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15 February 2019 Nonabelian Cohen–Lenstra moments
Melanie Matchett Wood
Duke Math. J. 168(3): 377-427 (15 February 2019). DOI: 10.1215/00127094-2018-0037

Abstract

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.

Citation

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Melanie Matchett Wood. "Nonabelian Cohen–Lenstra moments." Duke Math. J. 168 (3) 377 - 427, 15 February 2019. https://doi.org/10.1215/00127094-2018-0037

Information

Received: 22 February 2017; Revised: 6 July 2018; Published: 15 February 2019
First available in Project Euclid: 29 January 2019

zbMATH: 07040613
MathSciNet: MR3909900
Digital Object Identifier: 10.1215/00127094-2018-0037

Subjects:
Primary: 11R29
Secondary: 11R45

Keywords: class groups , class tower groups , Cohen–Lenstra , lifting invariants , unramified extensions

Rights: Copyright © 2019 Duke University Press

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Vol.168 • No. 3 • 15 February 2019
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