Modular Forms and Functions

by Robert A. Rankin

• ISBN13: 9780521091688
• ISBN10: 0521091683
• Edition: 1st
• Format: Paperback
• Copyright: 2008-12-04
• Publisher: Cambridge University Press

Summary

This book provides an introduction to the theory of elliptic modular functions and forms, a subject of increasing interest because of its connexions with the theory of elliptic curves. Modular forms are generalisations of functions like theta functions. They can be expressed as Fourier series, and the Fourier coefficients frequently possess multiplicative properties which lead to a correspondence between modular forms and Dirichlet series having Euler products. The Fourier coefficients also arise in certain representational problems in the theory of numbers, for example in the study of the number of ways in which a positive integer may be expressed as a sum of a given number of squares. The treatment of the theory presented here is fuller than is customary in a textbook on automorphic or modular forms, since it is not confined solely to modular forms of integral weight (dimension). It will be of interest to professional mathematicians as well as senior undergraduate and graduate students in pure mathematics.

Table of Contents

Groups of matrices and bilinear mappings

Mapping properties

Automorphic factors and multiplier systems

General properties of modular forms

Construction of modular forms

Functions belonging to the full modular group

Groups of level 2 and sums of squares

Modular forms of level N

Hecke operators and congruence groups

Applications

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