A predicted distribution for Galois groups of maximal unramified extensions

Abstract

We consider the distribution of the Galois groups \(\operatorname {Gal}(K^{\operatorname{un}}/K)\) of maximal unramified extensions as \(K\) ranges over \(\Gamma \)-extensions of ℚ or \({{\mathbb{F}}}_{q}(t)\). We prove two properties of \(\operatorname {Gal}(K^{\operatorname{un}}/K)\) coming from number theory, which we use as motivation to build a probability distribution on profinite groups with these properties. In Part I, we build such a distribution as a limit of distributions on \(n\)-generated profinite groups. In Part II, we prove as \(q\rightarrow \infty \), agreement of \(\operatorname {Gal}(K^{\operatorname{un}}/K)\) as \(K\) varies over totally real \(\Gamma \)-extensions of \({{\mathbb{F}}}_{q}(t)\) with our distribution from Part I, in the moments that are relatively prime to \(q(q-1)|\Gamma |\). In particular, we prove for every finite group \(\Gamma \), in the \(q\rightarrow \infty \) limit, the prime-to-\(q(q-1)|\Gamma |\)-moments of the distribution of class groups of totally real \(\Gamma \)-extensions of \({{\mathbb{F}}}_{q}(t)\) agree with the prediction of the Cohen–Lenstra–Martinet heuristics.