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Schur $σ$-groups of type $(3,3)$ for $p=3$

Eric Ahlqvist, Richard Pink

TL;DR

The paper investigates p-tower groups G_K for imaginary quadratic fields with p=3, focusing on Schur σ-groups of Zassenhaus type (3,3) and the finite quotient G_K/D_4(G_K). It proves that in 10 of 13 possible isomorphism classes, G_K is finite or an open subgroup of a PGL_2 form over Q_3, and together with Fontaine–Mazur or Cohen–Lenstra heuristics this supports the 'if' direction of McLeman's conjecture in this setting. A suite of methods—powerfulness criteria, explicit Massey-product presentations, and recursive generation of Schur quotients—along with heavy GAP/ANUPQ computations, underpin both the theoretical classification and the computational tests. The authors test the generalized Cohen–Lenstra heuristic via triple Massey products in étale cohomology for d(G_K)=2 across many K, finding good agreement with IPAD data and the predicted distributions, while also revealing a sparse, well-characterized subset of infinite cases. Overall, the work combines deep group-theoretic analysis with large-scale computations to advance understanding of p-tower groups for p=3 and to validate statistical predictions for their finite quotients.

Abstract

For any imaginary quadratic field $K$, the Galois group $G_K$ of its maximal unramified pro-$3$-extension is a Schur $σ$-group. If this has Zassenhaus type $(3,3)$, there are 13 possibilities for the isomorphism class of the finite quotient $G_K/D_4(G_K)$. We prove that for 10 of these 13 cases $G_K$ is either finite or isomorphic to an open subgroup of a form of $\mathop{\rm PGL}_2$ over $\mathbb{Q}_3$. Combined with the Fontaine-Mazur conjecture, or with earlier work on an analogue of the Cohen--Lenstra heuristic for Schur $σ$-groups, this lends credence to the "if" part of a conjecture of McLeman. Using explicit computations of triple Massey products, we also test the heuristic for all imaginary quadratic fields $K$ with $d(G_K)=2$ and discriminant $-10^8 < d_K < 0$ and find a reasonably good agreement.

Schur $σ$-groups of type $(3,3)$ for $p=3$

TL;DR

The paper investigates p-tower groups G_K for imaginary quadratic fields with p=3, focusing on Schur σ-groups of Zassenhaus type (3,3) and the finite quotient G_K/D_4(G_K). It proves that in 10 of 13 possible isomorphism classes, G_K is finite or an open subgroup of a PGL_2 form over Q_3, and together with Fontaine–Mazur or Cohen–Lenstra heuristics this supports the 'if' direction of McLeman's conjecture in this setting. A suite of methods—powerfulness criteria, explicit Massey-product presentations, and recursive generation of Schur quotients—along with heavy GAP/ANUPQ computations, underpin both the theoretical classification and the computational tests. The authors test the generalized Cohen–Lenstra heuristic via triple Massey products in étale cohomology for d(G_K)=2 across many K, finding good agreement with IPAD data and the predicted distributions, while also revealing a sparse, well-characterized subset of infinite cases. Overall, the work combines deep group-theoretic analysis with large-scale computations to advance understanding of p-tower groups for p=3 and to validate statistical predictions for their finite quotients.

Abstract

For any imaginary quadratic field , the Galois group of its maximal unramified pro--extension is a Schur -group. If this has Zassenhaus type , there are 13 possibilities for the isomorphism class of the finite quotient . We prove that for 10 of these 13 cases is either finite or isomorphic to an open subgroup of a form of over . Combined with the Fontaine-Mazur conjecture, or with earlier work on an analogue of the Cohen--Lenstra heuristic for Schur -groups, this lends credence to the "if" part of a conjecture of McLeman. Using explicit computations of triple Massey products, we also test the heuristic for all imaginary quadratic fields with and discriminant and find a reasonably good agreement.
Paper Structure (11 sections, 21 theorems, 48 equations)

This paper contains 11 sections, 21 theorems, 48 equations.

Table of Contents

  1. Introduction
  2. Pro-$p$-groups
  3. Criterion for an open normal subgroup to be power-ful
  4. $\sigma$-Pro-$p$-groups and weak Schur $\sigma$-groups
  5. Analytic strong Schur $\sigma$-groups
  6. Finite quotients of weak Schur $\sigma$-groups
  7. An explicit presentation via Massey products
  8. Weak Schur $\sigma$-groups with $d(G)=2$ for $p=3$
  9. Analytic Schur $\sigma$-groups of type $(3,3)$ for $p=3$
  10. $p$-Tower groups of imaginary quadratic fields
  11. Experimental results

Key Result

Proposition 2.2

Any automorphism of $G$ that acts trivially on $\mathop{\rm gr}\nolimits_1^D(G)$ also acts trivially on $\mathop{\rm gr}\nolimits_i^D(G)$ for every $i\geqslant1$.

Theorems & Definitions (23)

  • Proposition 2.2
  • Proposition 2.5
  • Proposition 2.6
  • Proposition 2.8
  • Proposition 2.10
  • Proposition 3.1
  • Proposition 3.4
  • Proposition 4.3
  • Proposition 5.1
  • Proposition 5.2
  • ...and 13 more