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  • © 1991

Number Theory III

Diophantine Geometry

Editors:

Part of the book series: Encyclopaedia of Mathematical Sciences (EMS, volume 60)

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Table of contents (10 chapters)

  1. Front Matter

    Pages i-xiii
  2. Some Qualitative Diophantine Statements

    • R. V. Gamkrelidze
    Pages 1-42
  3. Heights and Rational Points

    • R. V. Gamkrelidze
    Pages 43-67
  4. Abelian Varieties

    • R. V. Gamkrelidze
    Pages 68-100
  5. Modular Curves Over Q

    • R. V. Gamkrelidze
    Pages 123-142
  6. The Geometric Case of Mordell’s Conjecture

    • R. V. Gamkrelidze
    Pages 143-162
  7. Arakelov Theory

    • R. V. Gamkrelidze
    Pages 163-175
  8. Diophantine Problems and Complex Geometry

    • R. V. Gamkrelidze
    Pages 176-204
  9. Existence of (Many) Rational Points

    • R. V. Gamkrelidze
    Pages 244-262
  10. Back Matter

    Pages 263-296

About this book

In 1988 Shafarevich asked me to write a volume for the Encyclopaedia of Mathematical Sciences on Diophantine Geometry. I said yes, and here is the volume. By definition, diophantine problems concern the solutions of equations in integers, or rational numbers, or various generalizations, such as finitely generated rings over Z or finitely generated fields over Q. The word Geometry is tacked on to suggest geometric methods. This means that the present volume is not elementary. For a survey of some basic problems with a much more elementary approach, see [La 9Oc]. The field of diophantine geometry is now moving quite rapidly. Out­ standing conjectures ranging from decades back are being proved. I have tried to give the book some sort of coherence and permanence by em­ phasizing structural conjectures as much as results, so that one has a clear picture of the field. On the whole, I omit proofs, according to the boundary conditions of the encyclopedia. On some occasions I do give some ideas for the proofs when these are especially important. In any case, a lengthy bibliography refers to papers and books where proofs may be found. I have also followed Shafarevich's suggestion to give examples, and I have especially chosen these examples which show how some classical problems do or do not get solved by contemporary in­ sights. Fermat's last theorem occupies an intermediate position. Al­ though it is not proved, it is not an isolated problem any more.

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

Editors and Affiliations

  • Department of Mathematics, Yale University, New Haven, USA

    Serge Lang

Bibliographic Information

  • Book Title: Number Theory III

  • Book Subtitle: Diophantine Geometry

  • Editors: Serge Lang

  • Series Title: Encyclopaedia of Mathematical Sciences

  • DOI: https://doi.org/10.1007/978-3-642-58227-1

  • Publisher: Springer Berlin, Heidelberg

  • eBook Packages: Springer Book Archive

  • Copyright Information: Springer-Verlag Berlin Heidelberg 1991

  • Hardcover ISBN: 978-3-540-53004-6Published: 27 June 1991

  • Softcover ISBN: 978-3-540-61223-0Published: 14 April 1997

  • eBook ISBN: 978-3-642-58227-1Published: 01 December 2013

  • Series ISSN: 0938-0396

  • Edition Number: 1

  • Number of Pages: XIII, 296

  • Additional Information: Originally published as volume 60 in the series Encyclopaedia of Mathematical Sciences with the title: Number Theory 3

  • Topics: Number Theory, Algebraic Geometry

Buy it now

Buying options

eBook USD 84.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book USD 109.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Other ways to access