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Graduate Texts in Mathematics

Diophantine Geometry

An Introduction

Authors: Hindry, Marc, Silverman, Joseph H.

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eBook 39,99 €
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  • ISBN 978-1-4612-1210-2
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Hardcover 76,91 €
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  • ISBN 978-0-387-98975-4
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  • Usually dispatched within 3 to 5 business days.
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Softcover 50,91 €
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  • ISBN 978-0-387-98981-5
  • Free shipping for individuals worldwide
  • Usually dispatched within 3 to 5 business days.
  • The final prices may differ from the prices shown due to specifics of VAT rules
About this Textbook

This is an introduction to diophantine geometry at the advanced graduate level. The book contains a proof of the Mordell conjecture which will make it quite attractive to graduate students and professional mathematicians. In each part of the book, the reader will find numerous exercises.

Reviews

"In this excellent 500-page volume, the authors introduce the reader to four fundamental finiteness theorems in Diophantine geometry. After reviewing algebraic geometry and the theory of heights in Parts A and B, the Mordell-Weil theorem (the group of rational points on an abelian variety is finitely generated) is presented in Part C, Roth's theorem (an algebraic number has finitely many approximations of order $2 + \varepsilon$) and Siegel's theorem (an affine curve of genus $g \ge 1$ has finitely many integral points) are proved in Part D, and Faltings' theorem (a curve of genus $g \ge 2$ has finitely many rational points) is discussed in Part E.

Together, Parts C--E form the core of the book and can be read by any reader already acquainted with algebraic number theory, classical (i.e., not scheme-theoretical) algebraic geometry, and the height machine. The authors write clearly and strive to help the reader understand this difficult material. They provide insightful introductions, clear motivations for theorems, and helpful outlines of complicated proofs.

This volume will not only serve as a very useful reference for the advanced reader, but it will also be an invaluable tool for students attempting to study Diophantine geometry. Indeed, such students usually face the difficult task of having to acquire a sufficient grasp of algebraic geometry to be able to use algebraic-geometric tools to study Diophantine applications. Many beginners feel overwhelmed by the geometry before they read any of the beautiful arithmetic results. To help such students, the authors have devoted about a third of the volume, Part A, to a lengthy introduction to algebraic geometry, and suggest that the reader begin by skimming Part A, possibly reading more closely any material that covers gaps in the reader's knowledge. Then Part A should be used as a reference source for geometric facts as they are needed while reading the rest of the book.

The first arithmetic portion of the book is Part B, which deals with the theory of height functions, functions which measure the "size" of a point on an algebraic variety. These objects are a key tool for the Diophantine study in Parts C--E, and the authors, in their characteristically clear and insightful style, fully prove in Part B most results on heights later used in the book.

The book concludes in Part F with a survey of further results and open problems, such as the generalization of Mordell's conjecture to higher-dimensional subvarieties of abelian varieties and questions of quantitative and effective results on the solutions of Diophantine problems.

This book is a most welcome addition to the literature. It is well written and renders accessible to students of Diophantine geometry some of the most elegant and beautiful arithmetical results of the 20th century."  (Dino J. Lorenzini, Mathematical Reviews)

 


Table of contents (7 chapters)

  • Introduction

    Hindry, Marc (et al.)

    Pages 1-5

    Preview Buy Chapter 30,19 €
  • The Geometry of Curves and Abelian Varieties

    Hindry, Marc (et al.)

    Pages 6-167

    Preview Buy Chapter 30,19 €
  • Height Functions

    Hindry, Marc (et al.)

    Pages 168-256

    Preview Buy Chapter 30,19 €
  • Rational Points on Abelian Varieties

    Hindry, Marc (et al.)

    Pages 257-298

    Preview Buy Chapter 30,19 €
  • Diophantine Approximation and Integral Points on Curves

    Hindry, Marc (et al.)

    Pages 299-366

    Preview Buy Chapter 30,19 €

Buy this book

eBook 39,99 €
price for Spain (gross)
  • ISBN 978-1-4612-1210-2
  • Digitally watermarked, DRM-free
  • Included format: PDF
  • ebooks can be used on all reading devices
  • Immediate eBook download after purchase
Hardcover 76,91 €
price for Spain (gross)
  • ISBN 978-0-387-98975-4
  • Free shipping for individuals worldwide
  • Usually dispatched within 3 to 5 business days.
  • The final prices may differ from the prices shown due to specifics of VAT rules
Softcover 50,91 €
price for Spain (gross)
  • ISBN 978-0-387-98981-5
  • Free shipping for individuals worldwide
  • Usually dispatched within 3 to 5 business days.
  • The final prices may differ from the prices shown due to specifics of VAT rules
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Bibliographic Information

Bibliographic Information
Book Title
Diophantine Geometry
Book Subtitle
An Introduction
Authors
Series Title
Graduate Texts in Mathematics
Series Volume
201
Copyright
2000
Publisher
Springer-Verlag New York
Copyright Holder
Springer Science+Business Media New York
eBook ISBN
978-1-4612-1210-2
DOI
10.1007/978-1-4612-1210-2
Hardcover ISBN
978-0-387-98975-4
Softcover ISBN
978-0-387-98981-5
Series ISSN
0072-5285
Edition Number
1
Number of Pages
XIII, 561
Topics