Electronic ISBN:  9781470424626 
Product Code:  CBMS/101.E 
List Price:  $34.00 
Individual Price:  $27.20 

Book DetailsCBMS Regional Conference Series in MathematicsVolume: 101; 2004; 129 ppMSC: Primary 11;
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 7–9, 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.ReadershipGraduate students and research mathematicians interested in number theory and arithmetic algebraic geometry.

Table of Contents

Chapters

Chapter 1. Elliptic curves

Chapter 2. Modular forms

Chapter 3. Heegner points on $X_0(N)$

Chapter 4. Heegner points on Shimura curves

Chapter 5. Rigid analytic modular forms

Chapter 6. Rigid analytic modular parametrisations

Chapter 7. Totally real fields

Chapter 8. ATR points

Chapter 9. Integration on $\mathcal {H}_p\times \mathcal {H}$

Chapter 10. Kolyvagin’s theorem


Additional Material

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


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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 7–9, 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.
Graduate students and research mathematicians interested in number theory and arithmetic algebraic geometry.

Chapters

Chapter 1. Elliptic curves

Chapter 2. Modular forms

Chapter 3. Heegner points on $X_0(N)$

Chapter 4. Heegner points on Shimura curves

Chapter 5. Rigid analytic modular forms

Chapter 6. Rigid analytic modular parametrisations

Chapter 7. Totally real fields

Chapter 8. ATR points

Chapter 9. Integration on $\mathcal {H}_p\times \mathcal {H}$

Chapter 10. Kolyvagin’s theorem

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