Duke Mathematical Journal

Nonabelian Cohen–Lenstra moments

Melanie Matchett Wood

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In this article, we give a conjecture for the average number of unramified G-extensions of a quadratic field for any finite group G. The Cohen–Lenstra heuristics are the specialization of our conjecture to the case in which G is abelian of odd order. We prove a theorem toward the function field analogue of our conjecture and give additional motivations for the conjecture, including the construction of a lifting invariant for the unramified G-extensions that takes the same number of values as the predicted average and an argument using the Malle–Bhargava principle. We note that, for even |G|, corrections for the roots of unity in Q are required, which cannot be seen when G is abelian.

Article information

Duke Math. J., Volume 168, Number 3 (2019), 377-427.

Received: 22 February 2017
Revised: 6 July 2018
First available in Project Euclid: 29 January 2019

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 11R29: Class numbers, class groups, discriminants
Secondary: 11R45: Density theorems

unramified extensions Cohen–Lenstra class tower groups lifting invariants class groups


Wood, Melanie Matchett. Nonabelian Cohen–Lenstra moments. Duke Math. J. 168 (2019), no. 3, 377--427. doi:10.1215/00127094-2018-0037. https://projecteuclid.org/euclid.dmj/1548730815

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