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

TL;DR

This paper investigates whether the first Galois cohomology sets of reductive groups over global fields can be endowed with a group structure, revealing that such a structure exists only over fields without real embeddings.

Contribution

The paper proves that a functorial group structure on H^1(K,G) exists only over global fields without real embeddings, providing a clear criterion based on the nature of the field.

Findings

Group structure exists over global fields with no real embeddings.

No functorial group structure can be defined when real embeddings are present.

The result clarifies the limitations of Galois cohomology structures over different types of global fields.

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 H^1(K,G) for all reductive K-groups G in a functorial way.