On the distribution of equivalence classes of random symmetric p-adic matrices

Abstract

We consider random symmetric matrices with independent entries distributed according to the Haar measure on $Z_p$ for odd primes p and derive the distribution of their canonical form with respect to several equivalence relations. We give a few examples of applications including an alternative proof for the result of Bhargava, Cremona, Fisher, Jones and Keating on the probability that a random quadratic form over $Z_p$ has a non-trivial zero.