The density of integral quadratic forms having a $k$-dimensional totally isotropic subspace

TL;DR

This paper studies the likelihood that a random integral quadratic form contains a totally isotropic subspace of a specified dimension, deriving explicit formulas for local probabilities over p-adic fields.

Contribution

It provides explicit formulas for local probabilities of isotropic subspaces in quadratic forms over p-adic fields, advancing understanding of their distribution.

Findings

Global probability factors into local probabilities.

Derived rational function formulas invariant under p to 1/p substitution.

Computed explicit local probabilities over p-adics.

Abstract

We investigate the probability that a random quadratic form in ${\mathbb{Z}}[x_1,...,x_n]$ has a totally isotropic subspace of a given dimension. We show that this global probability is a product of local probabilities. Our main result computes these local probabilities for quadratic forms over the $p$-adics. The formulae we obtain are rational functions in $p$ invariant upon substituting $p \mapsto 1/p$.