Distribution of cokernels of ($n$+$u$) $\times$ $n$ matrices over $\mathbb{Z}_p$

TL;DR

This paper investigates the distribution of cokernels of certain matrices over rac{p} matrices over rac{p} and confirms that their probability distribution aligns with Cohen-Lenstra heuristics, extending understanding of random matrix cokernels.

Contribution

The paper derives exact probabilities for cokernels of (rac{p} matrices over rac{p} and shows they match Cohen-Lenstra predictions, including cases with positive and negative u.

Findings

Probability matches Cohen-Lenstra heuristics for large n

Distribution of cokernels includes rac{p}^u  G for finite abelian p-groups G

Results extend classical cases to new matrix dimensions

Abstract

Let $n, u \geq 0$, $M$ be a ($n$+$u$) $\times$ $n$ matrices over $\mathbb{Z}_p$, and $G$ be a finite abelian p-group group. We find that the probability that the cokernel of $M$ is isomorphic to $\mathbb{Z}_p^u \oplus G$ as $n$ goes to infinity is exactly what is expected from Cohen-Lenstra heuristics for the classical case when $u$ is negative.