My purpose is to make the subject accessible to those who find it hard to read more advanced or more algebraically oriented treatments. At the same time I want to introduce topics which are at the forefront of current research. Down-to-earth examples are given in the text and exercises, with the aim of making the material readable and interesting to mathematicians in fields far removed from the subject of the book.

Preface to the First Edition

Preface to the Second Edition

CHAPTER I From Congruent Numbers to Elliptic Curves

 1. Congruent numbers

 2. A certain cubic equation

 3. Elliptic curves

 4. Doubly periodic functions

 5. The field of elliptic functions

 6. Elliptic curves in Weierstrass form

 7. The additionclaw

 8. Points of finite order

 9. Points over finite fields, and the congruent number problem

CHAPTER II The Hasse-Weil L-Function of an Elliptic Curve

 I. The congruence zeta-function

 2. The zeta-function of E

 3. Varying the primep

 4. The prototype: the Riemann zeta-function

 5. The Hasse-Weil L-function and its functional equation

6. The critical value

CHAPTER III Modular forms

 1. SL2(Z) and its congruence subgroups

 2. Modular forms for SL2(Z)

 3. Modular forms for congruence subgroups

 4. Transformation formula for the theta-function

 5. The modular interpretation, and Hecke operators

CHAPTER IV Modular Forms of Half Integer Weight

 1. Definitio ns and examples

 2. Eisenstein series of half integer weight for (4)

 3. Hecke operators on forms of half integer weight

 4. The theorems of Shimura, Waldspurger, Tunnell, and the congruent number problem

Answers, Hints, and Referen es for Selected Exercises

Bibliography

Index