Small zeros of hermitian forms over a quaternion algebra

TL;DR

This paper extends classical results on small zeros of quadratic forms to hermitian forms over quaternion algebras, proving the existence of small-height isotropic bases in a non-commutative setting.

Contribution

It establishes a non-commutative analogue of Vaaler's theorem, providing a method to find small-height bases where hermitian forms vanish over quaternion algebras.

Findings

Existence of small-height isotropic bases over quaternion algebras.

Extension of Cassels' theorem to non-commutative quaternionic context.

Generalization of classical quadratic form results to hermitian forms over division algebras.

Abstract

Let $D$ be a positive definite quaternion algebra over a totally real number field $K$, $F(X,Y)$ a hermitian form in 2N variables over $D$, and $Z$ a right $D$-vector space which is isotropic with respect to $F$. We prove the existence of a small-height basis for $Z$ over $D$, such that $F(X,X)$ vanishes at each of the basis vectors. This constitutes a non-commutative analogue of a theorem of Vaaler, and presents an extension of the classical theorem of Cassels on small zeros of rational quadratic forms to the context of quaternion algebras.