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Quaternion Orders, Quadratic Forms, and Shimura Curves
Montserrat Alsina, Universitat Politècnica de Catalunya, Manresa, Spain, and Pilar Bayer, Universitat de Barcelona, Spain
A co-publication of the AMS and Centre de Recherches Mathématiques.
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CRM Monograph Series
2004; 196 pp; softcover
Volume: 22
ISBN-10: 0-8218-9019-0
ISBN-13: 978-0-8218-9019-6
List Price: US$65
Member Price: US$52
Order Code: CRMM/22.S
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Shimura curves are a far-reaching 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 worked-out examples, preparing the way for further research.

Titles in this series are co-published 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
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