A predicted distribution for Galois groups of maximal unramified extensions

Yuan Liu, Melanie Matchett Wood, David Zureick-Brown

Research output: Contribution to journalArticlepeer-review


We consider the distribution of the Galois groups Gal(Kun/K) of maximal unramified extensions as K ranges over Γ-extensions of ℚ or Fq(t). We prove two properties of Gal(Kun/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(Kun/K) as K varies over totally real Γ-extensions of Fq(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 Fq(t) agree with the prediction of the Cohen–Lenstra–Martinet heuristics.

Original languageEnglish (US)
Pages (from-to)49-116
Number of pages68
JournalInventiones Mathematicae
Issue number1
StatePublished - Jul 2024

ASJC Scopus subject areas

  • General Mathematics


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