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Modular Forms: A Classical and Computational Introduction, by Lloyd Kilford
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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; Results on Finite-Dimensionality; The Arithmetic of Modular Forms; Applications of Modular Forms; Modular Forms in Characteristic p; Computing with Modular Forms; Appendices:; MAGMA Code for Classical Modular Forms; SAGE Code for Classical Modular Forms; Hints and Answers to Selected Exercises.
- Sales Rank: #3642133 in Books
- Published on: 2008-08-11
- Original language: English
- Number of items: 1
- Dimensions: 8.80" h x .90" w x 6.10" l, 1.20 pounds
- Binding: Hardcover
- 236 pages
Review
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"
Most helpful customer reviews
5 of 6 people found the following review helpful.
Very readable introduction to modular forms
By TPC
A classical and computational introduction is a new book by L.J.P. Kilford. New enough that it even has reference to the resolution of Serre's conjecture. But this book is really an introduction to the classical aspects of the theory of modular forms and it does a great job. I took a few days to read through the book (of course, ignoring the details and proofs) and I would say it is very enjoyable. Kilford adds in lots of funny and quirky anecdotes, most of which I've read from different places but it's nice to have everything collected in one book. For example, he mentioned Lang's famous foreword:
"It is possible to write endlessly on elliptic curves. (This is not a threat.)"
I remembered being so tickled when I first saw this in Lang's book.
Back to this book. Even with the subject of modular forms it is the same. He knows he can't possibly explain everything and so is not afraid to be a little vague at times and cite the various references to where more in depth discussions can be found. Thus, he is able to accomplish much in this modest sized (200+ pages) book. My one small complaint is the title should not include "and computational". I found the last chapter on computational aspects too brief. He highlighted some history, discussed MAGMA and SAGE, giving some examples of the codes used, but I believe that this is not enough for someone interested in computing modular forms to get started on. And the appendixes on MAGMA and SAGE codes are each one page long with two longish lines of commands.
1 of 1 people found the following review helpful.
A Solid Introduction to the Subject
By Wil
As a master's student, I'm using Kilford's book for an introductory class on modular forms. I like Kilford's style and choice of material. He does a very good job of explaining the necessary material and referencing other sources for more complicated material.
However, the manuscript itself is ridden with typos. Some of the typos are humorous and provide an opportunity to test one's knowledge and understanding. For example, I could readily identify that Kilford switched even and odd in one of the homework exercises. However, some of the typos, particularly in definitions or proofs create imposing barriers to progress. Kilford's errata includes half a page of typos, while my instructor and the previous class compiled a list of 3 pages not already included in the errata.
As long as you have an instructor or adviser who can identify and correct any typos, the book is a good reference, but I find myself constantly checking the class notes for the correct definitions and statements, although it only takes a few minutes to edit all the typos yourself, it seems we are constantly finding new ones. I would expect that the second edition (I expect there will be a second edition) will quickly become a standard reference guide for those beginning to learn the subject.
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