Proceedings of Symposia in Pure Mathematics 1995; 737 pp; hardcover Volume: 58 ISBN10: 0821814982 ISBN13: 9780821814987 List Price: US$191 Member Price: US$152.80 Order Code: PSPUM/58
 During the 1980s, profound connections were discovered relating modern algebraic geometry and algebraic \(K\)theory to arithmetic problems. The term "arithmetic algebraic geometry" was coined during that period and is now used to denote an entire branch of modern number theory. These same developments in algebraic geometry and \(K\)theory greatly influenced research on the arithmetic of fields in general, and the algebraic theory of quadratic forms and the theory of finitedimensional division algebras in particular. This book contains papers presented at an AMS Summer Research Institute held in July 1992 at the University of California, Santa Barbara. The purpose of the conference was to provide a broad overview of the tools from algebraic geometry and \(K\)theory that have proved to be the most powerful in solving problems in the theory of quadratic forms and division algebras. In addition, the conference provided a venue for exposition of recent research. A substantial portion of the lectures of the major conference speakersColliotThélène, Merkurjev, Raskind, Saltman, Suslin, Swanare reproduced in the expository articles in this book. Readership Research mathematicians. Table of Contents Part 1  J.L. ColliotThélène  Birational invariants, purity, and the Gersten conjecture
 A. S. Merkurjev  \(K\)theory of simple algebras
 W. Raskind  Abelian class field theory of arithmetic schemes
 D. J. Saltman  Brauer groups of invariant fields, geometrically negligible classes, an equivariant Chow group, and unramified \(H^3\)
 R. G. Swan  Higher algebraic \(K\)theory
Part 2  J. Kr. Arason, R. Elman, and B. Jacob  On the Witt ring of elliptic curves
 R. Aravire and B. Jacob  \(p\)algebras over maximally complete fields
 J.P. Tignol  Appendix
 S. R. Ashford  Weil's additive characters and class number parity
 R. Baeza and M. I. Icaza  Decomposition of positive definite integral quadratic forms as sums of positive definite quadratic forms
 E. BayerFluckiger and J. Morales  Multiples of trace forms in number fields
 F. A. Bogomolov  On the structure of Galois groups of the fields of rational functions
 N. Childress and D. Grant  Formal groups of twisted multiplicative groups and \(L\)series
 M. D. Choi, T. Y. Lam, and B. Reznick  Sums of squares of real polynomials
 J.L. ColliotThélène and R. Sujatha  Unramified Witt groups of real anisotropic quadrics
 T. C. Craven  Orderings, valuations, and Hermitian forms over \(\ast\)fields
 B. Fein and M. Schacher  A conjecture about relative Brauer groups
 Y. Z. Flicker  Bernstein's isomorphism and good forms
 T. J. Ford  Examples of locally trivial Azumaya algebras
 D. W. Hoffmann  Isotropy of \(5\)dimensional quadratic forms over the function field of a quadric
 D. W. Hoffmann, D. W. Lewis, and J. Van Geel  Minimal forms for function fields of conics
 R. T. Hoobler  Generalized class field theory and cyclic algebras
 D. G. James  The number of embeddings of quadratic \(S\)lattices
 N. A. Karpenko  On topological filtration for SeveriBrauer varieties
 M.A. Knus, T. Y. Lam, D. B. Shapiro, and J.P. Tignol  Discriminants of involutions on biquaternion algebras
 S. Liedahl  \(p\)groups and \(K\)admissibility
 H. Li  Brauer groups over affine normal surfaces
 A. S. Merkurjev  Certain \(K\)cohomology groups of SeveriBrauer varieties
 J. Mináč and A. R. Wadsworth  The \(u\)invariant for algebraic extensions
 P. J. Morandi  On defective division algebras
 E. Peyre  Products of SeveriBrauer varieties and Galois cohomology
 S. Saito and R. Sujatha  A finiteness theorem for cohomology of surfaces over \(p\)adic fields and an application to Witt groups
 T. L. Smith  Witt rings and realizability of small \(2\)Galois groups
 R. Ware  Quadratic forms and solvable Galois groups
 V. I. Yanchevskiĭ  Symmetric and skewsymmetric elements of involutions, associated groups, and the problem of decomposability of involutions
