Small Zeros of Hermitian Forms over Quaternion Algebras

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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.


This lecture is the product of a joint work between W.K. Chan and Lenny Fukshansky, and is related to the article, "Small Zeros of Hermitian Forms over a Quaternion Algebra" from Acta Arithmetica Volume 142, Issue 3, pages 251-266.

The lecture was given during the Number Theory Seminar at Institut de Mathématiques de Jussieu in Paris, France in October 2010.

It was also meant to be given during the AMS-KMS Special Session: Arithmetic of quadratic forms, First Joint Meeting of AMS and Korean Mathematical Society, in Seoul, South Korea on 16-20 December 2009, but was canceled.

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© 2010 Lenny Fukshansky

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