TY - JOUR

T1 - What is the probability that a random integral quadratic form in nvariables has an integral zero?

AU - Bhargava, Manjul

AU - Cremona, John E.

AU - Fisher, Tom

AU - Jones, Nick G.

AU - Keating, Jonathan P.

N1 - Funding Information:
The first author (Bhargava) was supported by a Simons Investigator Grant and NSF grant DMS-1001828; the second (Cremona) and fifth (Keating) authors were supported by EPSRC Programme Grant EP/K034383/1 LMF: L-Functions and Modular Forms; the fifth author (Keating) was also supported by a grant from The Leverhulme Trust, a Royal Society Wolfson Merit Award, a Royal Society Leverhulme Senior Research Fellowship, and by the Air Force Office of Scientific Research, Air Force Material Command, USAF, under grant number FA8655-10-1-3088. Funding to pay the Open Access publication charges for this article was provided by the University of Warwick's RCUK Open Access Fund.

PY - 2016

Y1 - 2016

N2 - We show that the density of quadratic forms in nvariables 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 nvariables 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%.

AB - We show that the density of quadratic forms in nvariables 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 nvariables 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%.

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U2 - 10.1093/imrn/rnv251

DO - 10.1093/imrn/rnv251

M3 - Article

AN - SCOPUS:84981308718

VL - 2016

SP - 3828

EP - 3848

JO - International Mathematics Research Notices

JF - International Mathematics Research Notices

SN - 1073-7928

IS - 12

ER -