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 ideasfor 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.
Rezensionen / Stimmen
From the reviews: "Between number theory and geometry there have been several stimulating influences, and this book records these enterprises. This author, who has been at the centre of such research for many years, is one of the best guides a reader can hope for. The book is full of beautiful results, open questions, stimulating conjectures and suggestions where to look for future developments. This volume bears witness of the broad scope of knowledge of the author, and the influence of several people who have commented on the manuscript before publication... Although in the series of number theory, this volume is on diophantine geometry, the reader will notice that algebraic geometry is present in every chapter. ...The style of the book is clear. Ideas are well explained, and the author helps the reader to pass by several technicalities. Mededelingen van het wiskundig genootschap
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ISBN-13
978-3-642-58227-1 (9783642582271)
DOI
10.1007/978-3-642-58227-1
Schweitzer Klassifikation
I Some Qualitative Diophantine Statements.- §1. Basic Geometric Notions.- §2. The Canonical Class and the Genus.- §3. The Special Set.- §4. Abelian Varieties.- §5. Algebraic Equivalence and the Néron-Severi Group.- §6. Subvarieties of Abelian and Semiabelian Varieties.- §7. Hilbert Irreducibility.- II Heights and Rational Points.- §1. The Height for Rational Numbers and Rational Functions.- §2. The Height in Finite Extensions.- §3. The Height on Varieties and Divisor Classes.- §4. Bound for the Height of Algebraic Points.- III Abelian Varieties.- §0. Basic Facts About Algebraic Families and Néron Models.- §1, The Height as a Quadratic Function.- §2. Algebraic Families of Heights.- §3. Torsion Points and the l-Adic Representations.- §4. Principal Homogeneous Spaces and Infinite Descents.- §5. The Birch-Swinnerton-Dyer Conjecture.- §6. The Case of Elliptic Curves Over Q.- IV Faltings' Finiteness Theorems on Abelian Varieties and Curves.- §1. Torelli's Theorem.- §2. The Shafarevich Conjecture.- §3. The l-Adic Representations and Semisimplicity.- §4. The Finiteness of Certain l-Adic Representations. Finiteness I Implies Finiteness II.- §5. The Faltings Height and Isogenies: Finiteness I.- §6. The Masser-Wustholz Approach to Finiteness I.- V Modular Curves Over Q.- §1. Basic Definitions.- §2. Mazur's Theorems.- §3. Modular Elliptic Curves and Fermat's Last Theorem.- §4. Application to Pythagorean Triples.- §5. Modular Elliptic Curves of Rank 1.- VI The Geometric Case of Mordell's Conjecture.- §0. Basic Geometric Facts.- §1. The Function Field Case and Its Canonical Sheaf.- §2. Grauert's Construction and Vojta's Inequality.- §3. Parshin's Method with (?;2x/y).- §4. Manin's Method with Connections.- §5. Characteristic p and Voloch's Theorem.- VII Arakelov Theory.- §1. Admissible Metrics Over C.- §2. Arakelov Intersections.- §3. Higher Dimensional Arakelov Theory.- VIII Diophantine Problems and Complex Geometry.- §1. Definitions of Hyperbolicity.- §2. Chern Form and Curvature.- §3. Parshin's Hyperbolic Method.- §4. Hyperbolic Imbeddings and Noguchi's Theorems.- §5. Nevanlinna Theory.- IX Weil Functions. Integral Points and Diophantine Approximations.- §1. Weil Functions and Heights.- §2. The Theorems of Roth and Schmidt.- §3. Integral Points.- §4. Vojta's Conjectures.- §5. Connection with Hyperbolicity.- §6. From Thue-Siegel to Vojta and Faltings.- §7. Diophantine Approximation on Toruses.- X Existence of (Many) Rational Points.- §1. Forms in Many Variables.- §2. The Brauer Group of a Variety and Manin's Obstruction.- §3. Local Specialization Principle.- §4. Anti-Canonical Varieties and Rational Points.
