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Quaternion orders, quadratic forms, and Shimura curves. (English) Zbl 1073.11040
CRM Monograph Series 22. Providence, RI: American Mathematical Society (AMS) (ISBN 0-8218-3359-6/hbk). xvi, 196 p. (2004).
“The purpose of this monograph is to provide an introduction to Shimura curves from a theoretical and algorithmic perspective” states the introduction of this book.
The emphasis on the theoretical side lies here clearly on explicit and constructive methods for Shimura curves defined by the unit group of an indefinite quaternion algebra over \(\mathbb Q\).
The first three chapters of the book present the necessary background from the theory of Shimura curves and about (indefinite) quaternion algebras over \(\mathbb Q\) and quadratic forms; these chapters are mainly concerned with collecting known results and the notations to be used in the sequel; only few proofs are given here.
Chapter 4 gives a variety of technical results concerning embeddings of quadratic fields or orders into quaternion algebras and representations by ternary and quaternary quadratic forms associated to orders in indefinite quaternion algebras.
Chapters 5 and 6 contain then the main results: Procedures for the construction of fundamental domains in the upper half plane under the action of the quaternion unit group defining a Shimura curve and classification and explicit computation of the complex multiplication points on Shimura curves.
Chapter 7 describes the MAPLE package “Poincaré” that actually performs the computations described before. Appendices give tables and further material on Shimura curves.

11G18 Arithmetic aspects of modular and Shimura varieties
11-02 Research exposition (monographs, survey articles) pertaining to number theory
11R52 Quaternion and other division algebras: arithmetic, zeta functions
11F06 Structure of modular groups and generalizations; arithmetic groups
11E20 General ternary and quaternary quadratic forms; forms of more than two variables
14G35 Modular and Shimura varieties
30F35 Fuchsian groups and automorphic functions (aspects of compact Riemann surfaces and uniformization)