Splitting Varieties for Triple Massey Products

TL;DR

This paper constructs explicit splitting varieties for triple Massey products over fields of characteristic not 2, demonstrating that these products always contain zero for global fields, and establishing their relation to norm equations and the Hasse principle.

Contribution

It introduces explicit splitting varieties for triple Massey products and proves their Hasse principle, extending the understanding of Massey products over global fields.

Findings

Triple Massey products contain zero over global fields.

Splitting varieties are characterized by norm equations.

These varieties satisfy the Hasse principle.

Abstract

We construct splitting varieties for triple Massey products. For a,b,c in F^* the triple Massey product < a,b,c> of the corresponding elements of H^1(F, mu_2) contains 0 if and only if there is x in F^* and y in F[\sqrt{a}, \sqrt{c}]^* such that b x^2 = N_{F[\sqrt{a}, \sqrt{c}]/F}(y), where N_{F[\sqrt{a}, \sqrt{c}]/F} denotes the norm, and F is a field of characteristic different from 2. These varieties satisfy the Hasse principle by a result of D.B. Lee and A.R. Wadsworth. This shows that triple Massey products for global fields of characteristic different from 2 always contain 0.