Small Zeros of Hermitian Forms over a Quaternion Algebra

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Mathematics (CMC)

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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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