What is the probability that a random integral quadratic form in $n$ variables is isotropic?

Abstract

We show that the density of quadratic forms in $n$ variables over ${\mathbb{Z}}_p$ that are isotropic is a rational function in $p$, where the rational function is independent of $p$, and we determine this rational function explicitly. As a consequence, for each $n$, we determine the probability that a random integral quadratic form in $n$ variables is isotropic. In particular, we show that the probability that a random integral quaternary quadratic form is isotropic is $\approx 97.0\%$, in the case where the coefficients of the quadratic form are independently and uniformly distributed in the range $[-X,X]$ with $X\to \infty$. When random integral quaternary quadratic forms are chosen with respect to the Gaussian Orthogonal Ensemble (GOE), the probability of isotropy increases to $\approx 98.3\%$.

This paper is superceded by a newer paper to appear shortly, entitled "What is the probability that a random integral quadratic form in $n$ variables has an integral zero?" (by M. Bhargava, J. E. Cremona, T. A. Fisher, N. G. Jones, and J. P. Keating)

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