On the ‘Section Conjecture’ in anabelian geometry

By Jochen Koenigsmann. 2003

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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 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 π 1(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 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.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/e1d5ada3ebab51c397fb19436fa14aa44de8d34f

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