Abstract
For a smooth and geometrically irreducible variety X X over a field k k , the quotient of the absolute Galois group G k ( X ) G_{k(X)} by the commutator subgroup of G k ¯ ( X ) G_{\bar k(X)} projects onto G k 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 k is a number field and the Tate–Shafarevich group of the Picard variety of X 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 1 . Finally we show that the vanishing of the elementary obstruction is not preserved by extensions of scalars.
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