The Brauer group of an Abelian variety over a~finite field

Yu. G. Zarhin

Izvestiya. Mathematics , Tome 20 (1983) no. 2, pp. 203-234.

Résumé

The author presents a formula for the order of a component of the Brauer group of an Abelian variety over a finite field, where the order of the component in question is relatively prime to the characteristic of the field. For principally polarized Abelian surfaces this formula becomes the well-known Artin–Tate formula. A natural nondegenerate pairing between the components of the Brauer groups of an Abelian variety and its Picard variety is constructed. Bibliography: 27 titles.

Bibliographie

[1] Berkovich V. G., “O gruppe Brauera abelevykh mnogoobrazii”, Funkts. analiz i ego prilozh., 6:3 (1972), 10–15 | MR | Zbl

[2] Voskresenskii V. E., Algebraicheskie tory, Nauka, M, 1977 | MR

[3] Grothendieck A., “Le groupe de Brauer. I, II, III”, Dix exposes sur la cohomologie des schemas, North Holland, Amsterdam, 1968, 46–188 | MR

[4] Delin P., “Gipoteza Veilya. I”, Uspekhi matem. nauk, 30:5 (1975), 159–190 | MR | Zbl

[5] Hoobler R., “Brauer groups of abelian schemes”, Ann. Sci. École Norm. Sup. (4), 5:1 (1972), 45–70 | MR | Zbl

[6] Hoobler R., “Cohomology of purely inseparable Galois covering”, J. Reine Angew. Math., 266 (1974), 183–199 | MR | Zbl

[7] Lang S., Abelian varieties, Interscience, New York, 1959 | MR

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

[9] Manin Yu. I., Kubicheskie formy, Nauka, M., 1972 | MR

[10] Manin Yu. I., “Le groupe de Brauer–Grothendieck en Géometrie diophantienne”, Actes de Congres International de Mathematicians (Nice, 1970), 1, Gauthier–Villars, Paris, 1971, 401–411 | MR

[11] Milne J., Etale cohomology, Princeton University Press, Princeton, 1980 | MR | Zbl

[12] Milne J., “On a conjecture of Artin and Tate”, Ann. Math., 102:2 (1975), 517–533 | DOI | MR | Zbl

[13] Milne J., “The Tate–Safarevic group of a constant abelian variety”, Invent. Math., 6:1 (1968), 91–105 | DOI | MR | Zbl

[14] Milne J., “Extension of abelian varieties over finite fields”, Invent. Math., 5:1 (1968), 63–84 | DOI | MR | Zbl

[15] Oda T., “The first de Rham cohomology and Dieudonne modules”, Ann. scient. École Norm. Sup. (4), 2:1 (1969), 63–135 | MR | Zbl

[16] Cepp Zh.-P., Kogomologii Galua, Mir, M., 1968 | MR

[17] Tate J., “Algebraic cycles and poles of zeta-functions”, Arithmetical Algebraic Geometry, Harper and Row, New York, 1965, 93–110 | MR

[18] Teit Dzh., “O gipotezakh Bercha i Svinnertona-Daiera i ikh geometricheskom analoge”, Matematika, 12:6 (1968), 41–55

[19] Teit Dzh., “Endomorfizmy abelevykh mnogoobrazii nad konechnymi polyami”, Matematika, 12:6 (1968), 31–40

[20] Tate J., “Relations between $K_2$ and Galois cohomology”, Invent. Math., 36 (1976), 257–274 | DOI | MR | Zbl

[21] Artin M., Grothendieck A., Verdier J.-L., Theorie de topos et cohomologie étale de schemas (SGA4), tome 3, Lect. Notes Math., 305, Springer–Verlag, 1973 | MR | Zbl

[22] Artin M., Swinnerton-Dyer H. P. F., “The Shafarevich–Tate conjecture for pencils of elliptic curves on $K3$ surfaces”, Invent. Math., 20:3 (1973), 249–266 | DOI | MR | Zbl

[23] Ogus A., “Supersingular $K3$ crystals”, Asterisque, 64 (1979), 3–86 | MR | Zbl

[24] Khyuit E., Ross K., Abstraktnyi garmonicheskii analiz, t. I, Nauka, M., 1975

[25] Kleiman S. L., “Algebraic cycles and the Weyl conjectures”, Dix exposés sur la cohomologie des schémas, North–Holland, Amsterdam, 1968, 359–386 | MR

[26] Schneider P., On the values of the zeta-function of a variety over a finite field, IHES, preprint, 1981 | MR

[27] Ribet K. A., “Galois action on division points of abelian varieties with real multiplications”, Amer. J. Math., 98:3 (1976), 751–804 | DOI | MR | Zbl