## 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*are (up to conjugation) in one-to-one correspondence with the

_{K}*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.

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