xii Preface

at the end of each chapter. The reader should be warned, though,

that there are multiple references to mathematical objects and results

that we will not have enough space to discuss in full, and the student

will have to take these items on faith (we will provide references to

other texts, however, for those students who wish to deepen their

understanding). Some other objects and theorems are mentioned in

previous chapters but only explained fully in later chapters. To avoid

any confusion, we always try to clarify in the text which objects or

results the student should take on faith, which ones we expect the stu-

dent to be familiar with, and which will be explained in later chapters

(by providing references to later sections of the book).

The book begins with some motivating problems, such as the

congruent number problem, Fermat’s last theorem, and the represen-

tations of integers as sums of squares. Chapter 2 is a survey of the

algebraic theory of elliptic curves. In Section 2.9, we give a proof

of the weak Mordell-Weil theorem for elliptic curves with rational 2-

torsion and explain the method of 2-descent. The goal of Chapter

3 is to motivate the connection between elliptic curves and modular

forms. To that end, we discuss complex lattices, tori, modular curves

and how these objects relate to elliptic curves over the complex num-

bers. Chapter 4 introduces the spaces of modular forms for SL(2, Z)

and other congruence subgroups (e.g., Γ0(N)). In Chapter 5 we define

the L-functions attached to elliptic curves and modular forms. We

briefly discuss the Birch and Swinnerton-Dyer conjecture and other

related conjectures. Finally, in Section 5.4, we justify the statement

of the original conjecture of Taniyama-Shimura-Weil (which we usu-

ally refer to as the modularity theorem, since it was proved in 1999);

i.e., we explain the surprising connection between elliptic curves and

certain modular forms, and justify which modular forms correspond

to elliptic curves.

In order to make this book as self-contained as possible, I have

also included five appendices with concise introductions to topics that

some students may not have encountered in their classes yet. Appen-

dix A is a quick reference guide to two popular software packages:

PARI and Sage. Throughout the book, we strongly recommend that

the reader tries to find examples and do calculations using one of these