Schur $\sigma$-groups of type (3,3)

Richard Pink

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arXiv · 2505.05580

Schur $\sigma$-groups of type (3,3)

Richard Pink

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Abstract

For any odd prime $p$, the Galois group of the maximal unramified pro-$p$-extension of an imaginary quadratic field is a Schur $\sigma$-group. But Schur $\sigma$-groups can also be constructed and studied abstractly. We prove that if $p>3$, any Schur $\sigma$-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 $\sigma$-groups, or with the Fontaine-Mazur conjecture, this lends credence to the ``if'' part of a conjecture of McLeman.

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Richard Pink. 2025-05-08. Schur $\sigma$-groups of type (3,3). https://arxiv.org/abs/2505.05580

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