Schur $σ$-groups of type $(3,3)$ for $p=3$
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866911438047543296 |
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| author | Ahlqvist, Eric Pink, Richard |
| author_facet | Ahlqvist, Eric Pink, Richard |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_09889 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Schur $σ$-groups of type $(3,3)$ for $p=3$ Ahlqvist, Eric Pink, Richard Number Theory 11R11, (11R32, 11R34, 20D15) 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. |
| title | Schur $σ$-groups of type $(3,3)$ for $p=3$ |
| topic | Number Theory 11R11, (11R32, 11R34, 20D15) |
| url | https://arxiv.org/abs/2602.09889 |