Galois cohomology. (English) Zbl 0928.12003
Cornell, Gary (ed.) et al., Modular forms and Fermat’s last theorem. Papers from a conference, Boston, MA, USA, August 9–18, 1995. New York, NY: Springer. 101-120 (1997).
This paper gives a description of the Galois cohomology needed in Wiles’s proof of the Shimura-Taniyama-Weil conjecture for semi-stable elliptic curves.
First, the author briefly introduces the groups \(H^0\), \(H^1\), \(H^2\), and surveys some preliminary results as the inflation-restriction exact sequence and Tate’s local duality theorem. The main reason that Galois cohomology arises in Wiles’s proof is that certain cohomology groups can be used to classify deformations of Galois representations and the restrictions imposed to the type of deformation can be seen as cohomology classes lying in certain cohomological subsets, this is explained in section 4. Well chosen examples to illustrate the concepts and results are given.
Next, the author presents the generalized Selmer groups attached to a collection of local conditions and he proves the crucial formula relating the orders of finite Selmer groups and of cohomology groups, stated as theorem 2 in the paper. To show how this formula may be used, he gives an application in a concrete setting and he obtains a proof of the Kronecker-Weber theorem using these techniques, which, in fact, needs the power of class field theory to prove theorem 2.
For the entire collection see [Zbl 0878.11004].
First, the author briefly introduces the groups \(H^0\), \(H^1\), \(H^2\), and surveys some preliminary results as the inflation-restriction exact sequence and Tate’s local duality theorem. The main reason that Galois cohomology arises in Wiles’s proof is that certain cohomology groups can be used to classify deformations of Galois representations and the restrictions imposed to the type of deformation can be seen as cohomology classes lying in certain cohomological subsets, this is explained in section 4. Well chosen examples to illustrate the concepts and results are given.
Next, the author presents the generalized Selmer groups attached to a collection of local conditions and he proves the crucial formula relating the orders of finite Selmer groups and of cohomology groups, stated as theorem 2 in the paper. To show how this formula may be used, he gives an application in a concrete setting and he obtains a proof of the Kronecker-Weber theorem using these techniques, which, in fact, needs the power of class field theory to prove theorem 2.
For the entire collection see [Zbl 0878.11004].
Reviewer: N.Vila (Barcelona)
