Heights and quadratic forms: on Cassels' theorem and its generalizations

Details Heights and quadratic forms: on Cassels' theorem and its generalizations Heights and quadratic forms: on Cassels' theorem and its generalizations

2012-04-09

arxiv.org/abs/1204.1938v1

Mathematics

Number Theory

16 pages; to appear in the proceedings of the BIRS workshop on "Diophantine methods, lattices, and arithmetic theory of quadra

Scientific paper

In this survey paper, we discuss the classical Cassels' theorem on existence of small-height zeros of quadratic forms over Q and its many extensions, to different fields and rings, as well as to more general situations, such as existence of totally isotropic small-height subspaces. We also discuss related recent results on effective structural theorems for quadratic spaces, as well as Cassels'-type theorems for small-height zeros of quadratic forms with additional conditions. We conclude with a selection of open problems.

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