Elementary Dirichlet Series and Modular Forms

By: Goro Shimura

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Hardcover | 2 October 2007

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156 Pages

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23.39 x 15.6 x 1.12

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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.

Industry Reviews

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 ... ." (Juergen Elstrodt, Zentralblatt MATH, Vol. 1148, 2008)

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Preface	p. V
Introduction	p. 1
Preliminaries on Modular Forms and Dirichlet Series	p. 5
Basic symbols and the definition of modular forms	p. 5
Elementary Fourier analysis	p. 13
The functional equation of a Dirichlet series	p. 19
Critical Values of Dirichlet L-functions	p. 25
The values of elementary Dirichlet series at integers	p. 25
The class number of a cyclotomic field	p. 39
Some more formulas for L(k, x)	p. 45
The Case of Imaginary Quadratic Fields and Nearly Holomorphic Modular Forms	p. 53
Dirichlet series associated with an imaginary quadratic field	p. 53
Nearly holomorphic modular forms	p. 55
Eisenstein Series	p. 59
Fourier expansion of Eisenstein series	p. 59
Polynomial relations between Eisenstein series	p. 66
Recurrence formulas for the critical values of certain Dirichlet series	p. 75
Critical Values of Dirichlet Series Associated with Imaginary Quadratic Fields	p. 79
The singular values of nearly holomorphic forms	p. 79
The critical values of L-functions of an imaginary quadratic field	p. 84
The zeta function of a member of a one-parameter family of elliptic curves	p. 96
Supplementary Results	p. 113
Isomorphism classes of abelian varieties with complex multiplication	p. 113
The general case	p. 113
The case of elliptic curves	p. 117
Holomorphic differential operators on the upper half plane	p. 120
Appendix	p. 127
Integration and differentiation under the integral sign	p. 127
Fourier series with parameters	p. 130
The confluent hypergeometric function	p. 131
The Weierstrass [Characters not reproducible]-function	p. 136
The action of G[subscript A+] on modular forms	p. 139
References	p. 145
Index	p. 147
Table of Contents provided by Ingram. All Rights Reserved.

Elementary Dirichlet Series and Modular Forms

At a Glance

Hardcover

Industry Reviews

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How to return your order

Defective items

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