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

This article is cited in 18 scientific papers (total in 18 papers)

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
Received: 13.06.1972

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
    Related articles on Google Scholar: Russian articles, English articles

    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  mathnet  crossref  mathscinet  zmath
    2. Daniel Bertrand, “Sous-groupes à un paramètrep-adique de variétés de groupe”, Invent math, 40:2 (1977), 171  crossref  mathscinet  zmath
    3. O. N. Vvedenskii, “On pairings in elliptic curves over global fields”, Math. USSR-Izv., 12:2 (1978), 225–246  mathnet  crossref  mathscinet  zmath
    4. R. A. Sarkisyan, “Galois cohomology and some questions of the theory of algorithms”, Math. USSR-Sb., 39:4 (1981), 519–545  mathnet  crossref  mathscinet  zmath  isi
    5. D. Bertrand, “Kummer theory on the product of an elliptic curve by the multiplicative group”, Glasgow Math J, 22:1 (1981), 83  crossref  mathscinet  zmath
    6. Hans Opolka, “Eine Bemerkung zur Konstruktion von Galoisdarstellungen”, Arch Math, 39:6 (1982), 551  crossref  mathscinet  zmath  isi
    7. Rajiv Gupta, “Ramification in the Coates-Wiles tower”, Invent math, 81:1 (1985), 59  crossref  mathscinet  zmath  isi
    8. V. A. Kolyvagin, “Finiteness of $E(\mathbf Q)$ and $Ш(E,\mathbf Q)$ for a subclass of Weil curves”, Math. USSR-Izv., 32:3 (1989), 523–541  mathnet  crossref  mathscinet  zmath
    9. Bernadette Perrin-Riou, “Théorie d'Iwasawap-adique locale et globale”, Invent math, 99:1 (1990), 247  crossref  mathscinet  zmath  isi
    10. Hans Opolka, “Projective vectors of complex galois representations”, Communications in Algebra, 19:1 (1991), 125  crossref
    11. I. S. Rakhimov, “Arithmetic Invariants for a Class of Elliptic Curves”, Siberian Adv. Math., 14:2 (2004), 79–91  mathnet  mathscinet  zmath  elib
    12. Hoseog Yu, “On Tate-Shafarevich groups over galois extensions”, Isr J Math, 141:1 (2004), 211  crossref  mathscinet  zmath  isi
    13. I. S. Rakhimov, “O povedenii arifmeticheskikh invariantov nekotorogo klassa ellipticheskikh krivykh v krugovykh $\Gamma$-rasshireniyakh”, Matem. tr., 8:1 (2005), 122–134  mathnet  mathscinet  zmath  elib
    14. Hans Opolka, “A Note on Regular Crossed Products and Galois Representations”, Communications in Algebra, 35:5 (2007), 1469  crossref
    15. Misha Gavrilovich, “A remark on transitivity of Galois action on the set of uniquely divisible abelian extensions in
      $$Ext^1(E({\overline {{\mathbb {Q}}}}),\Lambda)$$
      ”, K-Theory, 38:2 (2008), 135  crossref  mathscinet  zmath  isi
    16. Ciperiani M., Krashen D., “Relative Brauer Groups of Genus 1 Curves”, Isr. J. Math., 192:2 (2012), 921–949  crossref  isi
    17. Mirela Çiperiani, Jakob Stix, “Weil–Châtelet divisible elements in Tate–Shafarevich groups I: The Bashmakov problem for elliptic curves over”, Compositio Math, 2013, 1  crossref
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