The Algebraic and Geometric Theory of Quadratic Forms

The theory of quadratic forms has a long and distinguished history, starting with Fermat and going through Minkowski, Siegel, Hasse and Eichler. With its emphasis on individual quadratic forms, the arithmetic theory is well explained in the books by Eichler (Springer, 1952) and O'Meara (Springer, 1963). The algebraic theory of quadratic forms, starting with the work of Witt in the 1930s through its rebirth in 1960s with the work of Pfister, shifts the emphasis from a particular quadratic form to the set of all such (non degenerate) forms over a fixed ground field, associating to this set an algebraic object, the Witt ring. Most of the algebraic theory of quadratic forms is in the well-known books by T. Y. Lam (Benjamin, 1973, now in its second edition AMS, 2005) and W. Scharlau (Springer, 1985). In both books, following tradition, the theory is developed for fields of characteristic different from 2, to avoid special cases, starting already with the correspondence between quadratic and symmetric bilinear forms given by the polarization identity of linear algebra. During the 1970s and 1980s, fueled by conjectures of Milnor relating his algebraic K-theory groups to the Galois cohomology of the ground field and the firsts results towards a proof of one of these conjectures by Merkurjev and Suslin, a geometric approach to the theory of quadratic forms started to develop, culminating with the recent proofs of Milnor’s conjectures by Voevodsky, Orlov and Vishik.

In this new book on quadratic forms, breaking with tradition and taking into account recent developments, the study of quadratic forms is not restricted to characteristic different from 2, and whenever possible characteristic-free proofs are given of classical or new results. This is done at the cost of splitting the study of quadratic forms from the study of symmetric bilinear forms, and the book focuses mainly on quadratic forms.

This characteristic-free approach is well suited for attaching geometric objects to a quadratic form, for example, the corresponding quadric or the variety of isotropic subspaces of fixed codimension in a projective space. Many properties of quadratic forms can now be formulated in terms of the associated geometric objects; for example, a quadratic form is isotropic if and only if the corresponding quadric has a rational point.

Another example comes when one considers a quadratic form q over a field F and a finitely generated field extension L/F. There is a variety X over F with function field L, and the given form q is isotropic over L if and only if there is a rational morphism from X to the quadric associated to the form q.

By identifying a morphism with its graph one has an important example of a correspondence, which brings in the category of Chow correspondences and provides new geometric tools for the study of quadratic forms. One can use this, for example, to study algebraic cycles on powers of a smooth projective quadric over a given field. The last part of the book is devoted to such topics, collecting results dispersed on the recent literature, providing a new valuable reference for the specialist.

The book does not include the recent proofs of Milnor’s conjectures, but it provides a new proof of a theorem of Merkurjev of the degree 2 case of Milnor’s conjecture: the norm-residue map gives an isomorphism between the quotient K₂(F)/2K₂(F) of Milnor’s K-group and the 2-torsion in the Brauer group of F, when the characteristic of F is not 2. The new proof uses an analogue of Hilbert’s theorem 90 for Milnor’s K₂.

This book is a welcome addition to the vast literature on quadratic forms, summarizing recent advances of the theory, highlighting the algebraic geometry approach and shedding new light on the classical results.

Felipe Zaldivar is Professor of Mathematics at the Universidad Autonoma Metropolitana-I, in Mexico City. His e-mail address is fzc@oso.izt.uam.mx