Preface xiii

two packages. Appendix B is a brief summary of complex analysis.

Due to space limitations we only include definitions, a few exam-

ples, and a list of the main theorems in complex analysis; for a full

treatment see [Ahl79], for instance. In Appendix C we introduce

the projective line and the projective plane. The p-adic integers and

the p-adic numbers are treated in Appendix D (for a complete refer-

ence, see [Gou97]). Finally, in Appendix E we list infinite families

of elliptic curves over Q, one family for each of the possible torsion

subgroups over Q.

I would like to emphasize once again that this book is, by no

means, a thorough treatment of elliptic curves and modular forms.

The theory is far too vast to be covered in one single volume, and the

proofs are far too technical for an undergraduate student. Therefore,

the humble goals of this text are to provide a big picture of the vast and

fast-growing theory, and to be an “advertisement” for undergraduates

of these very active and exciting areas of number theory. The author’s

only hope is that, after reading this text, students will feel compelled

to study elliptic curves and modular forms in depth, and in all their

full glory.

There are many excellent references that I would recommend to

the students, and that I have frequently consulted in the preparation

of this book:

(1) There are not that many books on these subjects at the

undergraduate level. However, Silverman and Tate’s book

[SiT92] is an excellent introduction to elliptic curves for

undergraduates. Washington’s book [Was08] is also acces-

sible for undergraduates and emphasizes the cryptography

applications of elliptic curves. Stein’s book [Ste08] also has

an interesting chapter on elliptic curves.

(2) There are several graduate-level texts on elliptic curves. Sil-

verman’s book [Sil86] is the standard reference, but Milne’s

[Mil06] is also an excellent introduction to the theory of el-

liptic curves (and also includes a chapter on modular forms).

Before reading Silverman or Milne, the reader would benefit