On copies of the absolute Galois group in $\mathrm{Out}\hat{F}_2$

TL;DR

This paper investigates how outer Galois actions on a free profinite group of rank two, arising from fundamental groups of certain algebraic curves, can uniquely determine the underlying curves and number fields.

Contribution

It demonstrates that the images of these Galois actions uniquely identify the associated algebraic curves and number fields under mild conditions.

Findings

Outer Galois actions encode enough information to distinguish curves and fields.

The images of these actions are sufficient to recover the geometric and arithmetic data.

Results apply to fundamental groups of the projective line minus three points and elliptic curves.

Abstract

In this article we consider outer Galois actions on a free profinite group of rank two, induced by the \'etale fundamental group of a projective line minus three points or of a pointed elliptic curve over a number field. Under mild technical assumptions their respective images uniquely determine the curves and the number fields.