The effective study of quadratic forms originated with a paper of Cassels in 1955, in which he proved that if an integral quadratic form is isotropic, then it has non-trivial zeros of bounded height. Here height stands for a certain measure of arithmetic complexity, which we will make precise. This theorem has since been generalized and extended in a number of different ways. We will discuss some of such generalizations for quadratic spaces over a fixed number field as well as over the field of algebraic numbers. Specifically, let K be either a number field or its algebraic closure, and let F be a quadratic form in N \geq 2 variables with coefficients in K. Let Z be a subspace of K^N, and suppose that the quadratic space (Z,F) is isotropic. We will discuss the existence of maximal totally isotropic subspaces of (Z,F) of bounded height, and orthogonal Witt decomposition of (Z,F) with components of bounded height. We will also discuss representations of isometries of (Z,F) as products of reflections of bounded height, which can be thought of as an effective version of Cartan-Dieudonne theorem. Lecture given at the Colloquium of the Graduiertenkolleg, June 2006.
© 2006 Lenny Fukshansky
Fukshansky, Lenny. "Quadratic forms and height functions." Colloquium of the Graduiertenkolleg, University of Göttingen, June 2006.