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

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 an 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. In particular, we provide a generalization of two results of Friedman and Washington about these random matrices. 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. Our results over ${\mathbb{F}}_q$ are about 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.