Resolving G-torsors by Abelian Base Extensions

Let G be a linear algebraic group defined over a field k. We prove that, under mild assumptions on k and G, there exists a finite k-subgroup S of G such that the natural map H(K,S) −→ H(K,G) is surjective for every field extension K/k. We give two applications of this result in the case where k an algebraically closed field of characteristic zero and K/k is finitely generated. First we show that for every α ∈ H(K,G) there exists an abelian field extension L/K such that αL ∈ H (L,G) is represented by a G-torsor over a projective variety. In particular, we prove that αL has trivial point obstruction. As a second application of our main theorem, we show that a (strong) variant of the algebraic form of Hilbert’s 13th problem implies that the maximal abelian extension of K has cohomological dimension ≤ 1. The last assertion, if true, would prove conjectures of Bogomolov and Königsmann, answer a question of Tits and establish an important case of Serre’s Conjecture II for the group E8.