Serge Lang is well-known as a mathematician and as an author of mathematical books, some of which have become standard texts. The current volume is the first to appear of the volumes covering number theory. The book gives an accessible presentation of an area of number theory that has recently had very important successes.

Klappentext

In 1988 Shafarevich asked me to write a volume for the Encyclopaedia of Mathematical Sciences on Diophantine Geometry. I said yes, and here is the volume. By definition, diophantine problems concern the solutions of equations in integers, or rational numbers, or various generalizations, such as finitely generated rings over Z or finitely generated fields over Q. The word Geometry is tacked on to suggest geometric methods. This means that the present volume is not elementary. For a survey of some basic problems with a much more elementary approach, see [La 9Oc]. The field of diophantine geometry is now moving quite rapidly. Out­ standing conjectures ranging from decades back are being proved. I have tried to give the book some sort of coherence and permanence by em­ phasizing structural conjectures as much as results, so that one has a clear picture of the field. On the whole, I omit proofs, according to the boundary conditions of the encyclopedia. On some occasions I do give some ideas for the proofs when these are especially important. In any case, a lengthy bibliography refers to papers and books where proofs may be found. I have also followed Shafarevich's suggestion to give examples, and I have especially chosen these examples which show how some classical problems do or do not get solved by contemporary in­ sights. Fermat's last theorem occupies an intermediate position. Al­ though it is not proved, it is not an isolated problem any more.

Inhalt

I Some Qualitative Diophantine Statements.- Basic Geometric Notions.- The Canonical Class and the Genus.- The Special Set.- Abelian Varieties.- Algebraic Equivalence and the Non-Severi Group.- Subvarieties of Abelian and Semiabelian Varieties.- Hilbert Irreducibility.- II Heights and Rational Points.- The Height for Rational Numbers and Rational Functions.- The Height in Finite Extensions.- The Height on Varieties and Divisor Classes.- Bound for the Height of Algebraic Points.- III Abelian Varieties.- Basic Facts About Algebraic Families and Non Models.- The Height as a Quadratic Function.- Algebraic Families of Heights.- Torsion Points and the l-Adic Representations.- Principal Homogeneous Spaces and Infinite Descents.- The Birch-Swinnerton-Dyer Conjecture.- The Case of Elliptic Curves Over Q.- IV Faltings' Finiteness Theorems on Abelian Varieties and Curves.- Torelli's Theorem.- The Shafarevich Conjecture.- The l-Adic Representations and Semisimplicity.- The Finiteness of Certain l-Adic Representations. Finiteness I Implies Finiteness II.- The Faltings Height and Isogenies: Finiteness I.- The Masser-Wustholz Approach to Finiteness I.- V Modular Curves Over Q.- Basic Definitions.- Mazur's Theorems.- Modular Elliptic Curves and Fermat's Last Theorem.- Application to Pythagorean Triples.- Modular Elliptic Curves of Rank 1.- VI The Geometric Case of Mordell's Conjecture.- Basic Geometric Facts.- The Function Field Case and Its Canonical Sheaf.- Grauert's Construction and Vojta's Inequality.- Parshin's Method with (?;2x/y).- Manin's Method with Connections.- Characteristic p and Voloch's Theorem.- VII Arakelov Theory.- Admissible Metrics Over C.- Arakelov Intersections.- Higher Dimensional Arakelov Theory.- VIII Diophantine Problems and Complex Geometry.- Definitions of Hyperbolicity.- Chern Form and Curvature.- Parshin's Hyperbolic Method.- Hyperbolic Imbeddings and Noguchi's Theorems.- Nevanlinna Theory.- IX Weil Functions. Integral Points and Diophantine Approximations.- Weil Functions and Heights.- The Theorems of Roth and Schmidt.- Integral Points.- Vojta's Conjectures.- Connection with Hyperbolicity.- From Thue-Siegel to Vojta and Faltings.- Diophantine Approximation on Toruses.- X Existence of (Many) Rational Points.- Forms in Many Variables.- The Brauer Group of a Variety and Manin's Obstruction.- Local Specialization Principle.- Anti-Canonical Varieties and Rational Points.