Abstract

Let X  be a smooth projective curve of genus > 1 over a field K  with function field K (X ), let π ₁(X ) be the arithmetic fundamental group of X  over K  and let G_F  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 π ₁(X ) ↠ G_K  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 G_{K  (X )} ↠ G_K .

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 byproduct 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.