The cokernel of a polynomial push-forward of a random integral matrix with concentrated residue

We prove new statistical results about the distribution of the cokernel of a random integral matrix with a concentrated residue. Given a prime $p$ and a positive integer $n$, consider a random $n\times n$ matrix $X_n$ over the ring ${\mathbb{Z}}_p$ of $p$-adic integers whose entries are independent. Previously, Wood showed that regardless of the distribution of $X_n$, as long as each entry of $X_n$ is not too concentrated on a single residue modulo $p$, the distribution of the cokernel $\mathrm{c}\mathrm{o}\mathrm{k}(X_n)$ of $X_n$, up to isomorphism, weakly converges to the Cohen–Lenstra distribution, as $n\to \mathrm{\infty }$. In this paper, we consider the case when $X_n$ has a concentrated residue $A_n$ so that $X_n=A_n+p{B}_n$, where $B_n$ is a random $n\times n$ matrix over ${\mathbb{Z}}_p$. We show that for every fixed $n$ and a non-constant monic polynomial $P(t)\in {\mathbb{Z}}_p[t]$, we can explicitly compute the distribution of $\mathrm{c}\mathrm{o}\mathrm{k}(P(X_n))$ when $B_n$ is a Haar-random matrix. Using this, we also show that for specific choices of $A_n$ a much wider class of random matrices $B_n$ gives the same distribution of $\mathrm{c}\mathrm{o}\mathrm{k}(P(X_n))$. For the Haar-random $B_n$, we deduce our result from an interesting equidistribution result for matrices over ${\mathbb{Z}}_p[t]\mathrm{/}(P(t))$, which we prove by establishing a version of the Weierstrass preparation theorem for the noncommutative ring ${\mathrm{M}}_n({\mathbb{Z}}_p)$ of $n\times n$ matrices over ${\mathbb{Z}}_p$.