Geodesic
On the Conjectures of Artin and Shafarevich--Tate
Informatics and Automation, Number theory, algebra, and algebraic geometry, Tome 241 (2003), pp. 254-264.

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

For an arithmetic model $\pi\colon X\to\operatorname{Spec}A$ of a smooth projective variety $V$ over a number field $k$, the interrelations between the conjecture of Artin about the finiteness of $\mathrm{Br}(X)$ and the conjecture of Shafarevich–Tate about the finiteness of $\text{III}(\operatorname {Spec}A,\mathrm{Pic}^0(V))$ are studied. 
@article{TRSPY_2003_241_a13,
     author = {S. G. Tankeev},
     title = {On the {Conjectures} of {Artin} and {Shafarevich--Tate}},
     journal = {Informatics and Automation},
     pages = {254--264},
     publisher = {mathdoc},
     volume = {241},
     year = {2003},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TRSPY_2003_241_a13/}
}
TY  - JOUR
AU  - S. G. Tankeev
TI  - On the Conjectures of Artin and Shafarevich--Tate
JO  - Informatics and Automation
PY  - 2003
SP  - 254
EP  - 264
VL  - 241
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/TRSPY_2003_241_a13/
LA  - ru
ID  - TRSPY_2003_241_a13
ER  - 
%0 Journal Article
%A S. G. Tankeev
%T On the Conjectures of Artin and Shafarevich--Tate
%J Informatics and Automation
%D 2003
%P 254-264
%V 241
%I mathdoc
%U http://geodesic.mathdoc.fr/item/TRSPY_2003_241_a13/
%G ru
%F TRSPY_2003_241_a13
S. G. Tankeev. On the Conjectures of Artin and Shafarevich--Tate. Informatics and Automation, Number theory, algebra, and algebraic geometry, Tome 241 (2003), pp. 254-264. http://geodesic.mathdoc.fr/item/TRSPY_2003_241_a13/

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

[2] Kassels Dzh., Frelikh A., Algebraicheskaya teoriya chisel, Mir, M., 1969 | MR

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

[4] Gonzalez-Aviles C. D., Brauer groups and Tate–Shafarevich groups, , 24 Apr. 2001 http:// arxiv.org/abs/math.AG/0104214 | MR

[5] Grothendieck A., “Le groupe de Brauer, III”, Dix exposés sur la cohomologie des schémas, North-Holland, Masson, Amsterdam, Paris, 1968, 88–188 | MR

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

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

[8] Milne J. S., “Comparison of the Brauer group with the Tate–Shafarevich group”, J. Fac. Univ. Tokyo. Sect. IA., 28:3 (1981), 735–743 | MR | Zbl

[9] Milne J. S., Arithmetic duality theorems, Acad. Press, New York, 1986 | MR | Zbl

[10] Serr Zh.-P., Kogomologii Galua, Mir, M., 1968 | MR

[11] Stuhler U., “On the connection between Brauer group and Tate–Shafarevich group”, J. Algebra, 70 (1981), 141–155 | DOI | MR | Zbl

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

[13] Tankeev S. G., “O gruppe Brauera arifmeticheskoi skhemy”, Izv. RAN. Ser. mat., 65:2 (2001), 155–186 | MR | Zbl

[14] Tankeev S. G., “O gruppe Brauera arifmeticheskoi skhemy, II”, Izv. RAN. Ser. mat., 67:5 (2003), 155–176 | MR | Zbl

[15] Tate J., On the conjectures of Birch and Swinnerton–Dyer and a geometric analog, Sém. Bourbaki 1965/66. Exp. 306, 26 pp.; Dix Exposes Cohomologie Schemas, Advanced Studies Pure Math., 3, 1968, 189–214

[16] Veil A., Osnovy teorii chisel, Mir, M., 1972 | MR