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 Uspekhi Mat. Nauk, 1972, Volume 27, Issue 6(168), Pages 25–66 (Mi umn5139)

The cohomology of Abelian varieties over a nuvber field

M. I. Bashmakov

Abstract: This article is a survey of results on the arithmetic of Abelian varieties that have been obtained by cohomological methods. It consists of an Introduction and six sections. In the Introduction the main facts to be proved in the article are stated. They are concentrated around two arithmetical problems: the determination of the rank of an Abelian variety over a number field and the related problem of the structure of the group of locally trivial principal homogeneous spaces (the Tate–Shafarevich group); also the investigation of the behaviour of points of finite order on an Abelian variety and the related problem of divisibility of principal homogeneous spaces. The first section recalls the proofs of the necessary facts from the Galois cohomology of finite modules. The basic results relating to the first of the problems mentioned are proved in §§ 3–4. The fifth and sixth sections are devoted to the problem of the divisibility of points and of principal homogeneous spaces; a certain cohomological fmiteness theorem is also proved here.

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English version:
Russian Mathematical Surveys, 1972, 27:6, 25–70

Bibliographic databases:

UDC: 513.013+513.83
MSC: 11R34, 11G15, 14K05, 14K22, 14K02

Citation: M. I. Bashmakov, “The cohomology of Abelian varieties over a nuvber field”, Uspekhi Mat. Nauk, 27:6(168) (1972), 25–66; Russian Math. Surveys, 27:6 (1972), 25–70

Citation in format AMSBIB
\Bibitem{Bas72} \by M.~I.~Bashmakov \paper The cohomology of Abelian varieties over a~nuvber field \jour Uspekhi Mat. Nauk \yr 1972 \vol 27 \issue 6(168) \pages 25--66 \mathnet{http://mi.mathnet.ru/umn5139} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=399110} \zmath{https://zbmath.org/?q=an:0256.14016} \transl \jour Russian Math. Surveys \yr 1972 \vol 27 \issue 6 \pages 25--70 \crossref{https://doi.org/10.1070/RM1972v027n06ABEH001392} 

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Citing articles on Google Scholar: Russian citations, English citations
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This publication is cited in the following articles:
1. O. Neumann, “On $p$-closed algebraic number fields with restricted ramification”, Math. USSR-Izv., 9:2 (1975), 243–254
2. Daniel Bertrand, “Sous-groupes à un paramètrep-adique de variétés de groupe”, Invent math, 40:2 (1977), 171
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4. R. A. Sarkisyan, “Galois cohomology and some questions of the theory of algorithms”, Math. USSR-Sb., 39:4 (1981), 519–545
5. D. Bertrand, “Kummer theory on the product of an elliptic curve by the multiplicative group”, Glasgow Math J, 22:1 (1981), 83
6. Hans Opolka, “Eine Bemerkung zur Konstruktion von Galoisdarstellungen”, Arch Math, 39:6 (1982), 551
7. Rajiv Gupta, “Ramification in the Coates-Wiles tower”, Invent math, 81:1 (1985), 59
8. V. A. Kolyvagin, “Finiteness of $E(\mathbf Q)$ and $Ø(E,\mathbf Q)$ for a subclass of Weil curves”, Math. USSR-Izv., 32:3 (1989), 523–541
9. Bernadette Perrin-Riou, “Théorie d'Iwasawap-adique locale et globale”, Invent math, 99:1 (1990), 247
10. Hans Opolka, “Projective vectors of complex galois representations”, Communications in Algebra, 19:1 (1991), 125
11. I. S. Rakhimov, “Arithmetic Invariants for a Class of Elliptic Curves”, Siberian Adv. Math., 14:2 (2004), 79–91
12. Hoseog Yu, “On Tate-Shafarevich groups over galois extensions”, Isr J Math, 141:1 (2004), 211
13. I. S. Rakhimov, “O povedenii arifmeticheskikh invariantov nekotorogo klassa ellipticheskikh krivykh v krugovykh $\Gamma$-rasshireniyakh”, Matem. tr., 8:1 (2005), 122–134
14. Hans Opolka, “A Note on Regular Crossed Products and Galois Representations”, Communications in Algebra, 35:5 (2007), 1469
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