Massey products <y,x,x,...,x,x,y> in Galois cohomology via rational points

TL;DR

This paper computes specific Massey products in Galois cohomology using rational points and embeddings into Picard varieties, providing new obstructions to sections of the projective line minus three points.

Contribution

It introduces a novel method to compute Massey products via embeddings into Picard varieties and Galois-equivariant maps, advancing understanding of obstructions in Galois cohomology.

Findings

Computed order n Massey products for specific elements.

Produced obstructions to -sections of  -  .

Extended computations to new Massey products involving  and .

Abstract

For $x$ an element of a field other than $0$ or $1$, we compute the order $n$ Massey products $$\langle (1-x)^{-1}, x^{-1}, \ldots, x^{-1}, (1-x)^{-1} \rangle$$ of $n-2$ factors of $x^{-1}$ and two factors of $(1-x)^{-1}$ by embedding $\mathbb{P}^1 - \{0,1,\infty\}$ into its Picard variety and constructing $\operatorname{Gal}(k^s/k)$ equivariant maps from $\pi_1$ applied to this embedding to unipotent matrix groups. This method produces obstructions to $\pi_1$-sections of $\mathbb{P}^1 - \{0,1,\infty\}$, partial computations of obstructions of Jordan Ellenberg, and also computes the Massey products $$\langle x^{-1} , (-x)^{-1}, \ldots, (-x)^{-1}, x^{-1} \rangle.$$