On cuspidal sections of algebraic fundamental groups

Abstract

Rational points in the boundary of a hyperbolic curve over a field with sufficiently nontrivial Kummer theory are the source for an abundance of sections of its fundamental group extension. We refine Nakamura's approach to these sections and obtain an anabelian theorem for hyperbolic genus 0 curves over quite general fields, for example $\mathbb{Q}^{\mathrm{ab}}$.