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Massey products in Galois cohomology and Pythagorean fields

Claudio Quadrelli

Abstract

We prove that a strengthened version of Minac-Tan's Massey Vanishing Conjecture holds true for fields with a finite number of square classes whose maximal pro-$2$ Galois group is of elementary type (as defined by I. Efrat). In particular, this proves Minac-Tan's Massey Vanishing Conjecture for Pythagorean fields with a finite number of square classes and their finite extensions.

Massey products in Galois cohomology and Pythagorean fields

Abstract

We prove that a strengthened version of Minac-Tan's Massey Vanishing Conjecture holds true for fields with a finite number of square classes whose maximal pro- Galois group is of elementary type (as defined by I. Efrat). In particular, this proves Minac-Tan's Massey Vanishing Conjecture for Pythagorean fields with a finite number of square classes and their finite extensions.
Paper Structure (17 sections, 15 theorems, 65 equations)

This paper contains 17 sections, 15 theorems, 65 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. $\mathbb{F}_p$-cohomology of pro-$p$ groups
  4. Massey products and upper unitriangular matrices
  5. Oriented pro-$p$ groups
  6. Orientations of pro-$p$ groups
  7. Demushkin pro-2 groups and orientations
  8. First case: $\mathop{\mathrm{Im}}\nolimits(\theta)\subseteq1+4\mathbb{Z}_2$
  9. Second case: $-1\in\mathop{\mathrm{Im}}\nolimits(\theta)$, $d$ even
  10. Third case: $\mathop{\mathrm{Im}}\nolimits(\theta)=\langle-1+2^f\rangle$
  11. Fourth case: $d$ odd
  12. Oriented pro-$p$ groups of elementary type
  13. Massey products and pro-2 groups of elementary type
  14. Demushkin pro-2 groups and Massey products
  15. The infinite dihedral pro-2 group
  16. ...and 2 more sections

Key Result

Theorem 1.2

Let $\mathbb{K}$ be a Pythagorean field with finitely many square classes, and let $\mathbb{L}/\mathbb{K}$ be an extension of finite degree. Then the maximal pro-2 Galois group $G_{\mathbb{L}}(2)$ of $\mathbb{L}$ satisfies the strong $n$-Massey vanishing property with respect to $\mathbb{F}_2$ for e

Theorems & Definitions (34)

  • Conjecture 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Corollary 1.4
  • Lemma 2.1
  • Remark 2.2
  • Proposition 2.3
  • Proposition 2.4
  • Proposition 2.5
  • Remark 2.6
  • ...and 24 more