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Schur $σ$-groups of type (3,3)

Richard Pink

TL;DR

The paper analyzes Schur $\sigma$-groups of Zassenhaus type $(3,3)$ for odd primes with $p>3$, proving that every infinite strong Schur $\sigma$-group of this type is isomorphic to an open subgroup of a form of $\mathrm{PGL}_2$ over ${\mathbb Q}_p$ and that the ambient algebraic group forces finiteness under Fontaine–Mazur. It develops a detailed framework: (i) a $p$-adic analytic structure via the Lazard/Zassenhaus filtration, (ii) a finite classification of ambient forms of $\mathrm{PGL}_2$ with involution, and (iii) a measure-theoretic perspective on the distribution of weak/strong Schur $\sigma$-groups and the corresponding $p$-tower groups. Consequently, the set of infinite strong Schur $\sigma$-groups of type $(3,3)$ is countable, and their infinite variants form a closed, measure-zero subset of the Sch topology. The results provide evidence for McLeman's conjecture in the $p>3$ regime on average and align with Fontaine–Mazur to imply finiteness of $p$-tower groups in this setting, consistent with the expected behavior of $p$-adic Galois groups of maximal unramified extensions.

Abstract

For any odd prime $p$, the Galois group of the maximal unramified pro-$p$-extension of an imaginary quadratic field is a Schur $σ$-group. But Schur $σ$-groups can also be constructed and studied abstractly. We prove that if $p>3$, any Schur $σ$-group of Zassenhaus type $(3,3)$, for which every open subgroup has finite abelianization, is isomorphic to an open subgroup of a form of ${\rm PGL}_2$ over ${\mathbb Q}_p$. Combined with earlier work on an analogue of the Cohen-Lenstra heuristic for Schur $σ$-groups, or with the Fontaine-Mazur conjecture, this lends credence to the ``if'' part of a conjecture of McLeman.

Schur $σ$-groups of type (3,3)

TL;DR

The paper analyzes Schur -groups of Zassenhaus type for odd primes with , proving that every infinite strong Schur -group of this type is isomorphic to an open subgroup of a form of over and that the ambient algebraic group forces finiteness under Fontaine–Mazur. It develops a detailed framework: (i) a -adic analytic structure via the Lazard/Zassenhaus filtration, (ii) a finite classification of ambient forms of with involution, and (iii) a measure-theoretic perspective on the distribution of weak/strong Schur -groups and the corresponding -tower groups. Consequently, the set of infinite strong Schur -groups of type is countable, and their infinite variants form a closed, measure-zero subset of the Sch topology. The results provide evidence for McLeman's conjecture in the regime on average and align with Fontaine–Mazur to imply finiteness of -tower groups in this setting, consistent with the expected behavior of -adic Galois groups of maximal unramified extensions.

Abstract

For any odd prime , the Galois group of the maximal unramified pro--extension of an imaginary quadratic field is a Schur -group. But Schur -groups can also be constructed and studied abstractly. We prove that if , any Schur -group of Zassenhaus type , for which every open subgroup has finite abelianization, is isomorphic to an open subgroup of a form of over . Combined with earlier work on an analogue of the Cohen-Lenstra heuristic for Schur -groups, or with the Fontaine-Mazur conjecture, this lends credence to the ``if'' part of a conjecture of McLeman.
Paper Structure (9 sections, 29 theorems, 25 equations)

This paper contains 9 sections, 29 theorems, 25 equations.

Table of Contents

  1. Introduction
  2. Pro-$p$-groups
  3. $\sigma$-Pro-$p$-groups
  4. Schur $\sigma$-groups
  5. Groups of type (3,3)
  6. Finite $p$-groups of type (3,3)
  7. Pro-$p$-groups of type (3,3)
  8. Infinite Schur $\sigma$-groups of Zassenhaus type (3,3)
  9. $p$-Tower groups

Key Result

Proposition 3.3

For any $\sigma$-invariant normal subgroup $N\mathrel{\triangleleft} G$ and any $\varepsilon\in\{\pm1\}$ the projection $G\twoheadrightarrow G/N$ induces a surjective map $G^\varepsilon \twoheadrightarrow (G/N)^\varepsilon$. If moreover $G$ is finite, then all fibers have the same cardinality $|N^\v

Theorems & Definitions (33)

  • Proposition 3.3
  • Proposition 3.4
  • Proposition 3.5
  • Proposition 3.6
  • Proposition 4.3
  • Proposition 4.5
  • Theorem 4.6
  • Proposition 4.7
  • Corollary 4.8
  • Remark 4.9
  • ...and 23 more