Elementary Dirichlet Series and Modular Forms

1. Front Matter

   Pages i-xii

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2. Preliminaries on Modular Forms and Dirichlet Series

   Pages 5-23

3. Critical Values of Dirichlet L-functions

   Pages 25-52

4. The Case of Imaginary Quadratic Fields and Nearly Holomorphic Modular Forms

   Pages 53-58

5. Eisenstein Series

   Pages 59-77

6. Critical Values of Dirichlet Series Associated with Imaginary Quadratic Fields

   Pages 79-111

7. Supplementary Results

   Pages 113-126

8. Back Matter

   Pages 127-147

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A book on any mathematical subject above textbook level is not of much value unless it contains new ideas and new perspectives. Also, the author may be encouraged to include new results, provided that they help the reader gain newinsightsandarepresentedalongwithknownoldresultsinaclearexposition. Itis with this philosophy that Iwrite this volume. The two subjects, Dirichlet series and modular forms, are traditional, but I treat them in both orthodox and unorthodox ways. However, I try to make the book accessible to those who are not familiar with such topics, by including plenty of expository material. More speci?c descriptions of the contents will be given in the Introduction. To some extent, this book has a supplementary nature to my previous book Introduction to the Arithmetic Theory of Automorphic Functions, published by Princeton University Press in 1971, though I do not write the present book with that intent. While the 1971 book grew out of my lectures in various places, the essential points of this new book have never been presented publicly or privately. I hope that it will draw an audience as large as that of the previous book.

• DEX
• Factor
• Grad
• Morphism
• Multiplication
• differential operator
• field
• form
• function
• functions
• knowledge
• modular form
• number theory
• review
• zeta function

From the reviews:

"This book contains new results, e.g., new formulas for special values of certain Dirichlet series. … Shimura’s exposition, shaped to his (celebrated and) distinctive viewpoint, serves as the ideal platform for the new material. … by dint of the prestige of the author and the subject, it undoubtedly deserves a place in a college library. … Summing Up: Recommended. Upper-division undergraduates through faculty." (D. V. Feldman, CHOICE, Vol. 45 (9), 2008)

"It will be of great interest for everybody who is interested in modular forms and/or L-series. … the monograph will be accessible to graduate students and will quickly lead them to frontiers of current research. The book is written in the well-known masterly style of the author … ." (Jürgen Elstrodt, Zentralblatt MATH, Vol. 1148, 2008)

• Princeton University, 08544-1000

  Goro Shimura

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