Modular Forms: Basics and Beyond

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.

• Dirichlet series
• Eisenstein series
• arithmeticity
• automorphic eigenforms
• modular forms of half-integral weight
• theta functions

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 Kühnlein, Zentralblatt MATH, Vol. 1234, 2012)

• Department of Mathematics, Princeton University, Princeton, USA

  Goro Shimura

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

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