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June 2021 Cohen–Lenstra distributions via random matrices over complete discrete valuation rings with finite residue fields
Gilyoung Cheong, Yifeng Huang
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Illinois J. Math. 65(2): 385-415 (June 2021). DOI: 10.1215/00192082-8939615

Abstract

Let (R,m) be a complete discrete valuation ring with the finite residue field Rm=Fq. Given a monic polynomial P(t)R[t] whose reduction modulo m gives an irreducible polynomial P(t)Fq[t], we initiate an investigation of the distribution of coker(P(A)), where AMatn(R) is randomly chosen with respect to the Haar probability measure on the additive group Matn(R) of n×n R-matrices. In particular, we provide a generalization of two results of Friedman and Washington about these random matrices. We use some concrete combinatorial connections between Matn(R) and Matn(Fq) to translate our problems about a Haar-random matrix in Matn(R) into problems about a random matrix in Matn(Fq) with respect to the uniform distribution. Our results over Fq are about the distribution of the P-part of a random matrix AMatn(Fq) with respect to the uniform distribution, and one of them generalizes a result of Fulman. We heuristically relate our results to a celebrated conjecture of Cohen and Lenstra, which predicts that given an odd prime p, any finite abelian p-group (i.e., Zp-module) H occurs as the p-part of the class group of a random imaginary quadratic field extension of Q with a probability inversely proportional to |AutZ(H)|. We review three different heuristics for the conjecture of Cohen and Lenstra, and they are all related to special cases of our main conjecture, which we prove as our main theorems.

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Gilyoung Cheong. Yifeng Huang. "Cohen–Lenstra distributions via random matrices over complete discrete valuation rings with finite residue fields." Illinois J. Math. 65 (2) 385 - 415, June 2021. https://doi.org/10.1215/00192082-8939615

Information

Received: 1 June 2020; Revised: 15 November 2020; Published: June 2021
First available in Project Euclid: 25 March 2021

Digital Object Identifier: 10.1215/00192082-8939615

Subjects:
Primary: 05E15
Secondary: 11C20

Rights: Copyright © 2021 by the University of Illinois at Urbana–Champaign

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Vol.65 • No. 2 • June 2021
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