The imaginary case of the nonabelian Cohen--Lenstra heuristics

2025/07/29 by Liu, Yuan, Willyard, Ken · 1 citation
#FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2507.21558

Abstract

For a finite group Γ, we study the distribution of the Galois group G_∅^#(K) of the maximal unramified extension of K that is split completely at ∞ and has degree prime to |Γ| and Char(K), as K varies over imaginary Γ-extensions of ℚ or \mathbbF_q(t). In the function field case, we compute the moments of the distribution of G_∅^#(K) by counting points on Hurwitz stacks. In order to understand the probability of the distribution, we prove that G_∅^#(K) admits presentations of a specific form, then use this presentation to build random groups to simulate the behavior of G_∅^#(K), and make the conjecture to predict the distribution using the probability measures of these random groups. Our results provide the imaginary analog of the work of Wood, Zureick-Brown, and the first author on the nonabelian Cohen--Lenstra heuristics.

Citations

• Homological stability for Hurwitz spaces and applications
• On the Distribution of Class Groups of Abelian Extensions
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• Conjectures for distributions of class groups of extensions of number fields containing roots of unity
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• A predicted distribution for Galois groups of maximal unramified extensions
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• Heuristics for p-class towers of imaginary quadratic fields, with an Appendix by Jonathan Blackhurst
• Homological stability for Hurwitz spaces and the Cohen-Lenstra conjecture over function fields
• Derived 𝑙-adic categories for algebraic stacks

Cited by

• The leading constant in Malle's conjecture over function fields

Related

• A predicted distribution for Galois groups of maximal unramified extensions
• Cohen-Lenstra Moments for Some Nonabelian Groups
• Non-abelian Cohen--Lenstra Heuristics in the presence of roots of unity
• Moments and interpretations of the Cohen-Lenstra-Martinet heuristics
• Heuristics for p-class towers of imaginary quadratic fields, with an Appendix by Jonathan Blackhurst