Title	Number Theory III [electronic resource] : Diophantine Geometry / edited by Serge Lang
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

From the reviews of the first printing of this book, published as Volume 60 of the Encyclopaedia of Mathematical Sciences: "Between number theory and geometry there have been several stimulating influences, and this book records of 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, and 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. Reading and rereading this book I noticed that the topics are treated in a nice, coherent way, however not in a historically logical order. ...The author writes "At the moment of writing, the situation is in flux...". That is clear from the scope of this book. In the area described many conjectures, important results, new developments took place in the last 30 years. And still new results come at a breathtaking speed in this rich field. In the introduction the author notices: "I have included several connections of diophantine geometry with other parts of mathematics, such as PDE and Laplacians, complex analysis, and differential geometry. A grand unification is going on, with multiple connections between these fields." Such a unification becomes clear in this beautiful book, which we recommend for mathematicians of all disciplines." Medelingen van het wiskundig genootschap, 1994 "... It is fascinating to see how geometry, arithmetic and complex analysis grow together!..." Monatshefte fรผr Mathematik, 1993

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{128}{153} Finiteness Theorems on Abelian Varieties and Curves -- ยง1. Torelliโ{128}{153}s 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. Mazurโ{128}{153}s Theorems -- ยง3. Modular Elliptic Curves and Fermatโ{128}{153}s Last Theorem -- ยง4. Application to Pythagorean Triples -- ยง5. Modular Elliptic Curves of Rank 1 -- VI The Geometric Case of Mordellโ{128}{153}s Conjecture -- ยง0. Basic Geometric Facts -- ยง1. The Function Field Case and Its Canonical Sheaf -- ยง2. Grauertโ{128}{153}s Construction and Vojtaโ{128}{153}s Inequality -- ยง3. Parshinโ{128}{153}s Method with (?;2x/y) -- ยง4. Maninโ{128}{153}s Method with Connections -- ยง5. Characteristic p and Volochโ{128}{153}s 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. Parshinโ{128}{153}s Hyperbolic Method -- ยง4. Hyperbolic Imbeddings and Noguchiโ{128}{153}s 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. Vojtaโ{128}{153}s 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 Maninโ{128}{153}s Obstruction -- ยง3. Local Specialization Principle -- ยง4. Anti-Canonical Varieties and Rational Points