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The 1-2-3 of Modular Forms

Jan Hendrik Bruinier, Gerard van der Geer, Günter Harder, and Don Zagier
Publication Date: 
Number of Pages: 
[Reviewed by
Suzanne Caulk
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If you ever wanted to gain a general understanding of modular forms or if you’re curious about the applications of this field, then you should check out The 1-2-3 of Modular Forms. This is a volume of lectures given at the 2004 summer school at the Sophus Lie Conference Center on “Modular Forms and Their Applications” by some of the leaders in the field. There are three sets of lectures, one for each of the most basic kinds of modular forms: elliptic, Hilbert and Siegel. There is an additional lecture on congruences between elliptic and Siegel modular forms. Each set of lectures includes standard introductory material as well as concrete examples and applications.

This text is not an encyclopedic listing of facts. Rather, it gives the reader a sense of the way the field developed and leaves some of the standard ideas for the reader to verify. The formatting makes it easy to distinguish fundamental theoretical content from applications so that readers can quickly find the sections of the book relevant to their own purpose. One could use this as a way to start a graduate student working on modular forms or it could be used to broaden a nonexpert’s view of the area.

Although not an exhaustive volume on the topic, most basic principal terms and relationships between terms are explained. Also, much of the notation is clarified which makes it fairly simple to move from this text to more advanced work in the field. Whether you are looking for a nice way to begin your study of modular forms or are already familiar with them, this is a book that you will enjoy.

Suzanne Caulk is an associate professor of mathematics at Regis University.  She is interested in all kinds of modular forms. You can reach her at