Elliptic Curves [Paperback]

J. S. Milne (Author)

Editorial Reviews

Review

J. S. Milne's lecture notes on elliptic curves are already well-known ... The book under review is a rewritten version of just these famous lecture notes from 1996, which appear here as a compact and inexpensive paperback that is now available worldwide. --
Zentralblatt MATH, Werner Kleinert

Indeed, the book is affordable (in fact, the most affordable of all references on the subject), but also a high quality work and a complete introduction to the rich theory of the arithmetic of elliptic curves, with numerous examples and exercises for the reader, many interesting remarks and an updated bibliography. --
Mathematical Reviews, Álvaro Lozano-Robledo

Indeed, the book is affordable (in fact, the most affordable of all references on the subject), but also a high quality work and a complete introduction to the rich theory of the arithmetic of elliptic curves, with numerous examples and exercises for the reader, many interesting remarks and an updated bibliography. -- Mathematical Reviews, Álvaro Lozano-Robledo

J. S. Milne's lecture notes on elliptic curves are already well-known ... The book under review is a rewritten version of just these famous lecture notes from 1996, which appear here as a compact and inexpensive paperback that is now available worldwide. -- Zentralblatt MATH, Werner Kleinert

Product Details

• Paperback: 246 pages
• Publisher: BookSurge Publishing (November 20, 2006)
• Language: English
• ISBN-10: 1419652575
• ISBN-13: 978-1419652578
• Product Dimensions: 8.8 x 6 x 0.7 inches
• Shipping Weight: 13.6 ounces (View shipping rates and policies)
• Average Customer Review: 5.0 out of 5 stars  See all reviews (1 customer review)
• Amazon Best Sellers Rank: #951,974 in Books (See Top 100 in Books)

More About the Author

Discover books, learn about writers, read author blogs, and more.

Customer Reviews

Most Helpful Customer Reviews

13 of 17 people found the following review helpful:

5.0 out of 5 stars An excellent overview, September 21, 2007

By 

Dr. Lee D. Carlson (Baltimore, Maryland USA) - See all my reviews
(VINE VOICE)    (HALL OF FAME REVIEWER)    (REAL NAME)   

Amazon Verified Purchase

This review is from: Elliptic Curves (Paperback)

Elliptic curves are so ubiquitous in mathematics and science and such beautiful objects that no author who expounds on them would do a bad job. This book is no exception to this axiom, and even though short the author, a noted expert on the subject, gives the reader important insights into the main properties of elliptic curves.

A highly interesting topic that is included in the book concerns Neron models, which the author motivates by considering an elliptic curve E over the p-adic number field Q(p). A change of variables to its Weierstrass equation is made so that ord() takes on its minimal value and the coefficients are in Z(p). The resulting elliptic curve over Z(p) is viewed as a minimal or "good" model of E with respect to plane projective curves. The idea of a Neron model is to generalize this strategy so that the dependence on plane projective curves is dropped. This involves the theory of schemes, a topic which the author only lightly touches on in this book. His motivation of the Neron model though is excellent, for he uses the work of the mathematician K. Kodaira on elliptic surfaces, which have the property that they are as "regular" as possible. This means that the "fibers" are elliptic curves that have the "minimal" number of singularities. By "blowing up" of points and "blowing down" of curves as much of the bad behavior of the fibers is removed as possible, a process sometimes called "good reduction." The price to be paid for this strategy is that the surface cannot be embedded into projective space.

Those readers familiar with the concept of smoothness in the classical theory of minimal surfaces will see the analogy with the concept of regularity in this case. In the more general theory of algebraic curves, if V is an algebraic curve over a ground field K, where K is a number field or a function field of a smooth projective curve C then one can construct a scheme using K and C. For a number field, S is the spectrum of the ring of integers in K, whereas for a function field it is C. The object is to construct the "best" model over S with the goal of understanding the arithmetic of V. Minimality of a (projective) model of the algebraic curve is unique up to a birational morphism, i.e. a projective minimal model of C is the same as another is every birational morphism between them is also an isomorphism. One can show that every algebraic curve over K with genus greater than or equal to one has a unique projective minimal model. An algebraic curve has a "good" reduction if the special fiber of its minimal model is smooth. An algebraic curve over K has a "bad" reduction if the special fiber of the minimal model has only ordinary double points as singularities.

Computing the rank of an elliptic curve E(Q) is still a major unsolved problem and as is the case in other books it is discussed in the context of the Selmer and Tate-Shafarevich groups. The Selmer group gives an upper bound for the rank, and the Tate-Shafarevich group measures the difference between the upper bound and the actual rank. The importance of these groups is illustrated via the proof of the (weak) Mordell-Weil theorem, which gives the finiteness of E(K)/nE(K) for any elliptic curve over a number field K and integer n.

In order to prove this theorem, the author takes the reader on a journey through group cohomology, starting first with the cohomology of finite groups and then with the cohomology of the infinite Galois group. As is well known in other contexts, cohomology theories essentially measure the obstruction to maps between spaces to be injective or surjective. For the case of group cohomology this is true also, where in this case the first cohomology group, at least the way it is described by the author, where the lack of surjectivity is measured by the `principal crossed' homomorphisms. Both the Selmer and the Tate-Shafarevich groups are defined in terms of the first cohomology group of an elliptic curve E(Q), and the author proves that the Selmer group is finite. Having done this, and using height theory, he proves the finiteness of E(K)/nE(K).

Because of its great complexity, a book of this size would not be able to include a detailed proof of Fermat's Last Theorem. The author discusses modular forms as a preparation for this theorem, but leaves the details to other works on the subject.