Elliptic Curves: Function Theory, Geometry, Arithmetic (Cambridge Tracts in Mathematics) [Hardcover]

Henry McKean (Author), Victor Moll (Author)

Book Description

ISBN-10: 0521582288 | ISBN-13: 978-0521582285 | Publication Date: May 28, 1997

The subject of elliptic curves is one of the jewels of nineteenth-century mathematics, whose masters were Abel, Gauss, Jacobi, and Legendre. This book presents an introductory account of the subject in the style of the original discoverers, with references to and comments about more recent and modern developments. It combines three of the fundamental themes of mathematics: complex function theory, geometry, and arithmetic. After an informal preparatory chapter, the book follows a historical path, beginning with the work of Abel and Gauss on elliptic integrals and elliptic functions. This is followed by chapters on theta functions, modular groups and modular functions, the quintic, the imaginary quadratic field, and on elliptic curves. The many exercises with hints scattered throughout the text give the reader a glimpse of further developments. Requiring only a first acquaintance with complex function theory, this book is an ideal introduction to the subject for graduate students and researchers in mathematics and physics.

Editorial Reviews

Review

"...undergraduates will find that McKean and Moll's book...offers more diverse viewpoints [than other books on elliptic curves]. Highly recommended."
Choice

"...this is a wonderful book that should reward those who have the background for it with immense joy and insight."
SIAM Review

Book Description

The subject of elliptic curves is one of the jewels of 19th century mathematics, whose masters were Abel, Gauss, Jacobi, and Legendre. This book presents account of the subject in the style of the original discoverers, with references to and comments about more modern developments. It combines three of the fundamental themes of mathematics: complex function theory, geometry, and arithmetic.Requiring only a first acquaintance with complex function theory, this book is an ideal introduction to the subject for graduate students and researchers in mathematics and physics. The many exercises with hints scattered throughout the text give the reader a glimpse of further developments.

Product Details

• Hardcover: 298 pages
• Publisher: Cambridge University Press (May 28, 1997)
• Language: English
• ISBN-10: 0521582288
• ISBN-13: 978-0521582285
• Product Dimensions: 9.3 x 6.4 x 0.9 inches
• Shipping Weight: 1.4 pounds (View shipping rates and policies)
• Average Customer Review: 4.8 out of 5 stars  See all reviews (5 customer reviews)
• Amazon Best Sellers Rank: #9,023,353 in Books (See Top 100 in Books)

Customer Reviews

Most Helpful Customer Reviews

38 of 38 people found the following review helpful:

5.0 out of 5 stars Highly recommended, May 23, 2000

By 

Mark Grindell "Mark Grindell" (Shipley,West Yorkshire) - See all my reviews
(VINE VOICE)   

This review is from: Elliptic Curves: Function Theory, Geometry, Arithmetic (Cambridge Tracts in Mathematics) (Hardcover)

This book avoids the traps which would make this subject so inaccessible. Rather than frightening the reader with group theory and the sort of very advanced material that would fit it into a post graduate slot, the book starts with very little beyond geometry and complex number theory. The book carefully progresses to discussions on the projective line, and Riemann surfaces (never too much at once) to the inevitable subjects of the Icosohedral group, and invariant theory. It manages to do this almost without you noticing the depth of maths that is being covered - quite a feat!

From here on, elliptic integrals are discussed, and the work of Jacobi, Gauss, Legendre and Abel discussed freely, with many examples and clear pictures. The text is interspersed with exercises (some of which you can do with a few moments thought, others more difficult). I enjoyed this section (and the remainder of the book) for several very interesting short accounts of subjects slightly tangential to the main material.

[One of my favorites was the account of a letter with a amazingly strange but elegant identity with a continued fraction sent by Ramanujan to Hardy, and Hardy's subsequent absolute amazement... You MUST NOT miss reading that, even if it isn't what you picked the book up for!]

Then the book goes into the area I bought the book for - modular groups, and the solution of the Quintic. This subject draws mostly on work by Hermite, and later, Klein, but is presented carefully and slowly.

I was very glad to find this book. It doesn't race through the subject at breakneck speed, which is what some books on Galois Theory or Algebraic Curves do, and has illuminated quite a few additional topics for me. I guess that now I will be able to recognize the origins of so much hard maths now (and all those entries in the tables of integrals I never understood)

After all, this subject is now very important. Elliptic curves occur in many subjects - Cryptography, Information Theory, and of course, the proof of Fermats last theorem.

15 of 16 people found the following review helpful:

5.0 out of 5 stars long on content, short on abstract nonsense, November 25, 2000

By 

"mumbojumbo" (Seattle, WA USA) - See all my reviews

This review is from: Elliptic Curves: Function Theory, Geometry, Arithmetic (Paperback)

This is a great book because it presents some of the neatest topics in mathematics, without the usual discouraging layers of abstraction and notation. It attacks the topics historically so you get some idea of the motivation and steps followed, instead of a compendium of the most general results and their most elegant proofs.

Also, as a previous reviewer mentioned, the book derives the bizarre and amazing continued fraction formula from Ramanujan's letter to Hardy. I had always wanted to see this, ever since reading "The Man Who Knew Infinity." It is satisfying to see this demystified, even if you don't fully master the argument.

If you literally have not seen most of these topics before, as I had not, you won't find this an easy read, but it's well worth while. I spent a long time on it, and couldn't absorb it all, but I plan to read it again one day.

12 of 14 people found the following review helpful:

4.0 out of 5 stars Makes the others Look bad, July 28, 2001

By 

R. Bagula "Roger L. Bagula" (Lakeside, Ca United States) - See all my reviews
(VINE VOICE)    (REAL NAME)   

This review is from: Elliptic Curves: Function Theory, Geometry, Arithmetic (Paperback)

I got this book as a gift from a long time friend. He had trouble with reading it. It is only for that reason I give it only 4 stars. These authors make others that I have read on this range of subjects look bad: Fields Medalists included! A lot of it is that they just bother to give you the real mathematics with examples. I think the initial miss definition of the Riemann surface gives a false impression, because the explanations of ramified covers and toral elliptic lattices is just wonderful. Reading this book makes Dr. Singerman's papers look so much better! I was disappointed in the treatment of triangle groups, but the treatment of modular functions and gamma1 and gamma2 makes up for that. It is a masterful work... the best I have seen by a modern author. It reminds me of books by Ulam or Russell. Sawyer's little book is not as good!