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

Abstract

In this paper, we study the joint distribution of the cokernels of random p-adic matrices. Let p be a prime and P1(t), • • • , P l (t) ∈ Zp[t] be monic polynomials whose reductions modulo p in Fp[t] are distinct and irreducible. We determine the limit of the joint distribution of the cokernels cok(P1(A)), • • • , cok(P l (A)) for a random n × n matrix A over Zp with respect to Haar measure as n → ∞. By applying the linearization of a random matrix model, we also provide a conjecture which generalizes this result. Finally, we provide a sufficient condition that the cokernels cok(A) and cok(A + Bn) become independent as n → ∞, where Bn is a fixed n × n matrix over Zp for each n and A is a random n × n matrix over Zp.

Universality of the cokernels of random $p$-adic Hermitian matrices

“…Another way is to generalize the cokernel condition. We refer to the introduction of [10] for the recent progress in this direction. The following theorem provides the joint distribution of the cokernels cok(P j (A)) (1 ≤ j ≤ l), where P 1 (t), • • • , P l (t) ∈ Z p [t] are monic polynomials with some mild assumptions and A is a Haar random matrix in M n (Z p ).…”

“…Assume that K/Q p is ramified. Then for every sequence of ε-balanced matrices (X n ) n≥1 (X n ∈ H n (O)), the limiting distribution of cok(X n ) is given by the equation (10).…”