Totally Isotropic Subspaces Of Small Height In Quadratic Spaces

Wai K. Chan, Wesleyan University
Lenny Fukshansky, Claremont McKenna College
Glenn R. Henshaw, Wesleyan University


Abstract. Let K be a global field or Q, F a nonzero quadratic form on K N , N ≥ 2, and V a subspace of K N . We prove the existence of an infinite collection of finite families of small-height maximal totally isotropic subspaces of (V, F ) such that each such family spans V as a K -vector space. This result generalizes and extends a well known theorem of J. Vaaler [16] and further contributes to the effective study of quadratic forms via height in the general spirit of Cassels’ theorem on small zeros of quadratic forms. All bounds on height are explicit.