**CRM Monograph Series**

Volume: 22;
2004;
196 pp;
Softcover

MSC: Primary 11;
Secondary 30; 51

Print ISBN: 978-0-8218-9019-6

Product Code: CRMM/22.S

List Price: $69.00

AMS Member Price: $55.20

MAA member Price: $62.10

**Electronic ISBN: 978-1-4704-1769-7
Product Code: CRMM/22.E**

List Price: $69.00

AMS Member Price: $55.20

MAA member Price: $62.10

#### Supplemental Materials

# Quaternion Orders, Quadratic Forms, and Shimura Curves

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*Montserrat Alsina; Pilar Bayer*

A co-publication of the AMS and Centre de Recherches Mathématiques

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

# Table of Contents

## Quaternion Orders, Quadratic Forms, and Shimura Curves

- Cover Cover11
- Title page iii4
- Contents v6
- List of tables vii8
- List of figures xi12
- Introduction xiii14
- Quaternion algebras and quaternion orders 118
- Introduction to Shimura curves 1734
- Quaternion algebras and quadratic forms 2744
- Embeddings and quadratic forms 5572
- Hyperbolic fundamental domains for Shimura curves 7794
- Complex multiplication points in Shimura curves 103120
- The Poincare package 123140
- Appendix A. Tables 145162
- Appendix B. Further contributions to the study of Shimura curves 187204
- Appendix C. Applications of Shimura curves 189206
- Bibliography 191208
- Index 195212
- Back Cover Back Cover1216