2003/05/15 by Jochen Koenigsmann, Koenigsmann, Jochen
Computer Science · Mathematics · #12E30 (Primary) #14H30 (Secondary) #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Number Theory (math.NT) #Polynomial and algebraic computation
paper · pdf · doi:10.48550/arxiv.math/0305226
openalex publication_date 2003/05/15 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let X be a smooth projective curve of genus >1 over a field K which is finitely generated over the rationals. The section conjecture in Grothendieck's anabelian geometry says that the sections of the canonical projection from the arithmetic fundamental group of X onto the absolute Galois group of K are (up to conjugation) in one-to-one correspondence with K-rational points of X. The birational variant conjectures a similar correspondence where the fundamental group is replaced by the absolute Galois group of the function field K(X). The present paper proves the birational section conjecture for all X when K is replaced e.g. by the field of p-adic numbers. It disproves both conjectures for the field of real or p-adic algebraic numbers. And it 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.