On abelian birational sections
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- by Hélène Esnault and Olivier Wittenberg;
- J. Amer. Math. Soc. 23 (2010), 713-724
- DOI: https://doi.org/10.1090/S0894-0347-10-00660-0
- Published electronically: January 22, 2010
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.References
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Bibliographic Information
- Hélène Esnault
- Affiliation: Universität Duisburg–Essen, Mathematik, 45117 Essen, Germany
- MR Author ID: 64210
- Email: esnault@uni-due.de
- Olivier Wittenberg
- Affiliation: Département de mathématiques et applications, École normale supérieure, 45 rue d’Ulm, 75320 Paris Cedex 05, France
- MR Author ID: 729226
- Email: wittenberg@dma.ens.fr
- 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
- © Copyright 2010 by H. Esnault and O. Wittenberg
- Journal: J. Amer. Math. Soc. 23 (2010), 713-724
- MSC (2010): Primary 14G32; Secondary 14C25, 14G25, 14G20
- DOI: https://doi.org/10.1090/S0894-0347-10-00660-0
- MathSciNet review: 2629985