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Descente de torseurs, gerbes et points rationnels

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 $\pi:X\rightarrow \textup(Spec)\;k$ provide an obstruction to the existence of $k$-rational points on $X$, since Leray spectral sequence gives rise (when $X$ is \textquotedblleft(nice)\textquotedblright, \textit(e.g.) $X$ smooth and proper) to an exact sequence of groups. This sequence gives an obstruction for a $\bar(G)_(X)$-torsor $\bar(P)\rightarrow\bar(X)$ with field of moduli $k$ to be defined over $k$, \textit(i.e.) to be obtained by extension of scalars to the algebraic closure $\bar(k)$ of $k$ from a $G_(X)$-torsor $P\rightarrow X$. 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.
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Contributor : Stephane Zahnd <>
Submitted on : Wednesday, January 14, 2004 - 7:44:13 AM
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Stephane Zahnd. Descente de torseurs, gerbes et points rationnels. Mathématiques [math]. Université des Sciences et Technologie de Lille - Lille I, 2003. Français. ⟨tel-00004163⟩

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