Galois cohomology revisited

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.