A predicted distribution for Galois groups of maximal unramified extensions.

We consider the distribution of the Galois groups Gal (K un / K) of maximal unramified extensions as K ranges over Γ -extensions of ℚ or F q (t) . We prove two properties of Gal (K 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 → ∞ , agreement of Gal (K un / K) as K varies over totally real Γ -extensions of F q (t) with our distribution from Part I, in the moments that are relatively prime to q (q − 1) | Γ | . In particular, we prove for every finite group Γ , in the q → ∞ limit, the prime-to- q (q − 1) | Γ | -moments of the distribution of class groups of totally real Γ -extensions of F q (t) agree with the prediction of the Cohen–Lenstra–Martinet heuristics. [ABSTRACT FROM AUTHOR]