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 underยญ standing of some basic tools used throughout mathematics. Graduate stuยญ dents wanting to work in number theory or algebraic geometry would get a motivational, example-oriented introduction. In addition, advanced underยญ graduates could use the book for independent study projects, senior theses, and seminar work
CONTENT
1 From Congruent Numbers to Elliptic Curves -- II The HasseโWeil L-Function of an Elliptic Curve -- III Modular forms -- IV Modular Forms of Half Integer Weight -- Answers, Hints, and References for Selected Exercises
