Modular forms : a classical approach
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Bibliographic Information
Modular forms : a classical approach
（Graduate studies in mathematics, v. 179）
American Mathematical Society, c2017
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Note
Includes bibliographical references (p. 679-691) and indexes
Description and Table of Contents
Description
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.
Table of Contents
Introduction
Elliptic functions, elliptic curves, and theta function
Basic tools
The modular group
General aspects of holomorphic and nonholomorphic modular forms
Sets of $2 \times 2$ integer matrices
Modular forms and functions on subgroups
Eisenstein and Poincare series
Fourier coefficients of modular forms
Hecke operators and Euler products
Dirichlet series, functional equations, and periods
Unfolding and kernels
Atkin-Lehner-Li theory
Theta functions
More general modular forms: An introduction
Bibliography
Index of notation
General index.
by "Nielsen BookData"
