Abstract

Serre’s Conjecture II states that if K is a field of cohomological dimension ≤ 2, G is a semisimple simply connected K-group and X is a G-torsor, then X is trivial. This conjecture has been proved for several families of fields, and alternatively for several families of groups, but it is still open in its full generality. The usual way of tackling this conjecture consists in “simplifying” the structure of the groups involved, proving that the torsors come from subgroups for which the conjecture has already been proved. In joint work with Diego Izquierdo, we rather focus on “simplifying” the structure of the fields involved. For this, we formulated certain “transfer principles”, which allow us to construct fields with simpler properties, while controlling at the same time their cohomological dimension. In this talk I will give an idea of the present situation of the conjecture and how some of these transfer principles allow us to reduce it to the case of countable fields of characteristic 0.