Schur $σ$-groups of type (3,3)
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866916739201105920 |
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| author | Pink, Richard |
| author_facet | Pink, Richard |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_05580 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Schur $σ$-groups of type (3,3) Pink, Richard Number Theory 11R11 (11R32, 20D15, 20E18, 20F05) 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. |
| title | Schur $σ$-groups of type (3,3) |
| topic | Number Theory 11R11 (11R32, 20D15, 20E18, 20F05) |
| url | https://arxiv.org/abs/2505.05580 |