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

TL;DR

This paper investigates the distribution of equivalence classes of random symmetric matrices over p-adic integers, providing new insights and applications including an alternative proof for quadratic form zero probability.

Contribution

It introduces a detailed distribution analysis of symmetric p-adic matrices' canonical forms and applies this to problems in quadratic form theory.

Findings

Distribution formulas for symmetric p-adic matrices' canonical forms

Application to probability of non-trivial zeros in quadratic forms

Alternative proof for known quadratic form zero probability result

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.