Elementary Dirichlet Series and Modular Forms (eBook)

Goro Shimura (Autor)

eBook Download: PDF

2007 | 1. Auflage
160 Seiten
Springer New York (Verlag)
978-0-387-72474-4 (ISBN)

Infos

The main topics of the book are the critical values of Dirichlet L-functions and Hecke L-functions of an imaginary quadratic field, and various problems on elliptic modular forms. As to the values of Dirichlet L-functions, all previous papers and books reiterate a single old result with a single old method. After a review of elementry Fourier analysis, we present completely new results with new methods, though old results will also be proved. No advanced knowledge of number theory is required up to this point. As applications, new formulas for the second factor of the class number of a cyclotomic field will be given. The second half of the book assumes familiarity with basic knowledge of modular forms. However, all definitions and facts are clearly stated, and precise references are to the determination of the critical values of Hecke L-functions of an imaginary quadratic field. Other notable features of the book are: (1) some new results on classical Eisenstein series; (2) the discussion of isomorphism classes of ellitic curves with complex multiplication in connection with their zeta function and periods; (3) a new class of holomorphic differential operators that send modular forms to those of a different weight.

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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.

Produkt-Details

• Erscheint lt. Verlag: 6.8.2007
• Sprache: englisch
• Themenwelt: Mathematik / Informatik ► Mathematik ► Arithmetik / Zahlentheorie
• Themenwelt: Mathematik / Informatik ► Mathematik ► Geometrie / Topologie
• Themenwelt: Technik
• ISBN-10: 0-387-72474-5 / 0387724745
• ISBN-13: 978-0-387-72474-4 / 9780387724744