Cohen-Lenstra distributions via random matrices over complete discrete valuation rings with finite residue fields

TL;DR

This paper explores the distribution of cokernels of polynomials evaluated at random matrices over complete discrete valuation rings, connecting these distributions to the Cohen-Lenstra heuristics on class groups.

Contribution

It generalizes previous results on matrix distributions and relates them to Cohen-Lenstra heuristics, providing new insights and proofs for conjectures about class groups of quadratic fields.

Findings

Generalizes Friedman and Washington's results on matrix distributions

Relates matrix cokernel distributions to Cohen-Lenstra heuristics

Proves main conjecture connecting random matrices and class group distributions

Abstract

Let $(R, \mathfrak{m})$ be a complete discrete valuation ring with the finite residue field $R/\mathfrak{m} = \mathbb{F}_{q}$. Given a monic polynomial $P(t) \in R[t]$ whose reduction modulo $\mathfrak{m}$ gives an irreducible polynomial $\bar{P}(t) \in \mathbb{F}_{q}[t]$, we initiate the investigation of the distribution of $\mathrm{coker}(P(A))$, where $A \in \mathrm{Mat}_{n}(R)$ is randomly chosen with respect to the Haar probability measure on the additive group $\mathrm{Mat}_{n}(R)$ of $n \times n$ $R$-matrices. One of our main results generalizes two results of Friedman and Washington. Our other results are related to the distribution of the $\bar{P}$-part of a random matrix $\bar{A} \in \mathrm{Mat}_{n}(\mathbb{F}_{q})$ 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., $\mathbb{Z}_{p}$-module) $H$ occurs as the $p$-part of the class group of a random imaginary quadratic field extension of $\mathbb{Q}$ with a probability inversely proportional to $|\mathrm{Aut}_{\mathbb{Z}}(H)|$. 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. For proofs, we use some concrete combinatorial connections between $\mathrm{Mat}_{n}(R)$ and $\mathrm{Mat}_{n}(\mathbb{F}_{q})$ to translate our problems about a Haar-random matrix in $\mathrm{Mat}_{n}(R)$ into problems about a random matrix in $\mathrm{Mat}_{n}(\mathbb{F}_{q})$ with respect to the uniform distribution.