Non-degenerate symmetric bilinear forms over any field, in particular function fields, have been studied extensively for many years. It is known that there is a correspondence between function fields and geometric objects, namely algebraic varieties over the ground field, but somehow these geometric objects carry more information than the corresponding function field. Therefore, it makes sense to study non-degenerate symmetric bilinear forms on algebraic varieties, or more generally, over schemes. The author of the book under review started this research in the late 1960s and an important tool that he found was the specialization of these forms, namely if K and L are fields and if f: K → L is a place, there is a way to assign to a quadratic form (or symmetric bilinear form) q over K, a form over L, at least if q has good reduction with respect to f, by simply applying the map f to the coefficients that define q (assuming, of course that these coefficients belong to the valuation ring of f ). For quadratic forms this specialization defines a unique form over L, but for bilinear forms it only defines a Witt class of forms over L. As usual when dealing with (quadratic or symmetric bilinear) forms. the theory works well if the characteristic of the fields is not 2. A successful theory has been developed using these tools, first for the local case and lately for global fields. Using these tools, the author obtained several important results on general splitting of quadratic forms.
The aim of the book under review is to give a detailed treatment of specialization without the usual restriction on the characteristic of the fields involved. The restriction to characteristic not equal to 2 when dealing with quadratic forms is quite natural, since the polarization identity of linear algebra has a ½ factor. But it becomes necessary to deal with characteristic 2 when one wants to attach a geometric object to a quadratic form. Perhaps an analogy with the arithmetic of elliptic curves (or more generally, Abelian varieties) is not out of place: To understand certain arithmetic aspects of an elliptic curve, even over the rational field, it is sometimes necessary to look at its reductions modulo a prime.
In fact, the specialization theory of quadratic and bilinear forms uses a concept of good reduction for these forms similar to the one used for elliptic or Abelian varieties. This is recalled and used in the first and second chapters of the book under review. It turns out that good reduction has some limitations and the author introduces, in chapter two, the concept of fair reduction of quadratic or bilinear forms to deal with these limitations. With these tools at hand the author obtains a more general specialization theory and with it several generic splittings of quadratic forms in chapter three. Chapter four is devoted to a further generalization of the theory by considering specializations with respect to quadratic places instead of the usual places.
This is an important monograph in which the author has done an excellent job putting together in one place many important results on specialization of quadratic and bilinear forms and generic splitting of quadratic forms that were previously dispersed in several research articles. It belongs on the shelf of any mathematician interested on the algebraic, geometric and arithmetic aspects of quadratic forms. The book under review joins the class of the ever expanding literature on quadratic forms that recently has seen also the addition of The Algebraic and Geometric Theory of Quadratic Forms by Elman, Karpenko and Merkurjev (AMS, 2008), Shimura’s monograph Arithmetic of Quadratic Forms (Springer, 2010), and the text-book Formes Quadratiques sur un Corps by B. Kahn (SMF, 2009).
Felipe Zaldivar is Professor of Mathematics at the Universidad Autonoma Metropolitana-I, in Mexico City. His e-mail address is email@example.com.