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This textbook covers the basic properties of elliptic curves and modular forms, with emphasis on certain connections with number theory. The ancient "congruent number problem" is the central motivating example for most of the book. 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. With numerous exercises (and answers) included, the textbook is also intended for graduate students who have completed the standard first-year courses in real and complex analysis and algebra. Such students would learn applications of techniques from those courses, thereby solidifying their under standing of some basic tools used throughout mathematics. Graduate stu dents wanting to work in number theory or algebraic geometry would get a motivational, example-oriented introduction. In addition, advanced under graduates could use the book for independent study projects, senior theses, and seminar work.
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 addition law.- 8. Points of finite order.- 9. Points over finite fields, and the congruent number problem.- II The Hasse-Weil L-Function of an Elliptic Curve.- 1. The congruence zeta-function.- 2. The zeta-function of En.- 3. Varying the prime p.- 4. The prototype: the Riemann zeta-function.- 5. The Hasse-Weil L-function and its functional equation.- 6. The critical value.- III Modular forms.- 1. $$S {{L}_{2}}(\mathbb{Z}) $$ and its congruence subgroups.- 2. Modular forms for $$ S{{L}_{2}}(\mathbb{Z}) $$.- 3. Modular forms for congruence subgroups.- 4. Transformation formula for the theta-function.- 5. The modular interpretation, and Hecke operators.- IV Modular Forms of Half Integer Weight.- 1. Definitions and examples.- 2. Eisenstein series of half integer weight for $$ {{\tilde{\Gamma }}_{0}}(4) $$.- 3. Hecke operators on forms of half integer weight.- 4. The theorems of Shimura, Waldspurger, Tunnell, and the congruent number problem.- Answers, Hints, and References for Selected Exercises.