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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 theprobability 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-Lenstraheuristics for the classical case when $u$ is negative.

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