This book presents a graduate student-level introduction to the classical theory of modular forms and computations involving modular forms, including modular functions and the theory of Hecke operators. It also includes applications of modular forms to such diverse subjects as the theory of quadratic forms, the proof of Fermat s last theorem and the approximation of pi. It provides a balanced overview of both the theoretical and computational sides of the subject, allowing a variety of courses to be taught from it.Contents: Historical Overview; Introduction to Modular Forms; Arithmetic of Modular Forms; Applications of Modular Forms; Mod p Modular Forms; p-adic Modular Forms; Computing with Modular Forms; Appendices on MAGMA Code for Classical Modular Forms; SAGE Code for Classical Modular Forms; Hints and Answers to the Exercises.

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This fascinating, contemporaneous, and even now unfolding story of current research in a historically brilliant part of mathematics is told with riveting attention to detail. -- The Mathematical Intelligencer "The Mathematical Intelligencer"

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