Joint distribution of the cokernels of random p-adic matrices II

Abstract

In this paper, we study the combinatorial relations between the cokernels cok(An + pxiIn) (1 ≤ i ≤ m), where An is an n × n matrix over the ring of p-adic integers ℤp, In is the n × n identity matrix and x1, ..., xm are elements of ℤp whose reductions modulo p are distinct. For a positive integer m ≤ 4 and given x1, ..., xm ∈ ℤp, we determine the set of m-tuples of finitely generated ℤp-modules (H1, ..., Hm) for which (cok(An + px1In), ..., cok(An + pxmIn)) = (H1, ..., Hm) for some matrix An. We also prove that if An is an n × n Haar random matrix over ℤp for each positive integer n, then the joint distribution of cok(An + pxiIn) (1 ≤ i ≤ m) converges as n → ∞.