Galois Cohomology

Galois cohomology is the basic language used for much research into algebraic aspects of the BSD conjecture. It was introduced by Lang and Tate in 1958 in [LT58]. This section contains a survey of the basic facts we will need in order to define Shafarevich-Tate groups, discuss descent, and construct Kolyvagin's cohomology classes.

The best basic reference on Galois cohomology is chapters VII and X of Serre's Local Fields [Ser79] or the (very similar!) article by Atiyah and Wall in Cassels-Frohlich [Cp86, Ch. IV]. See also the article by Gruenberg in [Cp86, Ch. V] for an introduction to profinite groups such as $\Gal (\overline{\mathbb{Q}}/\mathbb{Q})$. Since this section is only a survey, you should read one of the above two references in detail, if you haven't already. You might also want to read Chapter 1 of [CS00] by Coates and Sujatha, which contains an excellent summary of more advanced topics in Galois cohomology, and Serre's book Galois Cohomology [Ser97] discusses many general advanced topics in depth. The original article [LT58] is also well worth reading.