A new edition of a classical treatment of elliptic and modular functions with some of their number-theoretic applications, this text offers an updated bibliography and an alternative treatment of the transformation formula for the Dedekind eta function. It covers many topics, such as Hecke’s theory of entire forms with multiplicative Fourier coefficients, and the last chapter recounts Bohr’s theory of equivalence of general Dirichlet series.
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41 of 41 people found the following review helpful:
5.0 out of 5 stars
Eminently Readable Introduction to Modular Forms,
This review is from: Modular Functions and Dirichlet Series in Number Theory (Graduate Texts in Mathematics) (v. 41) (Hardcover)
The contents of this text have been clearly refined by having been taught. As a result, the book is clear, well presented and easy to learn from. This is in marked contrast to most technical books, where one is often left scratching ones head by page five, thinking "what the heck does that symbol mean?" Not here: each new concept is clearly defined and articulated with the basic theorems and lemmas surrounding it. No notation is used without first having been carefully defined. In addition, the exercises contain a number of additional goodies (again, better than the usual fare).
If you want to learn the material, learn it quickly, learn it in a way that free from roadblocks and detours, this is the book. It is an excellent intro to modular forms, modular functions, the j-invariant, the Weierstrass elliptic functions and the Hecke operators, in the context of the modular group SL(2,Z). The only criticisms would be: -- It is far from being an exhaustive treatment of SL(2,Z)-related ideas -- It fails to mention that the ideas of the j-invariant and modular forms can be generalized in many directions, e.g through fuchsian/kleinian groups -- It contains absolutely no adelic/p-adic material. But these are hardly faults: this book opens the doorway to these advanced topics.
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