Small zeros of hermitian forms over a quaternion algebra

Details Small zeros of hermitian forms over a quaternion algebra Small zeros of hermitian forms over a quaternion algebra

2009-11-20

arxiv.org/abs/0911.4104v1

Acta Arithmetica, vol. 142 no. 3 (2010), pg. 251--266

Mathematics

Number Theory

14 pages; to appear in Acta Arithmetica

Scientific paper

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.

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