Descente de torseurs, gerbes et points rationnels - Descent of torsors, gerbes and rational points

Abstract

Let $k$ be a field of characteristic 0 and $G$ a linear algebraic $k$-group. When $G$ is abelian, it is well known that torsors under $G_X$ over a $k$-scheme $\pi :X\to \text{Spec}k$ provide an obstruction to the existence of $k$-rational points on $X$, since Leray spectral sequence gives rise (when $X$ is 'nice', e.g. $X$ smooth and proper) to an exact sequence of groups (5-term exact sequence associated). This sequence gives an obstruction for a ${\bar{G}}_X$-torsor $\bar{P}\to \bar{X}$ with field of moduli $k$ to be defined over $k$, i.e. to be obtained by extension of scalars to the algebraic closure $\bar{k}$ of $k$ from a $G_X$-torsor $P\to X$. This obstruction is measured by a gerbe, which is neutral if $X$ possesses a $k$-rational point. We try to extend this result to the non-commutative case, and in some cases, we deduce non-abelian cohomological obstruction to the existence of $k$-rational points on $X$, and results about descent of torsors.

144 pages, uses xypic, in french, thesis