UNIVERSALITY OF THE COKERNELS OF RANDOM p-ADIC HERMITIAN MATRICES

Abstract

In this paper, we study the distribution of the cokernel of a general random Hermitian matrix over the ring of integers O of a quadratic extension K of Qp. For each positive integer n, let Xn be a random n × n Hermitian matrix over O whose upper triangular entries are independent and their reductions are not too concentrated on certain values. We show that the distribution of the cokernel of Xn always converges to the same distribution which does not depend on the choices of Xn as n→∞and provide an explicit formula for the limiting distribution. This answers Open Problem 3.16 from the ICM 2022 lecture note of Wood [Probability theory for random groups arising in number theory, 2022] in the case of the ring of integers of a quadratic extension of Qp.