The power operation in the Galois cohomology of a reductive group over a number field

For a number field K admitting an embedding into the field of real numbers R, it is impossible to construct a functorial in G group structure in the Galois cohomology pointed set H^1(K,G) for all connected reductive K-groups G. However, over an arbitrary number field K, we define a *power operation* of raising to power n (x,n) \mapsto x^n: H^1(K,G) \times Z ---> H^1(K,G). We show that our power operation has many functorial properties. When G is a torus, the set H^1(K,G) has a natural group structure, and our x^n coincides with the n-th power of x in this group. Such a power operation with good properties exists and is unique when K is a local or global field, and cannot be defined over an arbitrary field.For a cohomology class x in H^1(K,G), we define the period per(x) to be the least integer m>0 such that x^m=1, and the index ind(x) to be the greatest common divisor of the degrees [L:K] of finite separable extensions L/K splitting x. These period and index generalize the period and index of a central simple algebra over K. For an arbitrary reductive group G defined over a local or global field K, we show that per(x) divides ind(x), that per(x) and ind(x) have the same prime factors, but the equality per(x)=ind(x) may not hold. All terms will be explained and examples will be given. The talk is based on a joint work with Zinovy Reichstein and Philippe Gille.