Alsina M., Bayer P. Quaternion Orders, Quadratic Forms, and Shimura Curves

American Mathematical Society, 2004. — 210 p. — (CRM Monograph Series. Book 22) — ISSN: 0821833596The purpose of this monograph is to provide an introduction to 8himura curves from a theoretical and algorithmic perspective. 8himura curves lie at the crossroads of many areas, including complex analysis, p-adic analysis, arithmetic, Diophantine geometry, algebraic geometry, algebra, and noncommutative algebra. Our approach to them has two objectives: to construct fundamental domains in the Poincare half-plane, and to determine their complex multiplication points. Our presentation is based on a previous study of quadratic forms attached to orders in quaternion algebras. The algorithms needed for the computations have been compiled in a package, named Poincare, which has been implemented in Maple V.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
Appendixes
Tables
Further Contributions to the Study of Shimura Curves
Applications of Shimura Curves