On the ‘Section Conjecture’ in anabelian geometry
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Jochen Koenigsmann
Abstract
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 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.
Walter de Gruyter GmbH & Co. KG
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- Diophantine definability of infinite discrete nonarchimedean sets and Diophantine models over large subrings of number fields
- Generalized Hodge metrics and BCOV torsion on Calabi-Yau moduli
- Arcs, valuations and the Nash map
- On Artin’s L-functions. II: Dirichlet coefficients
- A complex ball uniformization of the moduli space of cubic surfaces via periods of K 3 surfaces
- The spectrum of prime ideals in tensor triangulated categories
- Characterization of the linear partial di¤erential equations that admit solution operators on Gevrey classes
- On the ‘Section Conjecture’ in anabelian geometry
Articles in the same Issue
- Semilattices of groups and inductive limits of Cuntz algebras
- Diophantine definability of infinite discrete nonarchimedean sets and Diophantine models over large subrings of number fields
- Generalized Hodge metrics and BCOV torsion on Calabi-Yau moduli
- Arcs, valuations and the Nash map
- On Artin’s L-functions. II: Dirichlet coefficients
- A complex ball uniformization of the moduli space of cubic surfaces via periods of K 3 surfaces
- The spectrum of prime ideals in tensor triangulated categories
- Characterization of the linear partial di¤erential equations that admit solution operators on Gevrey classes
- On the ‘Section Conjecture’ in anabelian geometry
