Lectures on Modular Forms. (AM-48) ( Annals of Mathematics Studies )

Publication series ：Annals of Mathematics Studies

Author： Gunning Robert C.  

Publisher： Princeton University Press‎

Publication year： 2016

E-ISBN: 9781400881666

P-ISBN(Paperback): 9780691079950

Subject： O141.4 model theory

Keyword： 复分析、复变函数,数学

Language： ENG

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Description

New interest in modular forms of one complex variable has been caused chiefly by the work of Selberg and of Eichler. But there has been no introductory work covering the background of these developments. H. C. Gunning's book surveys techniques and problems; only the simpler cases are treated-modular forms of even weights without multipliers, the principal congruence subgroups, and the Hecke operators for the full modular group alone.

Chapter

Cover

Title

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INTRODUCTION

CONTENTS

CHAPTER I. GEOMETRICAL BACKGROUND

§1. Linear Fractional Transformations

§2. The Modular Group

§3. The Principal Congruence Subgroups

§4. The Riemann Surfaces Associated to Subgroups of the Modular Group

§5. Examples

CHAPTER II. MODULAR FORMS

§6. Introduction

§7. A Review of Some Function Theory on Riemann Surfaces

§8. The Dimension of the Space of Modular Forms

CHAPTER III. POINCARÉ SERIES

§9. Construction of Modular Forms

§10. The Petersson Inner Product

§11. Completeness of Poincaré Series

§12. The Fourier Coefficients of Cusp Forms

CHAPTER IV. EISENSTEIN SERIES

§13. Construction of the Eisenstein Series.

§14. The Fourier Coefficients of Eisenstein Series

§15. The Modular Forms for the Modular Group

CHAPTER V. MODULAR CORRESPONDENCES

§16. The Hecke Operators

§17. Hecke Operators and Fourier Coefficients

§18. Arithmetic Properties of the Fourier Coefficients

CHAPTER VI. QUADRATIC FORMS

§19. Introduction

§20. The Generalized Jacobi Inversion Formula

§21. Representations by Sums of Squares

§22. Even Integral Quadratic Forms

§23. Arithmetic Applications

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