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

Mikhail Borovoi

TL;DR

This paper investigates whether the first Galois cohomology set ${\rm H}^1(K,G)$ of a reductive $K$-group can be endowed with a functorial group structure for global fields $K$. It leverages known positive results for non-archimedean local fields and for global fields with no real places, and provides explicit obstructions in the presence of real places, including $K={\mathds R}$ and number fields with a real embedding. The obstructions are derived from real cohomology, maximal compact tori, and local-global principles, with concrete counterexamples such as ${\rm SU}_4$ over ${\mathds R}$ and ${\rm SU}_{4,L/K}$ over real-placed number fields. The results delineate fundamental limits on extending non-abelian Galois cohomology to a functorial group-valued invariant in the global setting, highlighting the decisive role of real data in these obstruction phenomena.

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.

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 set of a reductive -group can be endowed with a functorial group structure for global fields . It leverages known positive results for non-archimedean local fields and for global fields with no real places, and provides explicit obstructions in the presence of real places, including and number fields with a real embedding. The obstructions are derived from real cohomology, maximal compact tori, and local-global principles, with concrete counterexamples such as over and over real-placed number fields. The results delineate fundamental limits on extending non-abelian Galois cohomology to a functorial group-valued invariant in the global setting, highlighting the decisive role of real data in these obstruction phenomena.

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.
Paper Structure (4 sections, 3 theorems, 16 equations)

This paper contains 4 sections, 3 theorems, 16 equations.

Table of Contents

  1. Introduction
  2. Generalities on Galois cohomology
  3. Galois cohomology over ${\mathds R}$
  4. Galois cohomology over a number field

Key Result

Theorem 1.2

In the cases it is impossible to define a functorial in $G$ group structure on the pointed sets ${\rm H}^1(K,G)$ for all reductive $K$-groups $G$ in such a way that when $G=T$ is a $K$-torus, the new group structure on ${\rm H}^1(K,T)$ coincides with the standard one.

Theorems & Definitions (7)

  • Theorem 1.2
  • Theorem 3.1
  • proof
  • Example 3.2
  • Theorem 4.1
  • proof
  • Example 4.2