704 pages • 1 5/16” spine

James ArthurDavid EllwoodRobert KottwitzEditors

Clay Mathematics ProceedingsVolume 4

American Mathematical Society

Clay Mathematics Institute

www.ams.org

www.claymath.org

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Proceedings of the Clay Mathematics Institute2003 Summer School, The Fields Institute Toronto, Canada, June 2–27, 2003

The modern theory of automorphic forms, embodied inwhat has come to be known as the Langlands program,is an extraordinary unifying force in mathematics. Itproposes fundamental relations that tie arithmeticinformation from number theory and algebraic geometrywith analytic information from harmonic analysis andgroup representations. These “reciprocity laws”,conjectured by Langlands, are still largely unproved.However, their capacity to unite large areas ofmathematics insures that they will be a central area of study for years to come.

The goal of this volume is to provide an entry point intothis exciting and challenging field. It is directed on theone hand at graduate students and professionalmathematicians who would like to work in the area.The longer articles in particular represent an attempt to enable a reader to master some of the more difficulttechniques. On the other hand, the book will also beuseful to mathematicians who would like simply tounderstand something of the subject. They will be ableto consult the expository portions of the various articles.

The volume is centered around the trace formula andShimura varieties. These areas are at the heart of thesubject, but they have been especially difficult to learnbecause of a lack of expository material. The volumeaims to rectify the problem. It is based on the coursesgiven at the 2003 Clay Mathematics Institute SummerSchool. However, many of the articles have beenexpanded into comprehensive introductions, either tothe trace formula or the theory of Shimura varieties, orto some aspect of the interplay and application of thetwo areas.

Clay Mathematics ProceedingsVolume 4

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