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 $\mathbb{R}$, 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 power operation of raising to power $n$:
$$(x,n) \mapsto x^n : H^1(K,G) \times \mathbb{Z} \longrightarrow H^1(K,G).$$
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 $\operatorname{per}(x)$ to be the least integer $m>0$ such that $x^m=1$, and the index $\operatorname{ind}(x)$ 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 $K$ (in the special case where $G$ is the projective linear group $\mathrm{PGL}_n$, 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 $\operatorname{per}(x)$ divides $\operatorname{ind}(x)$, that $\operatorname{per}(x)$ and $\operatorname{ind}(x)$ have the same prime factors, but the equality $\operatorname{per}(x) = \operatorname{ind}(x)$ may not hold.

The talk is based on joint work with Zinovy Reichstein and Philippe Gille.