Massey products in the \'etale cohomology of number fields

TL;DR

This paper develops formulas for 3-fold Massey products in the étale cohomology of number fields, leading to new insights into class field towers, class groups, and Galois representations of quadratic imaginary fields.

Contribution

It introduces explicit formulas for Massey products in étale cohomology and applies them to identify new examples of imaginary quadratic fields with infinite p-class field towers.

Findings

Found the first examples of imaginary quadratic fields with p-rank two class groups and infinite p-class field towers.

Provided a necessary and sufficient condition for the vanishing of 3-fold Massey products.

Disproved McLeman's (3,3)-conjecture.

Abstract

We give formulas for 3-fold Massey products in the \'etale cohomology of the ring of integers of a number field and use these to find the first known examples of imaginary quadratic fields with class group of $p$-rank two possessing an infinite $p$-class field tower, where $p$ is an odd prime. Furthermore, a necessary and sufficient condition, in terms of class groups of $p$-extensions, for the vanishing of 3-fold Massey products is given. As a consequence, we give an elementary and sufficient condition for the infinitude of class field towers of imaginary quadratic fields. We also disprove McLeman's $(3,3)$-conjecture. Lastly, we relate the vanishing of Massey products to the existence of Galois representations of $G_{\mathbb{Q},S}$ which realize an unexpectedly large class group for certain extensions of a quadratic imaginary number field.