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

TL;DR

This survey explores Cassels' theorem on small-height zeros of quadratic forms over Q, extending to various fields and rings, and discusses recent effective structural results and open problems in the area.

Contribution

It provides a comprehensive overview of classical and recent generalizations of Cassels' theorem, highlighting new effective results and open questions in quadratic forms theory.

Findings

Extensions of Cassels' theorem to different fields and rings

Recent effective structural theorems for quadratic spaces

Cassels'-type theorems with additional conditions

Abstract

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.