Small zeros of hermitian forms over a quaternion algebra

Wai Kiu Chan; Lenny Fukshansky

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arXiv · 0911.4104

Small zeros of hermitian forms over a quaternion algebra

Wai Kiu Chan·Lenny Fukshansky

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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.

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Wai Kiu Chan, Lenny Fukshansky. 2009-11-20. Small zeros of hermitian forms over a quaternion algebra. https://arxiv.org/abs/0911.4104

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