Modular forms: a classical approach
by Cohen, Henri
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 s…
Read More ↓Title Statements
- Full title :Modular forms : a classical approach / Henri Cohen, Fredrik Strömberg.
Summary
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 being given in a separate chapter.--Publisher website.
Read More ↓Edition Statements
- Publication Date:2017
Publication Statements
- Publisher(s):American mathematical Society
- Place of Publication:Providence, Rhode Island
- Language:eng
- Publication Date:2017
The Physical Object
- Description:xii, 700 pages ; : 27 cm.
Edition Identifiers
- ISBN:9780821849477, 0821849476
- oclc:ocn963439483
- LC Classification:QA243
- Dewey Decimal:512.7/3
Table of Contents
Introduction -- Elliptic Functions, Elliptic Curves, and Theta Functions -- Basic Tools -- The Modular Group -- General Aspects of Holomorphic and Nonholomorphic Modular Forms -- Sets of 2 x 2 Integer Matrices -- Modular Forms and Functions on Subgroups -- Eisenstein and Poincaré 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.
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