On the birational section conjecture with local conditions

Abstract

A birationally liftable Galois section s of a hyperbolic curve X/k over a number field k yields an adelic point x(s) in the smooth completion of X. We show that x(s) is X-integral outside a set of places of Dirichlet density 0, or s is cuspidal. The proof relies on $G{L}_2(F_{\ell })$-quotients of ${\pi }_1(U)$ for some open U of X. If k is totally real or imaginary quadratic, we prove that all birationally adelic, non-cuspidal Galois sections come from rational points as predicted by the section conjecture of anabelian geometry. As an aside we also obtain a strong approximation result for rational points on hyperbolic curves over Q or imaginary quadratic fields.

Theorem C (and Section 7) of the original version have been deleted due to a gap in the proof. This is the journal version

Cited by in corpus (1)

• On the birational section conjecture with strong birationality assumptions