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This book treats its two subjects, Dirichlet series and modular forms, in both orthodox and unorthodox ways. Regardless of the unorthodox treatment, the author has made the book accessible to those who are not familiar with such topics.
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…mehr

Produktbeschreibung
This book treats its two subjects, Dirichlet series and modular forms, in both orthodox and unorthodox ways. Regardless of the unorthodox treatment, the author has made the book accessible to those who are not familiar with such topics.
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.
  • Produktdetails
  • Springer Monographs in Mathematics
  • Verlag: Springer / Springer New York / Springer, Berlin
  • Artikelnr. des Verlages: 12054664
  • 2007
  • Seitenzahl: 156
  • Erscheinungstermin: 10. September 2007
  • Englisch
  • Abmessung: 241mm x 160mm x 13mm
  • Gewicht: 405g
  • ISBN-13: 9780387724737
  • ISBN-10: 0387724737
  • Artikelnr.: 22825645
Autorenporträt
Goro Shimura, Princeton University, Princeton , NJ, USA
Inhaltsangabe
Preliminaries on Modular Forms and Dirichlet Series.- Critical Values of Dirichlet L-functions.- The Case of Imaginary Quadratic Fields and Nearly Holomorphic Modular Forms.- Eisenstein Series.- Critical Values of Dirichlet Series Associated with Imaginary Quadratic Fields.- Supplementary Results.
Rezensionen
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)