Modular Forms

A Classical and Computational Introduction

By: Lloyd James Peter Kilford

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eText | 12 March 2015 | Edition Number 2

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Modular Forms is 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 various subjects, such as the theory of quadratic forms, the proof of Fermat's Last Theorem and the approximation of π. The text gives a balanced overview of both the theoretical and computational sides of its subject, allowing a variety of courses to be taught from it.

This second edition has been revised and updated. New material on the future of modular forms as well as a chapter about longer-form projects for students has also been added.

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Contents:

Historical Overview
Introduction to Modular Forms
Results on Finite-Dimensionality
The Arithmetic of Modular Forms
Applications of Modular Forms
Modular Forms in Characteristic p
Computing with Modular Forms
The Future of Modular Forms?
Modular Forms Projects
Appendices:
- Magma Code for Classical Modular Forms
- Sage Code for Classical Modular Forms
- Hints and Answers to Selected Exercises

Readership: Graduate students studying or taking a course in modular forms.
Key Features:

Covers the computational side together with the theory
Has a wide variety of exercises
Historical asides and references to modular forms in mathematical culture which help ground the subject and motivate student interest