I assume some familiarity with modular forms, although many of the basic facts
are given here without proof. There are many introductory textbooks which are
useful references such as [Apo, Hi1, Iw2, Kna, Kno, Kob2, La2, Mi2, Ogg1,
Ran, Sch, Se2, Shi1]. Thanks to the efforts of Stein and others, there are now
computer packages such as Magma which can carry out modular form calculations
which were inconceivable a few years ago. These new resources are also invaluable.
In this monograph I hope to provide ample motivation on some of the topics
in which modular forms play a role. I do not make an attempt to catalog all of the
results in these areas; rather I highlight results which give their flavor. At the end
of most chapters, I list some open problems and questions. I invite solutions, and
I look forward to substantial progress on these topics.
I owe a great debt to many people who were involved with the NSF-CBMS
Regional Conference at the University of Illinois at Urbana-Champaign, where these
lectures were presented. Scott Ahlgren and Bruce Berndt organized a wonderful
conference which ran very smoothly. I also thank Jan Bruinier, YoungJu Choie,
Masanobu Kaneko, Winfried Kohnen, and Ram Murty, the invited speakers, who
delivered exciting lectures on many of the topics described here. I would also
like to thank Scott Ahlgren, Matt Boylan, Jan Bruinier, Denis Charles, Rohit
Chatterjee, Ahmad El-Guindy, Jayce Getz, Masanobu Kaneko, Winfried Kohnen,
Jeremy Lovejoy, Karl Mahlburg, Bill McGraw, Jeremy Rouse, and Kathryn Zuhr
for their many helpful comments on early versions of this manuscript. I also thank
my wife Erika, and my children, Aspen and Sage, for their patience and tremendous
support. Without them this project would not have been possible. To all of these
people, my sincerest thanks.