A predicted distribution for Galois groups of maximal unramified extensions

Abstract

We consider the distribution of the Galois groups $\mathrm{Gal}(K^{\mathrm{un}}/K)$ of maximal unramified extensions as $K$ ranges over $\Gamma$-extensions of $\mathbb{Q}$ or ${\mathbb{F}}_q(t)$. We prove two properties of $\mathrm{Gal}(K^{\mathrm{un}}/K)$ 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 $q\to \infty$, agreement of $\mathrm{Gal}(K^{\mathrm{un}}/K)$ 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 $q\to \infty$ 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.

arXiv commentcontains minor corrections and updates from the previous version

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Cited by in corpus (1)

• Deficiency of p-Class Tower Groups and Minkowski Units