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

Abstract

For a connected reductive group $G$ over a local or global field $K$, we define a *diamond* (or *power*) operation $$(\xi,n)\mapsto \xi^{\Diamond n}\,\colon\, H^1(K,G)\times {\mathbb Z}\to H^1(K,G)$$ 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 $H^1(K,G)$ 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 $H^1(K,G)$, 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 this power operation, for a cohomology class $\xi$ in $H^1(K,G)$ over local or global field, we define the period ${\rm per}(\xi)$ to be the least integer $n\ge 1$ such that $\xi^{\Diamond n}=1$. We define the index ${\rm ind}(\xi)$ 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 $G$ defined over a local or global field $K$, we show that ${\rm per}(\xi)$ divides ${\rm ind}(\xi)$ and that ${\rm ind}(\xi)$ may be strictly greater than ${\rm per}(\xi)$, but they always have the same prime factors.