For a number field $K$ admitting an embedding into the field of real numbers $\mathbb{R}$, 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 *diamond* (or *power*) operation of raising to power $n$

 

$(x,n)  \mapsto  x^{\diamond n}:  H^1(K,G) \times Z   ---> H^1(K,G).$

We show that this operation has many functorial properties. When $G$ is a torus, the set $H^1(K,G)$ has a natural group structure, and $x^{\diamond n}$ coincides with the $n$-th power of $x$ in this group.

For a cohomology class $x$ in $H^1(K,G)$, we  define the period per$(x)$ to be the least $n>0$ such that $x^{\diamond n}=1$, 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 a central simple algebra over $K$ (in the special case where $G$ is the projective linear group $\mbox{PGL}_n$, the elements of $H^1(K, G)$ can be represented by central simple algebras). 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.

 

The talk is based on a joint work with Zinovy Reichstein. All necessary definitions will be given, including the definition of the Galois cohomology set $H^1(K,G)$.