| Title: |
Schur $σ$-groups of type $(3,3)$ for $p=3$ |
| Authors: |
Ahlqvist, Eric; Pink, Richard |
| Publication Year: |
2026 |
| Collection: |
ArXiv.org (Cornell University Library) |
| Subject Terms: |
Number Theory; 11R11; (11R32; 11R34; 20D15) |
| Description: |
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. ; 31 pages |
| Document Type: |
text |
| Language: |
unknown |
| Relation: |
http://arxiv.org/abs/2602.09889 |
| Availability: |
http://arxiv.org/abs/2602.09889 |
| Accession Number: |
edsbas.F1DF49C7 |
| Database: |
BASE |