Universality for Cokernels of Dedekind Domain Valued Random Matrices

TL;DR

This paper extends the universality results for cokernels of random matrices over Dedekind domains, showing that under certain conditions, their distribution aligns with conjectures on class groups, generalizing previous integral matrix results.

Contribution

It develops a new condition ensuring universality of cokernel distributions for matrices over Dedekind domains, broadening the scope beyond integers.

Findings

Establishes a universality condition for Dedekind domain matrices.

Shows the distribution matches Cohen-Lenstra conjecture under new conditions.

Generalizes previous integral matrix results to broader algebraic settings.

Abstract

We use the moment method of Wood to study the distribution of random finite modules over a countable Dedekind domain with finite quotients, generated by taking cokernels of random $n\times n$ matrices with entries valued in the domain. Previously, Wood found that when the entries of a random $n\times n$ integral matrix are not too concentrated modulo a prime, the asymptotic distribution (as $n\to\infty$) of the cokernel matches the Cohen and Lenstra conjecture on the distribution of class groups of imaginary quadratic fields. We develop and prove a condition that produces a similar universality result for random matrices with entries valued in a countable Dedekind domain with finite quotients.