CBMS Regional Conference Series in Mathematics 2004; 129 pp; softcover Number: 101 ISBN10: 0821828681 ISBN13: 9780821828687 List Price: US$34 Member Price: US$27.20 All Individuals: US$27.20 Order Code: CBMS/101
 The book surveys some recent developments in the arithmetic of modular elliptic curves. It places a special emphasis on the construction of rational points on elliptic curves, the Birch and SwinnertonDyer conjecture, and the crucial role played by modularity in shedding light on these two closely related issues. The main theme of the book is the theory of complex multiplication, Heegner points, and some conjectural variants. The first three chapters introduce the background and prerequisites: elliptic curves, modular forms and the ShimuraTaniyamaWeil conjecture, complex multiplication and the Heegner point construction. The next three chapters introduce variants of modular parametrizations in which modular curves are replaced by Shimura curves attached to certain indefinite quaternion algebras. The main new contributions are found in Chapters 79, which survey the author's attempts to extend the theory of Heegner points and complex multiplication to situations where the base field is not a CM field. Chapter 10 explains the proof of Kolyvagin's theorem, which relates Heegner points to the arithmetic of elliptic curves and leads to the best evidence so far for the Birch and SwinnertonDyer conjecture. Readership Graduate students and research mathematicians interested in number theory and arithmetic algebraic geometry. Reviews "The book is well written, and would be a good text to run a graduate seminar on, or for a graduate student to make independent study of, as the author has tried his best to make the material accessible."  Chandrashekhar Khare for Mathematical Reviews Table of Contents  Elliptic curves
 Modular forms
 Heegner points on \(X_0(N)\)
 Heegner points on Shimura curves
 Rigid analytic modular forms
 Rigid analytic modular parametrisations
 Totally real fields
 ATR points
 Integration on \(\mathcal{H}_p\times\mathcal{H}\)
 Kolyvagin's theorem
 Bibliography
