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Chasing maximal pro-p Galois groups via 1-cyclotomicity

Claudio Quadrelli

Abstract

Let $p$ be a prime. We prove that certain amalgamated free pro-$p$ products of Demushkin groups with pro-$p$-cyclic amalgam cannot give rise to a 1-cyclotomic oriented pro-$p$ group, and thus do not occur as maximal pro-$p$ Galois groups of fields containing a root of 1 of order $p$. We show that other cohomological obstructions which are used to detect pro-$p$ groups that are not maximal pro-$p$ Galois groups - the quadraticity of $\mathbb{Z}/p\mathbb{Z}$-cohomology and the vanishing of Massey products - fail with the above pro-$p$ groups. Finally, we prove that the Minač-Tân pro-$p$ group cannot give rise to a 1-cyclotomic oriented pro-$p$ group, and we conjecture that every 1-cyclotomic oriented pro-$p$ group satisfy the strong $n$-Massey vanishing property for $n>2$.

Chasing maximal pro-p Galois groups via 1-cyclotomicity

Abstract

Let be a prime. We prove that certain amalgamated free pro- products of Demushkin groups with pro--cyclic amalgam cannot give rise to a 1-cyclotomic oriented pro- group, and thus do not occur as maximal pro- Galois groups of fields containing a root of 1 of order . We show that other cohomological obstructions which are used to detect pro- groups that are not maximal pro- Galois groups - the quadraticity of -cohomology and the vanishing of Massey products - fail with the above pro- groups. Finally, we prove that the Minač-Tân pro- group cannot give rise to a 1-cyclotomic oriented pro- group, and we conjecture that every 1-cyclotomic oriented pro- group satisfy the strong -Massey vanishing property for .

Paper Structure

This paper contains 28 sections, 23 theorems, 129 equations.

Table of Contents

  1. Introduction
  2. Oriented pro-$p$ groups and cohomology
  3. Notation and preliminaries
  4. Oriented pro-$p$ groups
  5. Kummerianity and 1-cyclotomicity
  6. Examples
  7. Kummerianity and 1-cocyles
  8. The $\mathbb{Z}/p\mathbb{Z}$-cohomology of $G$
  9. Degree 1 and 2
  10. Amalgams
  11. Quadratic cohomology
  12. Proof of Theorem \ref{['thm:intro1']} case (1.1.a)
  13. Kummerianity
  14. The subgroup $U$
  15. The subgroup $H$
  16. ...and 13 more sections

Key Result

Theorem 1.1

Let $G$ be a pro-$p$ group with pro-$p$ presentation where $d_1,d_2$ are two positive odd integers, and either: Then there are no orientations $\theta\colon G\to1+p\mathbb{Z}_p$ such that the oriented pro-$p$ group $(G,\theta)$ is 1-cyclotomic.

Theorems & Definitions (50)

  • Theorem 1.1
  • Corollary 1.2
  • Proposition 1.3
  • Theorem 1.4
  • Conjecture 1.5
  • Proposition 2.2
  • Remark 2.3
  • Example 2.4
  • Example 2.5
  • Example 2.6
  • ...and 40 more