On the "Section Conjecture" in anabelian geometry
| dc.creator | Koenigsmann, Jochen | |
| dc.date | 2003-05-15 | |
| dc.date.accessioned | 2026-07-07T06:32:55Z | |
| dc.date.available | 2026-07-07T06:32:55Z | |
| dc.description | 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. | |
| dc.description | 21 pages, latex | |
| dc.identifier | https://arxiv.org/abs/math/0305226 | |
| dc.identifier | http://arxiv.org/abs/math/0305226 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/99023 | |
| dc.subject | Algebraic Geometry | |
| dc.subject | Number Theory | |
| dc.subject | 12E30 (Primary), 14H30 (Secondary) | |
| dc.title | On the "Section Conjecture" in anabelian geometry | |
| dc.type | text |