Schur $σ$-groups of type $(3,3)$ for $p=3$

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.

💡 Research Summary

The paper investigates the structure of Schur σ‑groups of Zassenhaus type (3, 3) when the prime p equals 3, with a focus on the maximal unramified pro‑3 extension G_K of an imaginary quadratic field K. For such groups the quotient G_K / D₄(G_K) can fall into one of 13 isomorphism classes. The authors develop a general criterion to decide whether a given open normal subgroup E(G) of a finitely generated pro‑p group G is powerful (i.e., its Frattini subgroup is generated by p‑th powers). This criterion depends only on the finite quotient G / E²(G) and can be checked algorithmically.

Using GAP together with the ANUPQ package, the authors enumerate all possible finite quotients G / E²(G) for the 13 cases and apply the criterion. They prove that in 10 of the 13 (3, 3) cases the subgroup D₂(G) is powerful. By Lazard’s theorem this implies that G is a p‑adic analytic group. Moreover, when G is a strong Schur σ‑group (as is the case for any p‑tower group G_K), they show that G must be isomorphic to an open subgroup of a form of PGL₂ over ℚ₃. Consequently, the unramified Fontaine–Mazur conjecture forces G_K to be finite for those 10 isomorphism classes, providing strong statistical support for the “if” direction of McLeman’s conjecture.

The second major component of the work is an explicit presentation of G_K / D₄(G_K) via triple Massey products. Building on Vögel’s description of Massey products in étale cohomology and on previous formulas by the first author and Carlson, the authors translate the cohomological data into concrete group presentations. This makes it possible to compute the isomorphism class of G_K / D₄(G_K) for every imaginary quadratic field K with discriminant –10⁸ < d_K < 0 and with d(G_K)=2. The resulting data match perfectly with the index‑p‑abelianization data (IPAD) computed by Boston, Bush, and Hajir.

In total, 19 distinct isomorphism classes of G_K / D₄(G_K) appear for d(G_K)=2, 13 of which correspond to Zassenhaus type (3, 3). The authors compare the observed frequencies of these 19 classes with the probabilities predicted by a Cohen–Lenstra‑type heuristic for weak Schur σ‑groups (as formalized in earlier work). The agreement is good, especially considering that the predicted probabilities span several orders of magnitude. The rarest class, where G_K / D₄(G_K) ≅ F₂ / D₄(F₂) (i.e., Zassenhaus type (a,b) with a,b ≥ 5), occurs for exactly 46 fields, consistent with the fact that such groups are known to be infinite by Ko‑ch–Venkov.

Overall, the paper achieves three main goals: (1) it establishes powerfulness of D₂(G) in most (3, 3) cases for p = 3, leading to p‑adic analytic structure and connections to PGL₂(ℚ₃); (2) it links these structural results to conjectural frameworks (Fontaine–Mazur, McLeman, Cohen–Lenstra) providing both theoretical and statistical evidence; and (3) it supplies extensive computational data confirming the heuristic predictions for a large set of imaginary quadratic fields. The methods introduced—particularly the powerfulness criterion and the Massey‑product based presentations—are likely to be useful for future investigations of Schur σ‑groups at other primes or with higher generator rank.