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Article Dans Une Revue Journal of the American Mathematical Society Année : 2010

On abelian birational sections

Résumé

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.

Dates et versions

hal-02370289 , version 1 (19-11-2019)

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Hélène Esnault, Olivier Wittenberg. On abelian birational sections. Journal of the American Mathematical Society, 2010, 23 (3), pp.713-724. ⟨10.1090/S0894-0347-10-00660-0⟩. ⟨hal-02370289⟩
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