# Small Zeros of Hermitian Forms over a Quaternion Algebra

## Document Type

Article

## Department

Mathematics (CMC)

## Publication Date

2010

## Abstract

Let *D* be a positive definite quaternion algebra over a totally real number field *K*, *F*(X,Y) a hermitian form in 2*N* 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.

## Rights Information

© 2010 IMPAN

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## Recommended Citation

Chan, Wai Kiu, and Lenny Fukshansky. "Small Zeros of Hermitian Forms over a Quaternion Algebra." Acta Arithmetica 142.3 (2010): 251-266. Web. 3 Apr. 2012. DOI:10.4064/aa142-3-3