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 ${\mathbb{Z}}_p$ 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 ${\mathbb{Z}}_p$ has a non-trivial zero.

23 pages, with additional references and minor corrections

References in corpus (4)

• Limits and fluctuations of $p$-adic random matrix products
• What is the probability that a random integral quadratic form in $n$ variables has an integral zero?
• On determinants of random symmetric matrices over Z_m
• Traces of powers of matrices over finite fields