Rational Points on Modular Elliptic Curves

by Darmon, Henri

• ISBN13: 9780821828687
• ISBN10: 0821828681
• Format: Paperback
• Copyright: 2003-12-01
• Publisher: Amer Mathematical Society

Summary

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

Preface	xi
Chapter 1. Elliptic curves	1	(12)
1.1. Elliptic curves	1	(2)
1.2. The Mordell-Weil theorem	3	(3)
1.3. The Birch and Swinnerton-Dyer conjecture	6	(1)
1.4. L-functions	7	(1)
1.5. Some known results	8	(1)
Further results and references	9	(1)
Exercises	10	(3)
Chapter 2. Modular forms	13	(16)
2.1. Modular forms	13	(1)
2.2. Hecke operators	14	(2)
2.3. Atkin-Lehner theory	16	(1)
2.4. L-series	17	(1)
2.5. Eichler-Shimura theory	18	(2)
2.6. Wiles' theorem	20	(1)
2.7. Modular symbols	21	(4)
Further results and references	25	(1)
Exercises	26	(3)
Chapter 3. Heegner points on Χ0(Ν)	29	(16)
3.1. Complex multiplication	29	(4)
3.2. Heegner points	33	(1)
3.3. Numerical examples	34	(1)
3.4. Properties of Heegner points	35	(1)
3.5. Heegner systems	36	(1)
3.6. Relation with the Birch and Swinnerton-Dyer conjecture	37	(2)
3.7. The Gross-Zagier formula	39	(1)
3.8. Kolyvagin's theorem	40	(1)
3.9. Proof of the Gross-Zagier-Kolyvagin theorem	40	(1)
Further results	41	(1)
Exercises	42	(3)
Chapter 4. Heegner points on Shimura curves	45	(12)
4.1. Quaternion algebras	46	(1)
4.2. Modular forms on quaternion algebras	47	(2)
4.3. Shimura curves	49	(1)
4.4. The Eichler-Shimura construction, revisited	50	(1)
4.5. The Jacquet-Langlands correspondence	50	(1)
4.6. The Shimura-Taniyama-Weil conjecture, revisited	51	(1)
4.7. Complex multiplication for Η/ΤΝ+,Ν-	51	(1)
4.8. Heegner systems	52	(1)
4.9. The Gross-Zagier formula	53	(1)
References	54	(1)
Exercises	54	(3)
Chapter 5. Rigid analytic modular forms	57	(10)
5.1. ρ-adic uniformisation	57	(3)
5.2. Rigid analytic modular forms	60	(3)
5.3. ρ-adic line integrals	63	(2)
Further results	65	(1)
Exercises	65	(2)
Chapter 6. Rigid analytic modular parametrisations	67	(12)
6.1. Rigid analytic modular forms on quaternion algebras	67	(1)
6.2. The Cerednik-Drinfeld theorem	68	(1)
6.3. The ρ-adic Shimura-Taniyama-Weil conjecture	68	(1)
6.4. Complex multiplication, revisited	69	(1)
6.5. An example	70	(3)
6.6. &rho-adic L-functions, d'après Schneider-Iovita-Spiess	73	(1)
6.7. A Gross-Zagier formula	74	(1)
Further results	75	(1)
Exercises	75	(4)
Chapter 7. Totally real fields	79	(8)
7.1. Elliptic curves over number fields	79	(1)
7.2. Hilbert modular forms	80	(2)
7.3. The Shimura-Taniyama-Weil conjecture	82	(1)
7.4. The Eichler-Shimura construction for totally real fields	83	(1)
7.5. The Heegner construction	84	(1)
7.6. A preview of Chapter 8	85	(1)
Further results	86	(1)
Chapter 8. ATR points	87	(10)
8.1. Period integrals	87	(1)
8.2. Generalities on group cohomology	88	(1)
8.3. The cohomology of Hilbert modular groups	89	(4)
8.4. ATR points	93	(2)
References	95	(1)
Exercises	95	(2)
Chapter 9. Integration on Ηρ x Η	97	(16)
9.1. Discrete arithmetic subgroups of SL2(Qp) x SL2(R)	98	(1)
9.2. Forms on Ηρ x Η	99	(2)
9.3. Periods	101	(3)
9.4. Some ρ-adic cocycles	104	(1)
9.5. Stark-Heegner points	105	(1)
9.6. Computing Stark-Heegner points	106	(3)
Further results	109	(1)
Exercises	109	(4)
Chapter 10. Kolyvagin's theorem	113	(12)
10.1. Bounding Selmer groups	114	(3)
10.2. Kolyvagin cohomology classes	117	(4)
10.3. Proof of Kolyvagin's theorem	121	(1)
References	122	(1)
Exercises	122	(3)
Bibliography	125