CRM Monograph Series 2004; 196 pp; softcover Volume: 22 ISBN10: 0821890190 ISBN13: 9780821890196 List Price: US$69 Member Price: US$55.20 Order Code: CRMM/22.S
 Shimura curves are a farreaching generalization of the classical modular curves. They lie at the crossroads of many areas, including complex analysis, hyperbolic geometry, algebraic geometry, algebra, and arithmetic. This monograph presents Shimura curves from a theoretical and algorithmic perspective. The main topics are Shimura curves defined over the rational number field, the construction of their fundamental domains, and the determination of their complex multiplication points. The study of complex multiplication points in Shimura curves leads to the study of families of binary quadratic forms with algebraic coefficients and to their classification by arithmetic Fuchsian groups. In this regard, the authors develop a theory full of new possibilities that parallels Gauss' theory on the classification of binary quadratic forms with integral coefficients by the action of the modular group. This is one of the few available books explaining the theory of Shimura curves at the graduate student level. Each topic covered in the book begins with a theoretical discussion followed by carefully workedout examples, preparing the way for further research. Titles in this series are copublished with the Centre de Recherches Mathématiques. Readership Graduate students and research mathematicians interested in number theory, algebra, algebraic geometry, and those interested in the tools used in Wiles' proof of Fermat's Last Theorem. Table of Contents  Quaternion algebras and quaternion orders
 Introduction to Shimura curves
 Quaternion algebras and quadratic forms
 Embeddings and quadratic forms
 Hyperbolic fundamental domains for Shimura curves
 Complex multiplication points in Shimura curves
 The Poincare package
 Tables
 Further contributions to the study of Shimura curves
 Applications of Shimura curves
 Bibliography
 Index
