A predicted distribution for Galois groups of maximal unramified extensions (Q6560721)

Fix a finite group \(\Gamma\). Let \(Q\) be equal to either the rational field \(\mathbb{Q}\) or the function field \(\mathbb{F}_q(t)\) for some prime power \(q\). A \(\Gamma\)-extension of \(Q\) is defined to be a Galois extension \(K/Q\) together with an isomorphism Gal\((K/Q)\cong\Gamma\). For a \(\Gamma\)-extension \(K/Q\) let \(\Delta=2|\Gamma|\) if \(Q=\mathbb{Q}\) and \(\Delta=q(q-1)|\Gamma|\) if \(Q=\mathbb{F}_q(t)\). Let \(K^{\#}/K\) be the maximal Galois extension such that for every finite subextension \(L/K\) of \(K^{\#}/K\), \(L/K\) is everywhere unramified, all infinite places of \(K\) split completely in \(L\), and \([L:K]\) is relatively prime to \(\Delta\). This paper considers the distribution of the groups Gal\((K^{\#}/K)\) as \(K\) varies over the \(\Gamma\)-extensions of \(Q\).\N\NA \(\Gamma\)-group is defined to be a profinite group \(H\) with a continuous action by \(\Gamma\). Say that \(H\) is an admissible \(\Gamma\)-group if \(H\) is generated (topologically) by the set \(\{h^{-1}\cdot\gamma(h):h\in H,\,\gamma\in\Gamma\}\). Say that the \(\Gamma\)-group \(H\) has Property E if for every prime \(p\) with \(p\nmid2|\Gamma|\) and every non-split central extension of \(\Gamma\)-groups \(1\rightarrow\mathbb{Z}/p\mathbb{Z}\rightarrow \widetilde{G}\rightarrow G\rightarrow1\), one can lift every surjection \(H\rightarrow G\) of \(\Gamma\)-groups to a surjection \(H\rightarrow\widetilde{G}\). It is proved here that for every \(\Gamma\)-extension \(K/Q\), the \(\Gamma\)-group Gal\((K^{\#}/K)\) is admissible and has Property E. For each \(n\ge1\) the authors construct a free admissible \(\Gamma\)-group \(\mathcal{F}_n\) on \(n\) generators. They then show that a quotient of \(\mathcal{F}_n\) has Property E if and only if it is of the form \(\mathcal{F}_n/[r^{-1}\cdot\gamma(r)]_{r\in S, \,\gamma\in\Gamma}\) for some \(S\subset\mathcal{F}_n\). Since \(\mathcal{F}_n\) is compact one can define a random admissible group with Property E by \(X_{\Gamma,n}=\mathcal{F}_n/[r^{-1}\cdot\gamma(r)]_{r\in S, \,\gamma\in\Gamma}\) where \(S=\{s_1,\dots,s_{n+1}\}\) is chosen randomly using the Haar measure on \(\mathcal{F}_n\). This leads to the first main result of the paper, which says that there is a probability measure \(\mu_{\Gamma}\) on the set of isomorphism classes of admissible \(\Gamma\)-groups such that as \(n\rightarrow\infty\), the distributions given by the groups \(X_{\Gamma,n}\) converge weakly to \(\mu_{\Gamma}\).\N\NThe authors conjecture that if we order the fields \(K\) by the norm of the radical of Disc\((K)\), the \(\Gamma\)-groups Gal\((K^{\#}/K)\) are equidistributed with respect to the measure \(\mu_{\Gamma}\). A weaker version of this conjecture is proved in the function field setting, with a second limit as \(q\rightarrow\infty\) applied to the conjectured formula. It is observed that the abelianized version of the conjecture would confirm the validity of the Cohen-Lenstra-Martinet heuristics, appropriately formulated. The construction of \(\mu_{\Gamma}\) uses methods developed in [\textit{Y. Liu} and \textit{M. M. Wood}, J. Reine Angew. Math. 762, 123--166 (2020; Zbl 1481.20227)]. The proof of the weakened form of the conjecture in the function field case is based on Hurwitz schemes.