The cohomology of Abelian varieties over a nuvber field

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.