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 $P_1(t),\cdots ,P_l(t)\in {\mathbb{Z}}_p[t]$ be monic polynomials whose reductions modulo $p$ in ${\mathbb{F}}_p[t]$ are distinct and irreducible. We determine the limit of the joint distribution of the cokernels $\text{cok}(P_1(A)),\cdots ,\text{cok}(P_l(A))$ for a random $n\times n$ matrix $A$ over ${\mathbb{Z}}_p$ with respect to Haar measure as $n\to \infty$. 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 $\text{cok}(A)$ and $\text{cok}(A+B_n)$ become independent as $n\to \infty$, where $B_n$ is a fixed $n\times n$ matrix over ${\mathbb{Z}}_p$ for each $n$ and $A$ is a random $n\times n$ matrix over ${\mathbb{Z}}_p$.

18 pages, to appear in Forum Math

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