On the ‘Section Conjecture’ in anabelian geometry

Let X be a smooth projective curve of genus > 1 over a field K with function field K(X), let π1(X) be the arithmetic fundamental group of X over K and let GF denote the absolute Galois group of a field F . The section conjecture in Grothendieck’s anabelian geometry says that the sections of the canonical projection π1(X) → GK are (up to conjugation) in one-to-one correspondence with the K-rational points of X, if K is finitely generated over Q. The birational variant conjectures a similar correspondence w.r.t. the sections of the projection GK(X) → GK . So far these conjectures were a complete mystery except for the obvious results over separably closed fields and some non-trivial results due to Sullivan and Huisman over the reals. The present paper proves — via model theory — the birational section conjecture for all local fields of characteristic 0 (except C), disproves both conjectures e.g. for the fields of all real or p-adic algebraic numbers, and gives a purely group theoretic characterization of the sections induced by K-rational points of X in the birational setting over almost arbitrary fields. As a biproduct we obtain Galois theoretic criteria for radical solvability of polynomial equations in more than one variable, and for a field to be PAC, to be large, or to be Hilbertian. Mathematics Subject Classification (2000): 12E30, 12F10, 14G05, 14H30 I would like to thank Jean-Louis Colliot-Thélène, Pierre Dèbes, Hendrik Lenstra, Hiroaki Nakamura, Florian Pop, Jean-Pierre Serre and Akio Tamagawa for very helpful discussions on the subject of this paper. Heisenberg-Stipendiat der Deutschen Forschungsgemeinschaft (KO 1962/1-2)