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Quaternion Orders, Quadratic Forms, and Shimura Curves

Montserrat Alsina Universitat Politècnica de Catalunya, Manresa, Spain
Pilar Bayer Universitat de Barcelona, Barcelona, Spain
A co-publication of the AMS and Centre de Recherches Mathématiques
Available Formats:
Softcover ISBN: 978-0-8218-9019-6
Product Code: CRMM/22.S
List Price: $74.00 MAA Member Price:$66.60
AMS Member Price: $59.20 Electronic ISBN: 978-1-4704-1769-7 Product Code: CRMM/22.E List Price:$69.00
MAA Member Price: $62.10 AMS Member Price:$55.20
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This product is available for purchase as a bundle. Purchasing as a bundle enables you to save on the electronic version.
List Price: $111.00 MAA Member Price:$99.90
AMS Member Price: $88.80 Click above image for expanded view Quaternion Orders, Quadratic Forms, and Shimura Curves Montserrat Alsina Universitat Politècnica de Catalunya, Manresa, Spain Pilar Bayer Universitat de Barcelona, Barcelona, Spain A co-publication of the AMS and Centre de Recherches Mathématiques Available Formats:  Softcover ISBN: 978-0-8218-9019-6 Product Code: CRMM/22.S  List Price:$74.00 MAA Member Price: $66.60 AMS Member Price:$59.20
 Electronic ISBN: 978-1-4704-1769-7 Product Code: CRMM/22.E
 List Price: $69.00 MAA Member Price:$62.10 AMS Member Price: $55.20 Bundle Print and Electronic Formats and Save! This product is available for purchase as a bundle. Purchasing as a bundle enables you to save on the electronic version.  List Price:$111.00 MAA Member Price: $99.90 AMS Member Price:$88.80
• Book Details

CRM Monograph Series
Volume: 222004; 196 pp
MSC: Primary 11; Secondary 30; 51;

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.

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.

• Chapters
• Quaternion algebras and quaternion orders
• Introduction to Shimura curves
• Quaternion algebras and quadratic forms
• Hyperbolic fundamental domains for Shimura curves
• Complex multiplication points in Shimura curves
• The Poincare package
• Appendix A. Tables
• Appendix B. Further contributions to the study of Shimura curves
• Appendix C. Applications of Shimura curves

• Requests

Review Copy – for reviewers who would like to review an AMS book
Accessibility – to request an alternate format of an AMS title
Volume: 222004; 196 pp
MSC: Primary 11; Secondary 30; 51;

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.

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.

• Chapters
• Quaternion algebras and quaternion orders
• Introduction to Shimura curves
• Quaternion algebras and quadratic forms