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
DOI:
https://doi.org/10.1090/S0894-0347-10-00660-0
Published electronically:
January 22, 2010
MathSciNet review:
2629985
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Abstract | References | Similar Articles | Additional Information
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
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
Article copyright:
© Copyright 2010
by H. Esnault and O. Wittenberg
