On abelian birational sections

Esnault, Hélène; Wittenberg, Olivier

doi:10.1090/S0894-0347-10-00660-0

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

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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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