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 $$\underline{x}(s) \in {\overline{X}}({\mathbb A}_k)_\bullet $$ of the smooth completion $$X \subseteq {\overline{X}}$$ . We show that $$\underline{x}(s)$$ is $$X$$ -integral outside a set of places of Dirichlet density $$0$$ , or $$s$$ is cuspidal. The proof relies on $${{\mathrm{GL}}}_2({\mathbb F}_\ell )$$ -quotients of $$\pi _1(U)$$ for some open $$U \subset X$$ . 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 $${\mathbb Q}$$ or imaginary quadratic fields.