Is there a group structure on the Galois cohomology of a reductive group over a global field?

Abstract

Let K be a global field, that is, a number field or a global function field. It is known that the answer to the question in the title over K is “Yes” when K has no real embeddings. We show that otherwise the answer is “No”. Namely, we show that when K is a number field admitting a real embedding, it is impossible to define a group structure on the first Galois cohomology sets \(\textrm{H}^1\hspace{-0.8pt}(K,G)\) for all reductive K-groups G in a functorial way.