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On abelian birational sections

Authors: Hélène Esnault and Olivier Wittenberg
Journal: J. Amer. Math. Soc. 23 (2010), 713-724
MSC (2010): Primary 14G32; Secondary 14C25, 14G25, 14G20
Published electronically: January 22, 2010
MathSciNet review: 2629985
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Abstract: For a smooth and geometrically irreducible variety $ X$ over a field $ k$, the quotient of the absolute Galois group $ G_{k(X)}$ by the commutator subgroup of $ G_{\bar k(X)}$ projects onto $ G_k$. We investigate the sections of this projection. We show that such sections correspond to ``infinite divisions'' of the elementary obstruction of Colliot-Thélène and Sansuc. If $ k$ is a number field and the Tate-Shafarevich group of the Picard variety of $ X$ is finite, then such sections exist if and only if the elementary obstruction vanishes. For curves this condition also amounts to the existence of divisors of degree $ 1$. Finally we show that the vanishing of the elementary obstruction is not preserved by extensions of scalars.

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Additional Information

Hélène Esnault
Affiliation: Universität Duisburg–Essen, Mathematik, 45117 Essen, Germany

Olivier Wittenberg
Affiliation: Département de mathématiques et applications, École normale supérieure, 45 rue d’Ulm, 75320 Paris Cedex 05, France

Received by editor(s): February 18, 2009
Received by editor(s) in revised form: November 28, 2009
Published electronically: January 22, 2010
Additional Notes: This research was supported in part by the DFG Leibniz Preis, the SFB/TR 45, and the ERC/Advanced Grant 226257
Article copyright: © Copyright 2010 by H. Esnault and O. Wittenberg

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