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×n matrix over Zp, the ring of p-adic integers. Let P1(t),…,Pl(t)∈Zp[t] be monic polynomials of degree at most 2 whose images modulo p are distinct and irreducible in Fp[t], where Fp denotes the finite field of p elements. For each j, let Gj be a finite module over Zp[t]/(Pj(t)). We show that as n goes to infinity, the probabilities that cok(Pj(X))≃Gj 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 cok(Pj(X))≃Gj for each j is a constant multiple of the probability that cok(Pj(X¯))≃Gj/pGj for each j, where X¯ is an n×n uniformly random matrix over Fp. These results generalize work of Friedman and Washington and prove new cases of a conjecture of Cheong and Huang.