Specialization of Quadratic and Symmetric Bilinear Forms

ISBN-13: 9781447125860 / Angielski / Miękka / 2012 / 192 str.

Manfred Knebusch; Thomas Unger

Opis

A Mathematician Said Who Can Quote Me a Theorem that's True? For the ones that I Know Are Simply not So, When the Characteristic is Two This pretty limerick ?rst came to my ears in May 1998 during a talk by T.Y. Lam 1 on ?eld invariants from the theory of quadratic forms. It is--poetic exaggeration allowed--a suitable motto for this monograph. What is it about? At the beginning of the seventies I drew up a specialization theoryofquadraticandsymmetricbilinear formsover ?elds 32].Let?: K? L be a place. Then one can assign a form? (?)toaform? over K in a meaningful way ? if? has "good reduction" with respect to? (see1.1). The basic idea is to simply apply the place? to the coe?cients of?, which must therefore be in the valuation ring of?. The specialization theory of that time was satisfactory as long as the ?eld L, and therefore also K, had characteristic 2. It served me in the ?rst place as the foundation for a theory of generic splitting of quadratic forms 33], 34]. After a very modest beginning, this theory is now in full bloom. It became important for the understanding of quadratic forms over ?elds, as can be seen from the book 26]of Izhboldin-Kahn-Karpenko-Vishik for instance. One should note that there exists a theoryof(partial)genericsplittingofcentralsimplealgebrasandreductivealgebraic groups, parallel to the theory of generic splitting of quadratic forms (see 29] and the literature cited there).

Recenzje

From the reviews:

"In the present book Knebusch develops a theory of specialization covering all the situations occurring by combining the cases when the characteristic of the field L is equal or unequal to 2 with the cases when the forms are either symmetric bilinear or quadratic and have a prescribed type of reduction under the place. ... The book is a welcome addition to the literature on quadratic forms supporting the tendency of presenting results for forms over all fields independently of the field characteristic." (K. Szymiczek, Mathematical Reviews, Issue 2011 h)

"The author developed a specialization theory and a subsequent generic splitting theory of quadratic and symmetric bilinear forms over fields. ... This book comprehensively covers the theory without the restriction of the characteristic. ... offers results on specialization with respect to quadratic places providing a look on further research. A book for the specialist!" (H. Mitsch, Monatshefte für Mathematik, Vol. 164 (3), November, 2011)

"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 ... ." (Feline Zaldivar, The Mathematical Association of America, January, 2011)

"The present book represents the state of the art in the theory of specialization of quadratic forms, containing a substantial amount of previously unpublished new results. It is a welcome addition to the literature as this is the first monograph in which this original and important approach to the algebraic theory of quadratic forms has been given such a comprehensive treatment." (Detlev Hoffmann, Zentralblatt MATH, Vol. 1203, 2011)

Spis treści

Fundamentals of Specialization Theory.- Generic Splitting Theory.- Some Applications.- Specialization with Respect to Quadratic Places.

Blurb

The specialization theory of quadratic and symmetric bilinear forms over fields and the subsequent generic splitting theory of quadratic forms were invented by the author in the mid-1970's. They came to fruition in the ensuing decades and have become an integral part of the geometric methods in quadratic form theory. This book comprehensively covers the specialization and generic splitting theories. These theories, originally developed mainly for fields of characteristic different from 2, are explored here without this restriction. In this book, a quadratic form φ over a field of characteristic 2 is allowed to have a big quasilinear part QL(φ) (defined as the restriction of φ to the radical of the bilinear form associated to φ), while in most of the literature QL(φ) is assumed to have dimension at most 1. Of course, in nature, quadratic forms with a big quasilinear part abound. In addition to chapters on specialization theory, generic splitting theory and their applications, the book's final chapter contains research never before published on specialization with respect to quadratic places and will provide the reader with a glimpse towards the future.