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

2022/12/20 by Drakengren, Lycka, Fisher, Tom
#11D09 #11E08 #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2212.10495

Abstract

We investigate the probability that a random quadratic form in ℤ[x₁,...,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 ↦ 1/p.

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