Soc., Providence, RI, 1966, 347–352 | MR

[12] Mamford D., Abelevy mnogoobraziya, Mir, M., 1971

[13] Ogus A., “Hodge cycles and crystalline cohomology”, Hodge cycles, motives, and Shimura varieties, Lecture Notes in Math., 900, 1982, 357–414 | MR | Zbl

[14] Pyatetskii-Shapiro I. I., “Vzaimootnosheniya mezhdu gipotezami Teita i Khodzha dlya abelevykh mnogoobrazii”, Matem. sb., 85 (1971), 610–620 | MR

[15] Serre J.-P., Abelian $l$-adic representations and elliptic curves, Benjamin, N. Y., 1968 | MR | Zbl

[16] Shimura G., “On analytic families of polarized abelian varieties and automorphic functions”, Ann. Math., 78 (1963), 149–192 | DOI | MR | Zbl

[17] Tankeev S. G., “Tsikly na prostykh abelevykh mnogoobraziyakh prostoi razmernosti nad chislovymi polyami”, Izv. AN SSSR. Ser. matem., 51:6 (1987), 1214–1227 | MR

[18] Tankeev S. G., “Algebraicheskie tsikly na abelevom mnogoobrazii bez kompleksnogo umnozheniya”, Izv. RAN. Ser. matem., 58:3 (1994), 103–126 | MR | Zbl

[19] Tankeev S. G., “Tsikly na abelevykh mnogoobraziyakh i isklyuchitelnye chisla”, Izv. RAN. Ser. matem., 60:2 (1996), 159–194 | MR | Zbl

[20] Tankeev S. G., Algebraicheskie tsikly na prostom $4p$-mernom abelevom mnogoobrazii nad chislovym polem (to appear)

[21] Tate J., “Algebraic cycles and poles of zeta functions”, Arithmetical Algebraic Geometry, Harper and Row, N. Y., 1965, 93–110 | MR

[22] Tate J., “Endomorphisms of abelian varieties over finite fields”, Invent. Math., 2 (1966), 134–144 | DOI | MR | Zbl

[23] Zarkhin Yu. G., “Vesa prostykh algebr Li v kogomologiyakh algebraicheskikh mnogoobrazii”, Izv. AN SSSR. Ser. matem., 48:2 (1984), 264–304 | MR | Zbl