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Diophantine geometry

Author: Serge Lang
Publisher: Berlin ; New York : Springer-Verlag, ©1991.
Series: Encyclopaedia of mathematical sciences, v. 60.; Teorii︠a︡ chisel., 3.
Edition/Format:   Print book : EnglishView all editions and formats
Database:WorldCat
Summary:

In 1988 Shafarevich asked me to write a volume for the Encyclopaedia of Mathematical Sciences on Diophantine Geometry. By definition, diophantine problems concern the solutions of equations in  Read more...

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Material Type: Internet resource
Document Type: Book, Internet Resource
All Authors / Contributors: Serge Lang
ISBN: 3540530045 9783540530046 0387530045 9780387530048
OCLC Number: 214988617
Description: xi, 296 pages ; 24 cm.
Contents: 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 Neron-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 Neron 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.
Series Title: Encyclopaedia of mathematical sciences, v. 60.; Teorii︠a︡ chisel., 3.
Responsibility: Serge Lang (ed.).
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