Quaternion Orders, Quadratic Forms, and Shimura Curves

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.