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This monograph provides a systematic treatment of the Brauer group
of schemes, from the foundational work of Grothendieck to recent
applications in arithmetic and algebraic geometry. The importance
of the cohomological Brauer group for applications to Diophantine
equations and algebraic geometry was discovered soon after this
group was introduced by Grothendieck. The Brauer-Manin obstruction
plays a crucial role in the study of rational points on varieties
over global fields. The birational invariance of the Brauer group
was recently used in a novel way to establish the irrationality of
many new classes of algebraic varieties. The book covers the vast
theory underpinning these and other applications. Intended as an
introduction to cohomological methods in algebraic geometry, most
of the book is accessible to readers with a knowledge of algebra,
algebraic geometry and algebraic number theory at graduate level.
Much of the more advanced material is not readily available in book
form elsewhere; notably, de Jong's proof of Gabber's theorem, the
specialisation method and applications of the Brauer group to
rationality questions, an in-depth study of the Brauer-Manin
obstruction, and proof of the finiteness theorem for the Brauer
group of abelian varieties and K3 surfaces over finitely generated
fields. The book surveys recent work but also gives detailed proofs
of basic theorems, maintaining a balance between general theory and
concrete examples. Over half a century after Grothendieck's
foundational seminars on the topic, The Brauer-Grothendieck Group
is a treatise that fills a longstanding gap in the literature,
providing researchers, including research students, with a valuable
reference on a central object of algebraic and arithmetic geometry.
We dedicate this volume to Professor Parimala on the occasion of
her 60th birthday. It contains a variety of papers related to the
themes of her research. Parimala's rst striking result was a
counterexample to a quadratic analogue of Serre's conjecture
(Bulletin of the American Mathematical Society, 1976). Her in uence
has cont- ued through her tenure at the Tata Institute of
Fundamental Research in Mumbai (1976-2006),and now her time at
Emory University in Atlanta (2005-present). A conference was held
from 30 December 2008 to 4 January 2009, at the U- versity of
Hyderabad, India, to celebrate Parimala's 60th birthday (see the
conf- ence's Web site at
http://mathstat.uohyd.ernet.in/conf/quadforms2008). The or- nizing
committee consisted of J.-L. Colliot-Thel ' en ' e, Skip Garibaldi,
R. Sujatha, and V. Suresh. The present volume is an outcome of this
event. We would like to thank all the participants of the
conference, the authors who have contributed to this volume, and
the referees who carefully examined the s- mitted papers. We would
also like to thank Springer-Verlag for readily accepting to publish
the volume. In addition, the other three editors of the volume
would like to place on record their deep appreciation of Skip
Garibaldi's untiring efforts toward the nal publication.
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Arithmetic Geometry - Lectures Given at the C.I.M.E. Summer School Held in Cetraro, Italy, September 10-15, 2007 (English, French, Paperback, 2010 ed.)
Jean-Louis Colliot-Thelene, Peter Swinnerton-Dyer, Paul Alan Vojta; Edited by Pietro Corvaja, Carlo Gasbarri
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R1,450
Discovery Miles 14 500
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Ships in 10 - 15 working days
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Arithmetic Geometry can be defined as the part of Algebraic
Geometry connected with the study of algebraic varieties through
arbitrary rings, in particular through non-algebraically closed
fields. It lies at the intersection between classical algebraic
geometry and number theory. A C.I.M.E. Summer School devoted to
arithmetic geometry was held in Cetraro, Italy in September 2007,
and presented some of the most interesting new developments in
arithmetic geometry. This book collects the lecture notes which
were written up by the speakers. The main topics concern
diophantine equations, local-global principles, diophantine
approximation and its relations to Nevanlinna theory, and
rationally connected varieties. The book is divided into three
parts, corresponding to the courses given by J-L Colliot-Thelene,
Peter Swinnerton Dyer and Paul Vojta.
This volume contains three long lecture series by J.L.
Colliot-Thelene, Kazuya Kato and P. Vojta. Their topics are
respectively the connection between algebraic K-theory and the
torsion algebraic cycles on an algebraic variety, a new approach to
Iwasawa theory for Hasse-Weil L-function, and the applications of
arithemetic geometry to Diophantine approximation. They contain
many new results at a very advanced level, but also surveys of the
state of the art on the subject with complete, detailed profs and a
lot of background. Hence they can be useful to readers with very
different background and experience. CONTENTS: J.L.
Colliot-Thelene: Cycles algebriques de torsion et K-theorie
algebrique.- K. Kato: Lectures on the approach to Iwasawa theory
for Hasse-Weil L-functions.- P. Vojta: Applications of arithmetic
algebraic geometry to diophantine approximations.
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Geometrie Algebrique Reelle et Formes Quadratiques - Journees S.M.F., Universite De Rennes 1, Mai 1981 (English, German, French, Paperback, 1982 ed.)
Jean-Louis Colliot-Thelene, Michel Coste, Louis Mah e, Marie-Francoise Roy
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R1,761
Discovery Miles 17 610
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Ships in 10 - 15 working days
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