Galois sections for abelianized fundamental groups

Given a smooth projective curve X of genus at least 2 over a number field k, Grothendieck's Section Conjecture predicts that the canonical projection from the étale fundamental group of X onto the absolute Galois group of k has a section if and only if the curve has a rational point. We show that there exist curves where the above map has a section over each completion of k but not over k. In the appendix Victor Flynn gives explicit examples in genus 2. Our result is a consequence of a more general investigation of the existence of sections for the projection of the étale fundamental group 'with abelianized geometric part' onto the Galois group. We also point out the relation to the elementary obstruction of Colliot-Thélène and Sansuc. © 2009 Springer-Verlag.