Modular Forms: Basics and Beyond

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Springer Science & Business Media, Nov 18, 2011 - Mathematics - 178 pages
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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.

 

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Contents

CHAPTER I PRELIMINARIES
1
CHAPTER II THETA FUNCTIONS AND FACTORS OF AUTOMORPHY
15
CHAPTER III THE RATIONALITY AND EISENSTEIN SERIES
39
CHAPTER IV THE CORRESPONDENCE BETWEEN FORMSOF INTEGRAL AND HALFINTEGRAL WEIGHT
89
CHAPTER V THE ARITHMETICITY OF CRITICAL VALUES OF DIRICHLET SERIES
128
APPENDIX
151
REFERENCES
171
INDEX
174
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About the author (2011)

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

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