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Massey products in the étale cohomology of number fields

Eric Ahlqvist, Magnus Carlson

TL;DR

The paper develops explicit formulas for $3$-fold Massey products in the étale cohomology of number fields and uses them to connect cohomological obstructions with the structure of unramified pro-$p$-extensions. By bridging group cohomology, étale cohomology, and arithmetic duality, it derives computable invariants that predict when the $p$-class field tower is infinite or finite. It delivers the first known examples of imaginary quadratic fields with $p$-rank two yet infinite $p$-class towers, and it disproves McLeman's $(3,3)$-conjecture, while also providing a clean vanishing criterion in terms of class groups of unramified extensions. The work blends deep theoretical dualities with extensive computational verification (via PARI), producing new criteria and concrete instances that sharpen our understanding of class field towers and their Galois representations.

Abstract

We give formulas for 3-fold Massey products in the étale 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.

Massey products in the étale cohomology of number fields

TL;DR

The paper develops explicit formulas for -fold Massey products in the étale cohomology of number fields and uses them to connect cohomological obstructions with the structure of unramified pro--extensions. By bridging group cohomology, étale cohomology, and arithmetic duality, it derives computable invariants that predict when the -class field tower is infinite or finite. It delivers the first known examples of imaginary quadratic fields with -rank two yet infinite -class towers, and it disproves McLeman's -conjecture, while also providing a clean vanishing criterion in terms of class groups of unramified extensions. The work blends deep theoretical dualities with extensive computational verification (via PARI), producing new criteria and concrete instances that sharpen our understanding of class field towers and their Galois representations.

Abstract

We give formulas for 3-fold Massey products in the étale 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 -rank two possessing an infinite -class field tower, where is an odd prime. Furthermore, a necessary and sufficient condition, in terms of class groups of -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 -conjecture. Lastly, we relate the vanishing of Massey products to the existence of Galois representations of which realize an unexpectedly large class group for certain extensions of a quadratic imaginary number field.
Paper Structure (15 sections, 25 theorems, 238 equations)

This paper contains 15 sections, 25 theorems, 238 equations.

Table of Contents

  1. Introduction
  2. Class field towers
  3. Massey products
  4. Overview of main results
  5. Massey products in Galois cohomology
  6. Massey products in étale cohomology
  7. Massey products for number fields
  8. Background on arithmetic duality
  9. The intersection of the kernel of norms
  10. Computations of Massey products
  11. Computation of the connecting homomorphism
  12. A necessary and sufficient condition for the vanishing of Massey products
  13. Class field towers
  14. Resolutions
  15. Cohomology and duality for locally constant sheaves

Key Result

Lemma 2.2

The datum of a trivializing cochain $k_{yz}$ of $-y\smallsmile z$ is equivalent to a lift $w_z\in Z^1(G, V_y)$ of $z\in H^1(G,{\mathbb Z}/p{\mathbb Z})$.

Theorems & Definitions (60)

  • Remark 2.1
  • Lemma 2.2
  • proof
  • Proposition 2.3
  • proof
  • Proposition 2.4
  • proof
  • Proposition 2.5
  • proof
  • Proposition 3.1
  • ...and 50 more