TY - JOUR

T1 - A predicted distribution for Galois groups of maximal unramified extensions

AU - Liu, Yuan

AU - Wood, Melanie Matchett

AU - Zureick-Brown, David

N1 - Publisher Copyright:
© The Author(s), under exclusive licence to Springer-Verlag GmbH Germany, part of Springer Nature 2024. corrected publication 2024.

PY - 2024/7

Y1 - 2024/7

N2 - 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.

AB - 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.

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U2 - 10.1007/s00222-024-01257-1

DO - 10.1007/s00222-024-01257-1

M3 - Article

AN - SCOPUS:85189949205

SN - 0020-9910

VL - 237

SP - 49

EP - 116

JO - Inventiones Mathematicae

JF - Inventiones Mathematicae

IS - 1

ER -