Geodesic
On the Brauer group of an arithmetic scheme
Izvestiya. Mathematics , Tome 65 (2001) no. 2, pp. 357-388.

Voir la notice de l'article provenant de la source Math-Net.Ru

For an Enriques surface $V$ over a number field $k$ with a $k$-rational point we prove that the $l$-component of $\operatorname{Br}(V)/{\operatorname{Br}(k)}$ is finite if and only if $l\ne 2$. For a regular projective smooth variety satisfying the Tate conjecture for divisors over a number field, we find a simple criterion for the finiteness of the $l$-component of $\operatorname{Br}'(V)/{\operatorname{Br}(k)}$. Moreover, for an arithmetic model $X$ of $V$ we prove a variant of Artin's conjecture on the finiteness of the Brauer group of $X$. Applications to the finiteness of the $l$-components of Shafarevich–Tate groups are given. 
@article{IM2_2001_65_2_a4,
     author = {S. G. Tankeev},
     title = {On the {Brauer} group of an arithmetic scheme},
     journal = {Izvestiya. Mathematics },
     pages = {357--388},
     publisher = {mathdoc},
     volume = {65},
     number = {2},
     year = {2001},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/IM2_2001_65_2_a4/}
}
TY  - JOUR
AU  - S. G. Tankeev
TI  - On the Brauer group of an arithmetic scheme
JO  - Izvestiya. Mathematics 
PY  - 2001
SP  - 357
EP  - 388
VL  - 65
IS  - 2
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/IM2_2001_65_2_a4/
LA  - en
ID  - IM2_2001_65_2_a4
ER  - 
%0 Journal Article
%A S. G. Tankeev
%T On the Brauer group of an arithmetic scheme
%J Izvestiya. Mathematics 
%D 2001
%P 357-388
%V 65
%N 2
%I mathdoc
%U http://geodesic.mathdoc.fr/item/IM2_2001_65_2_a4/
%G en
%F IM2_2001_65_2_a4
S. G. Tankeev. On the Brauer group of an arithmetic scheme. Izvestiya. Mathematics , Tome 65 (2001) no. 2, pp. 357-388. http://geodesic.mathdoc.fr/item/IM2_2001_65_2_a4/

[1] Burbaki N., Kommutativnaya algebra, Mir, M., 1971 | MR

[2] Burbaki N., Gomologicheskaya algebra, Nauka, M., 1987 | MR

[3] Cassels J. W. S., “Diophantine equations with special reference to elliptic curves. Survey article”, J. London Math. Soc., 41 (1966), 193–291 | DOI | MR

[4] Kassels Dzh., Frelikh A., Algebraicheskaya teoriya chisel, Mir, M., 1969

[5] Greenberg M., “Rational points in Henselian discrete valuation rings”, Publ. Math. IHES, 31 (1966), 59–64 | MR

[6] Griffits F., Kharris Dzh., Printsipy algebraicheskoi geometrii, Mir, M., 1982 | MR

[7] Grothendieck A., “Eléments de géométrie algébrique. IV: Étude locale des schémas et des morphismes des schémas”, Publ. Math. IHES, 32 (1967) | Zbl

[8] Khartskhorn R., Algebraicheskaya geometriya, Mir, M., 1981 | MR | Zbl

[9] Kokh Kh., Teoriya Galua $p$-rasshirenii, Mir, M., 1973 | MR

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

[11] Miln Dzh., Etalnye kogomologii, Mir, M., 1983 | MR | Zbl

[12] Milne J. S., Arithmetic duality theorems, Academic Press, Inc., N.Y., 1986 | MR | Zbl

[13] Mamford D., Lektsii o krivykh na algebraicheskoi poverkhnosti, Mir, M., 1968

[14] Serr Zh. -P., Algebraicheskie gruppy i polya klassov, Mir, M., 1968 | MR

[15] Serr Zh. -P., Kurs arifmetiki, Mir, M., 1972 | MR | Zbl

[16] Tankeev S. G., “Ob algebraicheskikh tsiklakh na poverkhnostyakh i abelevykh mnogoobraziyakh”, Izv. AN SSSR. Ser. matem., 45:2 (1981), 398–434 | MR | Zbl

[17] Tankeev S. G., “Poverkhnosti tipa $K3$ nad chislovymi polyami i $l$-adicheskie predstavleniya”, Izv. AN SSSR. Ser. matem., 52:6 (1988), 1252–1271

[18] Tankeev S. G., On the Brauer group of arithmetic scheme, Preprint / Manuskripte der Forshergruppe Arithmetik, 7, Universität Mannheim–Universität Heidelberg, 1999 | MR

[19] Tankeev S. G., “O gruppe Brauera”, Izv. RAN. Ser. matem., 64:4 (2000), 141–162 | MR | Zbl

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

[21] Tate J., “On the conjectures of Birch and Swinnerton–Dyer and a geometric analog”, Séminaire Bourbaki, Exposé 306, 1965/66, 1–26 | Zbl