Galois cohomology revisited

TL;DR

This paper links Galois cohomology of varieties over number fields to the K-theory of associated $C^*$-algebras, providing a new perspective on isomorphisms and twists, especially for elliptic curves.

Contribution

It introduces a novel approach connecting Galois cohomology with $C^*$-algebra K-theory, offering a new classification method for varieties and their twists.

Findings

Varieties are characterized by isomorphism or Morita equivalence of associated $C^*$-algebras.

Morita equivalent $C^*$-algebras parametrize twists of the variety.

Detailed analysis of rational elliptic curves within this framework.

Abstract

We recast the Galois cohomology of the variety $V$ over a number field $k$ in terms of the K-theory of a $C^*$-algebra $\mathscr{A}_V$ connected to $V$. It is proved that $V$ is isomorphic to $V'$ over $k$ (algebraic closure of $k$, resp.) if and only if $\mathscr{A}_V$ is isomorphic (Morita equivalent, resp.) to $\mathscr{A}_{V'}$. In particular, the Morita equivalent $C^*$-algebras $\mathscr{A}_V$ parametrize twists of the variety $V$. The case of rational elliptic curves is considered in detail.