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$$K$$-Theory and Algebraic Geometry: Connections with Quadratic Forms and Division Algebras
Edited by: Bill Jacob and Alex Rosenberg
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Proceedings of Symposia in Pure Mathematics
1995; 737 pp; hardcover
Volume: 58
ISBN-10: 0-8218-1498-2
ISBN-13: 978-0-8218-1498-7
List Price: US$183 Member Price: US$146.40
Order Code: PSPUM/58

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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 finite-dimensional 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 speakers--Colliot-Thélène, Merkurjev, Raskind, Saltman, Suslin, Swan--are reproduced in the expository articles in this book.

Research mathematicians.

Part 1
• J.-L. Colliot-Thélène -- 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. Bayer-Fluckiger 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. Colliot-Thélène 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 Severi-Brauer 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 Severi-Brauer varieties
• J. Mináč and A. R. Wadsworth -- The $$u$$-invariant for algebraic extensions
• P. J. Morandi -- On defective division algebras
• E. Peyre -- Products of Severi-Brauer 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 skew-symmetric elements of involutions, associated groups, and the problem of decomposability of involutions