**CBMS Regional Conference Series in Mathematics**

Volume: 101;
2004;
129 pp;
Softcover

MSC: Primary 11;

Print ISBN: 978-0-8218-2868-7

Product Code: CBMS/101

List Price: $34.00

Individual Price: $27.20

**Electronic ISBN: 978-1-4704-2462-6
Product Code: CBMS/101.E**

List Price: $34.00

Individual Price: $27.20

#### Supplemental Materials

# Rational Points on Modular Elliptic Curves

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*Henri Darmon*

A co-publication of the AMS and CBMS

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 Swinnerton-Dyer 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
Shimura-Taniyama-Weil 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 Swinnerton-Dyer
conjecture.

#### Table of Contents

# Table of Contents

## Rational Points on Modular Elliptic Curves

- Cover Cover11 free
- Title iii4 free
- Copyright iv5 free
- Contents vii8 free
- Preface xi12 free
- Chapter 1 Elliptic curves 114 free
- Chapter 2 Modular forms 1326
- Chapter 3 Heegner points on X[sub(0)](N) 2942
- 3.1. Complex multiplication 2942
- 3.2. Heegner points 3346
- 3.3. Numerical examples 3447
- 3.4. Properties of Heegner points 3548
- 3.5. Heegner systems 3649
- 3.6. Relation with the Birch and Swinnerton-Dyer conjecture 3750
- 3.7. The Gross-Zagier formula 3952
- 3.8. Kolyvagin's theorem 4053
- 3.9. Proof of the Gross-Zagier-Kolyvagin theorem 4053
- Further Results 4154
- Exercises 4255

- Chapter 4 Heegner points on Shimura curves 4558
- 4.1. Quaternion algebras 4659
- 4.2. Modular forms on quaternion algebras 4760
- 4.3. Shimura curves 4962
- 4.4. The Eichler-Shimura construction, revisited 5063
- 4.5. The Jacquet-Langlands correspondence 5063
- 4.6. The Shimura-Taniyama-Weil conjecture, revisited 5164
- 4.7. Complex multiplication for H/Γ[sub(N[sup(+)],N[sup(…)]) 5164
- 4.8. Heegner systems 5265
- 4.9. The Gross-Zagier formula 5366
- References 5467
- Exercises 5467

- Chapter 5. Rigid analytic modular forms 5770
- Chapter 6. Rigid analytic modular parametrisations 6780
- 6.1. Rigid analytic modular forms on quaternion algebras 6780
- 6.2. The Cerednik-Drinfeld theorem 6881
- 6.3. The p-adic Shimura-Taniyama-Weil conjecture 6881
- 6.4. Complex multiplication, revisited 6982
- 6.5. An example 7083
- 6.6. p-adic L-functions, d'apres Schneider-Iovita-Spiess 7386
- 6.7. A Gross-Zagier formula 7487
- Further results 7588
- Exercises 7588

- Chapter 7. Totally real fields 7992
- Chapter 8. ATR points 87100
- Chapter 9. Integration on H[sub(p)] × H 97110
- Chapter 10. Kolyvagin's theorem 113126
- Bibliography 125138
- Back Cover Back Cover1146

#### 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