Modular Forms: Basics and Beyond - Goro Shimura

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Résumé :
This is an advanced book on modular forms. While there are many books published about modular forms, they are written at an elementary level, and not so interesting from the viewpoint of a reader who already knows the?basics. This book offers something new, which may satisfy the desire of such a reader. However, we state every definition and every essential fact concerning classical modular forms of one variable. One of the principal new features of this book is the theory of modular forms of half-integral weight, another being the discussion of theta functions and Eisenstein series of holomorphic and nonholomorphic types. Thus the book is presented so that the reader can learn such theories systematically. Ultimately, we concentrate on the following two themes: (I) The correspondence between the forms of half-integral weight and those of integral weight. (II) The arithmeticity of various Dirichlet series associated with modular forms of integral or half-integral weight.

Biographie:
Goro Shimura is currently a professor emeritus of mathematics at Princeton University....

Sommaire:
Preface.- Notation and Terminology.- Chapter I. Preliminaries.- Chapter II. Theta Functions and Factors of Automorphy.- Chapter III. The Rationality and Eisenstein Series.- Chapter IV. The Correspondence between Forms of Integral and Half-integral Weight.- Chapter V. The Arithmeticity of Critical Values of Dirichlet Series.- Appendix.- References.- Index.

From the reviews: The book under review is a welcome addition to the existing literature on modular forms. Being written by one of the main contributors to this subject it is designed with great care and love to detailed proofs. ... The book is complete in the sense that all statements are clearly formulated and proven in a very rigorous way. ... use this book as a guided tour through much of Shimura's work. Without any doubt this journey is rewarding. (Stefan K?hnlein, Zentralblatt MATH, Vol. 1234, 2012)