On the birational section conjecture with local conditions

TL;DR

This paper investigates the birational section conjecture for hyperbolic curves over number fields, establishing conditions under which Galois sections correspond to rational points and analyzing their local properties.

Contribution

It proves that birationally liftable Galois sections are mostly integral outside a density zero set or cuspidal, and confirms the conjecture for totally real or imaginary quadratic fields.

Findings

Galois sections are integral outside a density zero set or cuspidal.

All non-cuspidal birationally adelic sections over certain fields come from rational points.

A strong approximation result for rational points on hyperbolic curves over Q and imaginary quadratic fields.

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 $GL_2(F_\ell)$-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.