joint with Manjul Bhargava, John Cremona, Nick Jones and Jon Keating
We show that the density of quadratic forms in n variables over Zp that are isotropic is a rational function of p, where the rational function is independent of p, and we determine this rational function explicitly. When real quadratic forms in n variables are distributed according to the Gaussian Orthogonal Ensemble (GOE) of random matrix theory, we determine explicitly the probability that a random such real quadratic form is isotropic (i.e., indefinite).
As a consequence, for each n, we determine an exact expression for the probability that a random integral quadratic form in n variables is isotropic (i.e., has a nontrivial zero over Z), when these integral quadratic forms are chosen according to the GOE distribution. In particular, we find an exact expression for the probability that a random integral quaternary quadratic form is isotropic; numerically, this probability of isotropy is approximately 98.3%.