Let K be a number field, G the absolute Galois group of K. Let M be a finite G-module. Let H^1(G, M ) be the first cohomology group of G with coefficients in M . In general, the group H^1(G, M ) is not finitely generated, which can make

it difficult to work with. A Selmer group is a particular subgroup of H^1(G, M ), finitely generated, and defined in such a way that it keeps some local properties of H^1(G, M ).

In this talk, I will give a definition of Selmer groups, and some examples of applications. Then I will explain a method to compute Selmer groups, using the properties of Hecke operators