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 \mathcal {O} of a quadratic extension K K of Q p \mathbb {Q}_p . For each positive integer n n , let X n X_n be a random n × n n \times n Hermitian matrix over O \mathcal {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 X n X_n always converges to the same distribution which does not depend on the choices of X n X_n as n → ∞ n \rightarrow \infty 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 Q p \mathbb {Q}_p .