Diophantine problems concern the solutions of equations in integers, rational numbers, or various generalizations. The book is an encyclopaedic survey of diophantine geometry. For the most part no proofs are given, but references are given where proofs may be found. There are some exceptions, notably the proof for a large part of Faltings' theorems is given. The survey puts together, from a unified point of view, the field of diophantine geometry which has developed since the early 1950s, after its origins in Mordell, Weil and Siegel's papers in the 1920s. The basic approach is that of algebraic geometry, but examples are given which show how this approach deals with (and sometimes solves!) classical problems phrased in very elementary terms. For instance, the Fermat problem is not solved, but it is shown to fit in to two great structural approaches, so that it is not an isolated problem any more. This monograph on number theory, algebraic geometry, several complex variables and differential geometry is intended for graduate students and researchers.
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
I Some Qualitative Diophantine Statements.- x1. Basic Geometric Notions.- x2. The Canonical Class and the Genus.- x3. The Special Set.- x4. Abelian Varieties.- x5. Algebraic Equivalence and the Neron-Severi Group.- x6. Subvarieties of Abelian and Semiabelian Varieties.- x7. Hilbert Irreducibility.- II Heights and Rational Points.- x1. The Height for Rational Numbers and Rational Functions.- x2. The Height in Finite Extensions.- x3. The Height on Varieties and Divisor Classes.- x4. Bound for the Height of Algebraic Points.- III Abelian Varieties.- x0. Basic Facts About Algebraic Families and Neron Models.- x1, The Height as a Quadratic Function.- x2. Algebraic Families of Heights.- x3. Torsion Points and the l-Adic Representations.- x4. Principal Homogeneous Spaces and Infinite Descents.- x5. The Birch-Swinnerton-Dyer Conjecture.- x6. The Case of Elliptic Curves Over Q.- IV Faltings' Finiteness Theorems on Abelian Varieties and Curves.- x1. Torelli's Theorem.- x2. The Shafarevich Conjecture.- x3. The l-Adic Representations and Semisimplicity.- x4. The Finiteness of Certain l-Adic Representations. Finiteness I Implies Finiteness II.- x5. The Faltings Height and Isogenies: Finiteness I.- x6. The Masser-Wustholz Approach to Finiteness I.- V Modular Curves Over Q.- x1. Basic Definitions.- x2. Mazur's Theorems.- x3. Modular Elliptic Curves and Fermat's Last Theorem.- x4. Application to Pythagorean Triples.- x5. Modular Elliptic Curves of Rank 1.- VI The Geometric Case of Mordell's Conjecture.- x0. Basic Geometric Facts.- x1. The Function Field Case and Its Canonical Sheaf.- x2. Grauert's Construction and Vojta's Inequality.- x3. Parshin's Method with (?;2x/y).- x4. Manin's Method with Connections.- x5. Characteristic p and Voloch's Theorem.- VII Arakelov Theory.- x1. Admissible Metrics Over C.- x2. Arakelov Intersections.- x3. Higher Dimensional Arakelov Theory.- VIII Diophantine Problems and Complex Geometry.- x1. Definitions of Hyperbolicity.- x2. Chern Form and Curvature.- x3. Parshin's Hyperbolic Method.- x4. Hyperbolic Imbeddings and Noguchi's Theorems.- x5. Nevanlinna Theory.- IX Weil Functions. Integral Points and Diophantine Approximations.- x1. Weil Functions and Heights.- x2. The Theorems of Roth and Schmidt.- x3. Integral Points.- x4. Vojta's Conjectures.- x5. Connection with Hyperbolicity.- x6. From Thue-Siegel to Vojta and Faltings.- x7. Diophantine Approximation on Toruses.- X Existence of (Many) Rational Points.- x1. Forms in Many Variables.- x2. The Brauer Group of a Variety and Manin's Obstruction.- x3. Local Specialization Principle.- x4. Anti-Canonical Varieties and Rational Points.
Series: Encyclopaedia of Mathematical Sciences
Number Of Pages: 296
Published: 27th June 1991
Publisher: Springer-Verlag Berlin and Heidelberg Gmbh & Co. Kg
Country of Publication: DE
Dimensions (cm): 24.13 x 15.88
Weight (kg): 0.62