Joint distribution of the cokernels of random p-adic matrices

Abstract

Abstract In this paper, we study the joint distribution of the cokernels of random p-adic matrices. Let p be a prime and let P 1 ⁢ ( t ) , … , P l ⁢ ( t ) ∈ ℤ p ⁢ [ t ] {P_{1}(t),\ldots,P_{l}(t)\in\mathbb{Z}_{p}[t]} be monic polynomials whose reductions modulo p in 𝔽 p ⁢ [ t ] {\mathbb{F}_{p}[t]} are distinct and irreducible. We determine the limit of the joint distribution of the cokernels cok ⁡ ( P 1 ⁢ ( A ) ) , … , cok ⁡ ( P l ⁢ ( A ) ) {{\operatorname{cok}}(P_{1}(A)),\ldots,{\operatorname{cok}}(P_{l}(A))} for a random n × n {n\times n} matrix A over ℤ p {\mathbb{Z}_{p}} with respect to the Haar measure as n → ∞ {n\rightarrow\infty} . 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 ) {{\operatorname{cok}}(A)} and cok ⁡ ( A + B n ) {{\operatorname{cok}}(A+B_{n})} become independent as n → ∞ {n\rightarrow\infty} , where B n {B_{n}} is a fixed n × n {n\times n} matrix over ℤ p {\mathbb{Z}_{p}} for each n and A is a random n × n {n\times n} matrix over ℤ p {\mathbb{Z}_{p}} .