Title | Number Theory III [electronic resource] : Diophantine Geometry / edited by Serge Lang |
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Imprint | Berlin, Heidelberg : Springer Berlin Heidelberg : Imprint: Springer, 1991 |
Connect to | http://dx.doi.org/10.1007/978-3-642-58227-1 |
Descript | XIII, 296 p. online resource |
I Some Qualitative Diophantine Statements -- ยง1. Basic Geometric Notions -- ยง2. The Canonical Class and the Genus -- ยง3. The Special Set -- ยง4. Abelian Varieties -- ยง5. Algebraic Equivalence and the Nรฉron-Severi Group -- ยง6. Subvarieties of Abelian and Semiabelian Varieties -- ยง7. Hilbert Irreducibility -- II Heights and Rational Points -- ยง1. The Height for Rational Numbers and Rational Functions -- ยง2. The Height in Finite Extensions -- ยง3. The Height on Varieties and Divisor Classes -- ยง4. Bound for the Height of Algebraic Points -- III Abelian Varieties -- ยง0. Basic Facts About Algebraic Families and Nรฉron Models -- ยง1, The Height as a Quadratic Function -- ยง2. Algebraic Families of Heights -- ยง3. Torsion Points and the l-Adic Representations -- ยง4. Principal Homogeneous Spaces and Infinite Descents -- ยง5. The Birch-Swinnerton-Dyer Conjecture -- ยง6. The Case of Elliptic Curves Over Q -- IV Faltingsโ Finiteness Theorems on Abelian Varieties and Curves -- ยง1. Torelliโs Theorem -- ยง2. The Shafarevich Conjecture -- ยง3. The l-Adic Representations and Semisimplicity -- ยง4. The Finiteness of Certain l-Adic Representations. Finiteness I Implies Finiteness II -- ยง5. The Faltings Height and Isogenies: Finiteness I -- ยง6. The Masser-Wustholz Approach to Finiteness I -- V Modular Curves Over Q -- ยง1. Basic Definitions -- ยง2. Mazurโs Theorems -- ยง3. Modular Elliptic Curves and Fermatโs Last Theorem -- ยง4. Application to Pythagorean Triples -- ยง5. Modular Elliptic Curves of Rank 1 -- VI The Geometric Case of Mordellโs Conjecture -- ยง0. Basic Geometric Facts -- ยง1. The Function Field Case and Its Canonical Sheaf -- ยง2. Grauertโs Construction and Vojtaโs Inequality -- ยง3. Parshinโs Method with (?;2x/y) -- ยง4. Maninโs Method with Connections -- ยง5. Characteristic p and Volochโs Theorem -- VII Arakelov Theory -- ยง1. Admissible Metrics Over C -- ยง2. Arakelov Intersections -- ยง3. Higher Dimensional Arakelov Theory -- VIII Diophantine Problems and Complex Geometry -- ยง1. Definitions of Hyperbolicity -- ยง2. Chern Form and Curvature -- ยง3. Parshinโs Hyperbolic Method -- ยง4. Hyperbolic Imbeddings and Noguchiโs Theorems -- ยง5. Nevanlinna Theory -- IX Weil Functions. Integral Points and Diophantine Approximations -- ยง1. Weil Functions and Heights -- ยง2. The Theorems of Roth and Schmidt -- ยง3. Integral Points -- ยง4. Vojtaโs Conjectures -- ยง5. Connection with Hyperbolicity -- ยง6. From Thue-Siegel to Vojta and Faltings -- ยง7. Diophantine Approximation on Toruses -- X Existence of (Many) Rational Points -- ยง1. Forms in Many Variables -- ยง2. The Brauer Group of a Variety and Maninโs Obstruction -- ยง3. Local Specialization Principle -- ยง4. Anti-Canonical Varieties and Rational Points