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

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