Universality of the Galois action on the fundamental group of $\mathbb{P}^1\setminus\{0,1,\infty\}$

TL;DR

This paper demonstrates that all semi-simple geometric Galois representations can be realized within the algebraic structure of the fundamental group of the thrice-punctured projective line, revealing a universal property.

Contribution

It establishes a universality result for Galois actions on the fundamental group of the thrice-punctured projective line, connecting geometric Galois representations to algebraic functions.

Findings

Semi-simple Galois representations from geometry are subquotients of the ring of regular functions.

The fundamental group of the thrice-punctured projective line has a universal property.

Galois actions are deeply connected to algebraic functions on this fundamental group.

Abstract

We prove that any semi-simple representation of the Galois group of a number field coming from geometry appears as a subquotient of the ring of regular functions on the pro-algebraic completion of the fundamental group of the projective line with $3$ punctures.