Galois sections for abelianized fundamental groups
Abstract
Description
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 give a criterion for the existence of sections in arbitrary dimension and over arbitrary perfect fields, and then study the case of curves over local and global fields more closely. We also point out the relation to the elementary obstruction of Colliot-Thélène and Sansuc.
This is the published version, except for a characteristic 0 assumption added in Section 5 which was unfortunately omitted there. Thanks to O. Wittenberg for noticing it
This is the published version, except for a characteristic 0 assumption added in Section 5 which was unfortunately omitted there. Thanks to O. Wittenberg for noticing it