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Fundamentals of Diophantine Geometry

  • Textbook
  • © 1983

Overview

Authors:
  1. Serge Lang

    1. Department of Mathematics, Yale University, New Haven, USA

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Table of contents (13 chapters)

  1. Front Matter

    Pages i-xviii
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  2. Absolute Values

    • Serge Lang
    Pages 1-17

  3. Proper Sets of Absolute Values. Divisors and Units

    • Serge Lang
    Pages 18-49

  4. Heights

    • Serge Lang
    Pages 50-75

  5. Geometric Properties of Heights

    • Serge Lang
    Pages 76-94

  6. Heights on Abelian Varieties

    • Serge Lang
    Pages 95-137

  7. The Mordell-Weil Theorem

    • Serge Lang
    Pages 138-157

  8. The Thue-Siegel-Roth Theorem

    • Serge Lang
    Pages 158-187

  9. Siegel’s Theorem and Integral Points

    • Serge Lang
    Pages 188-224

  10. Hilbert’s Irreducibility Theorem

    • Serge Lang
    Pages 225-246

  11. Weil Functions and Néron Divisors

    • Serge Lang
    Pages 247-265

  12. Néron Functions on Abelian Varieties

    • Serge Lang
    Pages 266-295

  13. Algebraic Families of Néron Functions

    • Serge Lang
    Pages 296-323

  14. Néron Functions Over the Complex Numbers

    • Serge Lang
    Pages 324-345

  15. Back Matter

    Pages 347-370
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Keywords

About this book

Diophantine problems represent some of the strongest aesthetic attractions to algebraic geometry. They consist in giving criteria for the existence of solutions of algebraic equations in rings and fields, and eventually for the number of such solutions. The fundamental ring of interest is the ring of ordinary integers Z, and the fundamental field of interest is the field Q of rational numbers. One discovers rapidly that to have all the technical freedom needed in handling general problems, one must consider rings and fields of finite type over the integers and rationals. Furthermore, one is led to consider also finite fields, p-adic fields (including the real and complex numbers) as representing a localization of the problems under consideration. We shall deal with global problems, all of which will be of a qualitative nature. On the one hand we have curves defined over say the rational numbers. Ifthe curve is affine one may ask for its points in Z, and thanks to Siegel, one can classify all curves which have infinitely many integral points. This problem is treated in Chapter VII. One may ask also for those which have infinitely many rational points, and for this, there is only Mordell's conjecture that if the genus is :;;; 2, then there is only a finite number of rational points.

Authors and Affiliations

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

    Serge Lang

Bibliographic Information

  • Book Title: Fundamentals of Diophantine Geometry

  • Authors: Serge Lang

  • DOI: https://doi.org/10.1007/978-1-4757-1810-2

  • Publisher: Springer New York, NY

  • eBook Packages: Springer Book Archive

  • Copyright Information: Springer Science+Business Media New York 1983

  • Hardcover ISBN: 978-0-387-90837-3Published: 29 August 1983

  • Softcover ISBN: 978-1-4419-2818-4Published: 03 December 2010

  • eBook ISBN: 978-1-4757-1810-2Published: 29 June 2013

  • Edition Number: 1

  • Number of Pages: XVIII, 370

  • Additional Information: Originally published by Wiley-Interscience

  • Topics: Geometry

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