Softcover ISBN:  9780821890196 
Product Code:  CRMM/22.S 
List Price:  $74.00 
MAA Member Price:  $66.60 
AMS Member Price:  $59.20 
Electronic ISBN:  9781470417697 
Product Code:  CRMM/22.E 
List Price:  $69.00 
MAA Member Price:  $62.10 
AMS Member Price:  $55.20 

Book DetailsCRM Monograph SeriesVolume: 22; 2004; 196 ppMSC: Primary 11; Secondary 30; 51;
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.ReadershipGraduate 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

Chapters

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

Appendix A. Tables

Appendix B. Further contributions to the study of Shimura curves

Appendix C. Applications of Shimura curves


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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.
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

Embeddings 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