Generalizations of results of Friedman and Washington on cokernels of random $p$-adic matrices

Abstract

Let $p$ be prime and $X$ be a Haar-random $n\times n$ matrix over ${\mathbb{Z}}_p$, the ring of $p$-adic integers. Let $P_1(t),\dots ,P_l(t)\in {\mathbb{Z}}_p[t]$ be monic polynomials of degree at most $2$ whose images modulo $p$ are distinct and irreducible in ${\mathbb{F}}_p[t]$. For each $j$, let $G_j$ be a finite module over ${\mathbb{Z}}_p[t]/(P_j(t))$. We show that as $n$ goes to infinity, the probabilities that $\mathrm{cok}(P_j(X))\simeq G_j$ are independent, and each probability can be described in terms of a Cohen-Lenstra distribution. We also show that for any fixed $n$, the probability that $\mathrm{cok}(P_j(X))\simeq G_j$ for each $j$ is a constant multiple of the probability that that $\mathrm{cok}(P_j(\bar{X}))\simeq G_j/p{G}_j$ for each $j$, where $\bar{X}$ is an $n\times n$ uniformly random matrix over ${\mathbb{F}}_p$. These results generalize work of Friedman and Washington and prove new cases of a conjecture of Cheong and Huang.

17 pages