Overview

"From the reviews: ""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 braod 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"

Full Product Details

Author:   Serge Lang ,  Serge Lang
Publisher:   Springer-Verlag Berlin and Heidelberg GmbH & Co. KG
Imprint:   Springer-Verlag Berlin and Heidelberg GmbH & Co. K
Edition:   Softcover reprint of the original 1st ed. 1991
Dimensions:   Width: 15.50cm , Height: 1.60cm , Length: 23.50cm
Weight:   0.980kg
ISBN:  

9783540612230

ISBN 10:   3540612238
Pages:   296
Publication Date:   14 April 1997
Audience:   College/higher education ,  Professional and scholarly ,  Undergraduate ,  Postgraduate, Research & Scholarly
Format:   Paperback
Publisher's Status:   Active
Availability:   In Print  
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Table of 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 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.

Reviews

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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