3-nilpotent obstructions to pi_1 sections for P^1_Q - {0,1,infty}

TL;DR

This paper investigates obstructions to lifting rational points to sections of the etale fundamental group quotients for the projective line minus three points, using Massey products and Galois cohomology, with explicit calculations over various fields.

Contribution

It provides a detailed analysis of 3-nilpotent obstructions and complete mod 2 calculations for specific fields, extending understanding of Galois cohomology obstructions.

Findings

Complete mod 2 calculations for K=Q_p and R.

Partial mod 2 calculations for K=Q.

Identification of obstructions via triple Massey products.

Abstract

We study which rational points of the Jacobian of P^1_K -{0,1,infty} can be lifted to sections of geometrically 3 nilpotent quotients of etale pi_1 over the absolute Galois group. This is equivalent to evaluating certain triple Massey products of elements of H^1(G_K). For K=Q_p or R, we give a complete mod 2 calculation. This permits some mod 2 calculations for K = Q. These are computations of obstructions of Jordan Ellenberg.