Graduate Topics Course: Quadratic Forms, Fall 2012
Unique number 56485, meeting MWF 10-11 in RLM 10.176.
Daniel Allcock, office RLM 9.112.
(my last name)@math.utexas.edu.

Here is homework 1.

Here are some class notes, Rational Quadratic Forms.

A quadratic form means a symmetric bilinear pairing, except in characteristic 2 where the structure is richer. The basic questions are when two vector spaces (or Z-modules or whatever) are isometric, and what the structure of the group of self-isometries of a given quadratic form is.

The major applications beyond number theory are in topology and algebraic geometry. For example, Freedman completely classified the simply connected topological 4-manifolds. The essential invariant is the second homology, equipped with its intersection pairing, which is an integral quadratic form. Except for another Z/2 invariant which is not always needed, this is the classifying invariant. In algebraic geometry the automorphism groups of integer quadratic forms play key roles in the study of moduli of algebraic varieties, especially surfaces.

Our goals in this class are (1) to treat some but not a great deal of the general theory over fields, including the oddities in characteristic two, (2) develop the theory of rational quadratic forms up through the Hasse-Minkowski theory, and (3) develop as much of the integral theory (quadratic forms over Z) as possible, with an emphasis on practical computation of things like the genus and spinor genus of a given quadratic form. The focus on integer and rational quadratic forms derives from our interest in applications.

My favorite reference for this area is ch. 15 of Conway-Sloane's Sphere Packings, Lattices and Groups, although it contains no proofs for the integral theory. Cassels' Rational Quadratic Forms is a classic introduction that covers most of the material in the course from as elementary a viewpoint as possible. (But for computation, the Conway-Sloane approach is superior.) Finally, O. T. O'Meara's Introduction to Quadratic Forms is a classic monograph which is much more than an introduction.