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

Mikhail Borovoi, Zinovy Reichstein

Published 2024-03-12Version 1

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 operation has many functorial properties. When $G$ is a torus, the ponted set $H^1(K,G)$ has a natural group structure, and our new operation coincides with the usual power operation $(\xi,n)\mapsto \xi^n$. For a cohomology class $\xi$ in $H^1(K,G)$, we define the period ${\rm per}(\xi)$ to be the greatest common divisor of $n\ge 1$ such that $\xi^{\Diamond n}=1$, and the index ${\rm ind}(\xi)$ to be the greatest common divisor of the degrees $[L:K]$ of finite separable extensions $L/K$ splitting $\xi$. We show that ${\rm per}(\xi)$ divides ${\rm ind}(\xi)$, but they need not be equal. However, ${\rm per}(\xi)$ and ${\rm ind}(\xi)$ have the same set of prime factors.