This book introduces the reader to the fascinating world of modular forms through a problem-solving approach. As such, it can be used by the undergraduate and graduate student for self-instruction. The topics covered include q-series, the modular group, the upper half-plane, modular forms of level one and higher level, the Ramanujan τ -function, the Petersson inner product, Hecke operators, Dirichlet series attached to modular forms and further special topics. It can be viewed as a gentle introduction for a deeper study of the subject. Thus, it is ideal for non-experts seeking an entry into the field.
Part I Problems
1 Jacobi’s q-series
2 The Modular Group
3 The Upper Half-Plane
4 Modular Forms of Level One
5 The Ramanujan τ-function
6 Modular Forms of Higher Level
7 The Petersson Inner Product
8 Hecke Operators of Higher Level
9 Dirichlet Series and Modular Forms
10 Special Topics
Part II Solutions
1 Jacobi’s q-series
2 The Modular Group
3 The Upper Half-Plane
4 Modular Forms of Level One
5 The Ramanujan τ-function
6 Modular Forms of Higher Level
7 The Petersson Inner Product
8 Hecke Operators of Higher Level
9 Dirichlet Series and Modular Forms
10 Special Topics
A short guide for further reading