The Power Operation in the Galois Cohomology of a Reductive Group Over a Global Field

Abstract For a connected reductive group $G$ over a local or global field $K$, we define a diamond (or power) operation $$ \begin{align*} &(\xi,n)\mapsto \xi^{\Diamond n}\,\colon\, \mathrm{H}^1\kern -0.8pt(K,G)\times{\mathbb Z}\to \mathrm{H}^1\kern -0.8pt(K,G)\end{align*} $$ of raising to power $n$ in the Galois cohomology pointed set. This operation is new when $K$ is a number field. We show that this power operation has many good properties. When $G$ is a torus, the set $\mathrm{H}^{1}\kern -0.8pt(K,G)$ has a natural group structure, and $\xi ^{\Diamond n}$ then coincides with the $n$-th power of $\xi $ in this group. On the other hand, we show that a power operation on $\mathrm{H}^{1}\kern -0.8pt(K,G)$, functorial in $G$, which we define over local and global fields, cannot be defined for an arbitrary field $K$. Our proof of this assertion relies on the results of Appendix B written by Philippe Gille. Using the power operation, for a cohomology class $\xi $ in $\mathrm{H}^{1}\kern -0.8pt(K,G)$ over local or global field, we define the period $\operatorname{per}(\xi )$ to be the least integer $n\geqslant 1$ such that $\xi ^{\Diamond n}=1$. We define the index $\operatorname{ind}(\xi )$ to be the greatest common divisor of the degrees $[L:K]$ of finite extensions $L/K$ splitting $\xi $. The period and index of a cohomology class generalize the period and index a central simple algebra over $K$. For any connected reductive group $G$ over a local or global field $K$, we show that $\operatorname{per}(\xi )$ divides $\operatorname{ind}(\xi )$ and that $\operatorname{ind}(\xi )$ may be strictly greater than $\operatorname{per}(\xi )$, but they always have the same prime factors.