Diophantine Geometry: An Introduction (Graduate Texts in Mathematics, 201)

I'd buy again! excellent business! The book is exactly as they said it would. Such a nice offer.

The subject of this book is now over 1900 years old, and it grows more fascinating year after year. The study of solutions of polynomial equations over the integers is now called Diophantine geometry, and is brilliantly outlined by the authors in this book. Avoiding the use of schemes, the author's goal in the book is to prove the Mordell-Weil theorem, Roth's theorem, Siegel's theorem, and Falting's theorem. The first part of the book gives a review of the geometry of curves and Abelian varieties for the reader who has only an elementary knowledge of algebraic geometry. But the authors emphatically recommend that the first part NOT be read, as this will prevent the reader from reaching the important ideas in the later parts. But all the standard results from algebraic geometry, such as local rings, heights, divisors, intersection numbers, ampleness, sheaves, Riemann surfaces, and a brief discussion of schemes is given. This is followed by a very well-written presentation of the theory of height functions. The concept of a height function is explained very carefully, as well as its role in translating geometric information about a variety into arithmetic information about the rational points on the variety. A brief but introduction to Arakelov theory is given in this part also. The Mordell-Weil theorem, which states that the group of rational points on an Abelian variety is finitely generated, is proven in the next part of the book. This theorem, the generalization of the famous tangent and chord construction for elliptic curves, is proven using the theory of heights and Fermat descent, via the weak Mordell-Weil theorem. This theorem treats the case where the group of rational points has been factored out by dividing its points by an integer greater than or equal to 2. The resulting quotient group is shown to be finite and this then implies the stronger version of the theorem. In an appendix, they give a very interesting analysis of the proof in terms of Galois cohomology and the Selmer and Tate-Shaferevich groups. In the next part of the book, the authors take a look at Diophantine approximation and give a proof of Roth's theorem, namely that an algebraic number has finitely many approximations or order 2 + epsilon. What is unique about the discussion is the clarity of the author's presentation, I have not seen it done as clearly as is done in this book. They summarize the main points behind the proof of the theorem before getting into the details. This makes it much easier for the reader to appreciate the constructions involved in the proof. Then, using the techniques of this part and the theory of heights, also prove in this part the theorem of Siegel, that every curve of genus greater than or equal to one has only finitely many rational points. The next part is dedicated to proving Falting's theorem, which states that every curve of genus greater than or equal to 2 has finitely many rational points. The proof the authors discuss though is not based on Faltings original proof, which used highly sophisticated techniques from scheme and stack theory, but the proof of E. Bombieri, which uses classical Diophantine approximation theory, height theory, and concepts from the classical theory of surfaces. The discussion is fascinating and very clearly presented. The authors close the book with a discussion on generalizations and open problems in the field of Diophantine geometry. The interesting abc-conjecture is discussed and shown to imply Falting's theorem and an asymptotic version of Fermat's last theorem. Most interestingly though, and one of the major reasons why I purchased this book, is the discussion on the effective computation of the relevant finite groups, such as the Mordell-Weil group. The four main theorems in the book are qualitative statements about the finiteness of the groups, they do not attempt to give an algorithm to compute the elements of the finite set. The case of elliptic curves and their torsion subgroups is presented as a theorem, but the proof is not given unfortunately. The authors do discuss the case for the rank of the Mordell-Weil group in terms of an algorithm given by Y. Manin, which is dependent on the resolution of a number of unproven conjectures which they discuss in this part. The authors also discuss the effective computation of rational points on curves in terms of moduli spaces, Arakelov theory, Mordell-Weil groups, and the abc conjecture/uniformization. Although brief, they do give many references for further reading. In addition, a short discussion on finding quantitative bounds on the number of elements in the groups is given. The authors then wrap up with a brief discussion of the Bombieri-Lang and Vojta conjectures. It is intriguing in all of this discussion on the role elliptic curves have furnished as a testing ground for most of the conjectures and results.

Perfect