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El. knyga: Modular Forms

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The theory of modular forms is a fundamental tool used in many areas of mathematics and physics. It is also a very concrete and "fun" subject in itself and abounds with an amazing number of surprising identities.

This comprehensive textbook, which includes numerous exercises, aims to give a complete picture of the classical aspects of the subject, with an emphasis on explicit formulas. After a number of motivating examples such as elliptic functions and theta functions, the modular group, its subgroups, and general aspects of holomorphic and nonholomorphic modular forms are explained, with an emphasis on explicit examples. The heart of the book is the classical theory developed by Hecke and continued up to the Atkin-Lehner-Li theory of newforms and including the theory of Eisenstein series, Rankin-Selberg theory, and a more general theory of theta series including the Weil representation. The final chapter explores in some detail more general types of modular forms such as half-integral weight, Hilbert, Jacobi, Maass, and Siegel modular forms.

Some "gems" of the book are an immediately implementable trace formula for Hecke operators, generalizations of Haberland's formulas for the computation of Petersson inner products, W. Li's little-known theorem on the diagonalization of the full space of modular forms, and explicit algorithms due to the second author for computing Maass forms.

This book is essentially self-contained; the necessary tools such as gamma and Bessel functions, Bernoulli numbers, and so on are given in a separate chapter.

Recenzijos

According to the preface, the authors expect the main use of this book to be for advanced graduate students to learn about the classical theory of modular forms. However, given the tremendous amount of detail provided, the book should also be useful as a reference for established researchers in the area. Further, it can undoubtedly be mined by instructors for a graduate course on modular forms. Sander Zwegers, Mathematical Reviews

This book gives a beautiful introduction to the theory of modular forms, with a delicate balance of analytic and arithmetic perspectives. Cohen and Strömberg start with a foundational collection of tools in analysis and number theory, which they use while guiding the reader through a vast landscape of results. They finish by showing us the frontiers of modern research, covering topics generalizing the classical theory in a variety of directions. Throughout, the authors expertly weave fine details with broad perspective. The target readership for this text is graduate students in number theory, though it will also be accessible to advanced undergraduates and will, no doubt, serve as a valuable reference for researchers for years to come. Jennifer Balakrishnan, Boston University

This marvelous book is a gift to the mathematical community and more specifically to anyone wanting to learn modular forms. The authors take a classical view of the material offering extremely helpful explanations in a generous conversational manner and covering such an impressive range of this beautiful, deep, and important subject. Barry Mazur, Harvard University

This book is an almost encyclopedic textbook on modular forms. There are already numerous and some excellent books on the subject. But none of the existing books by themselves contain this much and this detailed information. The authors' knowledge of the subject matter and the experience in writing books are clearly reflected in the end product. I would not only be very happy to use this book as a textbook next time I teach a course on modular forms, but I am also looking forward to having a hard copy in my library as an extensive reference book. Imamoglu Özlem, ETH Zurich

Modular forms are central to many different fields of mathematics and mathematical physics. Having a detailed and complete treatment of all aspects of the theory by two world experts is a very welcome addition to the literature. Peter Sarnak, Princeton University

Preface xi
Chapter 1 Introduction 1(16)
1.1 Modularity
1(7)
1.2 Examples of Modular Forms from Different Sources
8(6)
1.3 Notation
14(1)
Exercises
15(2)
Chapter 2 Elliptic Functions, Elliptic Curves, and Theta Functions 17(54)
2.1 Elliptic Functions
17(19)
2.2 Elliptic Curves
36(3)
2.3 Theta Functions
39(22)
2.4 Concluding Remarks
61(1)
Exercises
62(9)
Chapter 3 Basic Tools 71(44)
3.1 Classical Analytic Tools
71(9)
3.2 Bessel Functions
80(11)
3.3 Bernoulli Numbers and the Gamma and Zeta Functions
91(6)
3.4 Classical Arithmetic Tools
97(7)
3.5 The Lipschitz Summation Formulas
104(7)
3.6 Whittaker functions
111(1)
Exercises
112(3)
Chapter 4 The Modular Group 115(14)
4.1 The Extended Upper Half-Plane
115(1)
4.2 The Modular Group
116(1)
4.3 Fundamental Domains of the Modular Group
117(4)
4.4 Topology of Gamma
121(3)
4.5 Metrics and Measures on Gamma
124(1)
4.6 Fuchsian Groups of the First Kind
125(2)
Exercises
127(2)
Chapter 5 General Aspects of Holomorphic and Nonholomorphic Modular Forms 129(86)
5.1 Introduction
129(14)
5.2 Examples of Modular Forms: Eisenstein Series
143(9)
5.3 Differential Operators
152(17)
5.4 Taylor Coefficients of Modular Forms
169(5)
5.5 Modular Forms on the Modular Group and Its Subgroups
174(3)
5.6 Zeros, Poles, and Dimension Formulas
177(11)
5.7 The Modular Invariant j
188(2)
5.8 The Dedekind eta-Function and the Product Formula for Delta
190(2)
5.9 Eta Quotients
192(7)
5.10 A Brief Introduction to Complex Multiplication
199(4)
Exercises
203(12)
Chapter 6 Sets of 2 x 2 Integer Matrices 215(38)
6.1 Basic Tools
215(5)
6.2 Subgroups of Gamma
220(13)
6.3 Action on P1 (Q): Cusps
233(11)
6.4 Action on
244(1)
6.5 Sets of Integer Matrices of Given Determinant
245(4)
6.6 The Atkin-Lehner Involutions WQ
249(3)
Exercises
252(1)
Chapter 7 Modular Forms and Functions on Subgroups 253(16)
7.1 General Definitions
253(3)
7.2 The Case G = Gammao(N)
256(2)
7.3 Links Between Forms on Different Groups
258(3)
7.4 Dimensions of Spaces of Modular Forms on Gammao(N)
261(5)
7.5 Computational Aspects
266(1)
Exercises
266(3)
Chapter 8 Eisenstein and Poincare Series 269(42)
8.1 Definitions
269(4)
8.2 Basic Results on Poincare and Eisenstein Series
273(6)
8.3 Poincare and Eisenstein Series for Congruence Subgroups
279(2)
8.4 Fourier Expansions
281(7)
8.5 Eisenstein and Poincare Series in Mk(Gammao(N), X)
288(15)
8.6 Generalization of the Petersson Scalar Product
303(4)
Exercises
307(4)
Chapter 9 Fourier Coefficients of Modular Forms 311(30)
9.1 Introduction
311(1)
9.2 The Hecke Bounds for Fourier Coefficients
312(3)
9.3 Kloosterman Sums and Applications
315(11)
9.4 Petersson Products Involving Eisenstein Series
326(7)
9.5 A Theorem of Siegel
333(3)
Exercises
336(5)
Chapter 10 Hecke Operators and Euler Products 341(42)
10.1 Introduction
341(2)
10.2 Introduction to Hecke Operators
343(6)
10.3 The Hecke Operators Are Hermitian
349(11)
10.4 Eigenvalues and Eigenfunctions of Hecke Operators on Gamma
360(2)
10.5 Double Coset Operators
362(1)
10.6 Bases of Modular Forms for the Full Modular Group
363(6)
10.7 Euler Products
369(5)
10.8 Convolutions
374(8)
Exercises
382(1)
Chapter 11 Dirichlet Series, Functional Equations, and Periods 383(58)
11.1 Introduction
384(2)
11.2 The Main Theorem
386(4)
11.3 Weil's Theorem
390(7)
11.4 Application to the Riemann Zeta Function
397(1)
11.5 Periods and Antiderivatives of Modular Forms
398(4)
11.6 The Case of Eisenstein Series
402(2)
11.7 Transformation under an Arbitrary gamma belongs to Gamma
404(2)
11.8 Eichler Cohomology
406(8)
11.9 Interpretation in Terms of Periods
414(5)
11.10 Action of Hecke Operators on Periods
419(6)
11.11 Rationality and Parity Theorems
425(6)
11.12 Rankin-Selberg Theory
431(7)
Exercises
438(3)
Chapter 12 Unfolding and Kernels 441(74)
12.1 Nonholomorphic Eisenstein Series
441(3)
12.2 The Spaces M(jk) and M(epsilonk)
444(10)
12.3 Unfolding
454(16)
12.4 Kernels and the Trace Formula
470(20)
12.5 Generalization of Haberland's Formula
490(10)
12.6 Computation of Petersson Inner Products
500(10)
Exercises
510(5)
Chapter 13 Atkin-Lehner-Li Theory 515(42)
13.1 Introduction
515(2)
13.2 Preliminary Results
517(9)
13.3 The Theory of Newforms
526(11)
13.4 Diagonalizing the Full Space of Modular Forms
537(1)
13.5 The Trace Formula for Newforms
538(12)
13.6 Computing Spaces of Modular Forms
550(3)
Exercises
553(4)
Chapter 14 Theta Functions 557(36)
14.1 Introduction and Motivation
557(1)
14.2 The Fundamental Theorem
558(4)
14.3 Lattices and Theta Functions
562(15)
14.4 Vector-Valued Modular Forms and Representations of the Modular Group
577(2)
14.5 Finite Quadratic Modules and Weil Representations
579(8)
Exercises
587(6)
Chapter 15 More General Modular Forms: An Introduction 593(86)
15.1 Modular Forms of Half-Integral Weight
593(18)
15.2 Jacobi Forms
611(14)
15.3 Maass Forms
625(30)
15.4 Hilbert Modular Forms
655(10)
15.5 Bianchi Modular Forms
665(4)
15.6 Siegel Modular Forms
669(4)
Exercises
673(6)
Bibliography 679(14)
Index of Notation 693(4)
General Index 697
Henri Cohen, Universite Bordeaux, France.

Fredrik Stromberg, University of Nottingham, United Kingdom.
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