Elementary Dirichlet Series and Modular Forms

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.

A book on any mathematical subject beyond the textbook level is of little value unless it contains new ideas and new perspectives. Of course it helps to include new results, provided that they give the reader new insights and are presented along with known old results in a clear exposition. It is with this philosophy that the author writes this volume. The two subjects, Dirichlet series and modular forms, are traditional subjects, but here Goro Shimura treats them 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, by including plenty of expository material. The book contains never before published elementary proofs. It is self-contained, and suitable for use in a classroom setting.

Author writes in a clear and engaging style Contains never before published elementary proofs Author provides new results and detailed exposition Self-contained, and suitable for use in a classroom setting or for self-study A highly creative contribution to the theory of modular forms and dirichlet series Includes supplementary material: sn.pub/extras

Texte du rabat

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 elementary Fourier analysis, the author presents 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 given. The notion of nearly holomorphic modular forms is introduced and applied 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 elliptic 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.

The book will be of interest to graduate students and researchers who are interested in special values of L-functions, class number formulae, arithmetic properties of modular forms (especially their values), and the arithmetic properties of Dirichlet series. It treats in detail, from an elementary viewpoint, the simplest cases of a fundamental area of ongoing research, the only prerequisite being a basic course in algebraic number theory.

Contenu
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.