© 1978

Elliptic Curves

Diophantine Analysis


Part of the Grundlehren der mathematischen Wissenschaften book series (GL, volume 231)

Table of contents

  1. Front Matter
    Pages i-xi
  2. General Algebraic Theory

    1. Front Matter
      Pages 1-1
    2. Serge Lang
      Pages 3-32
    3. Serge Lang
      Pages 33-46
    4. Serge Lang
      Pages 47-76
    5. Serge Lang
      Pages 77-100
    6. Serge Lang
      Pages 101-127
    7. Serge Lang
      Pages 128-153
  3. Approximation of Logarithms

    1. Front Matter
      Pages 155-158
    2. Serge Lang
      Pages 159-180
    3. Serge Lang
      Pages 181-192
    4. Serge Lang
      Pages 218-233
    5. Serge Lang
      Pages 234-252
  4. Back Matter
    Pages 253-264

About this book


It is possible to write endlessly on elliptic curves. (This is not a threat.) We deal here with diophantine problems, and we lay the foundations, especially for the theory of integral points. We review briefly the analytic theory of the Weierstrass function, and then deal with the arithmetic aspects of the addition formula, over complete fields and over number fields, giving rise to the theory of the height and its quadraticity. We apply this to integral points, covering the inequalities of diophantine approximation both on the multiplicative group and on the elliptic curve directly. Thus the book splits naturally in two parts. The first part deals with the ordinary arithmetic of the elliptic curve: The transcendental parametrization, the p-adic parametrization, points of finite order and the group of rational points, and the reduction of certain diophantine problems by the theory of heights to diophantine inequalities involving logarithms. The second part deals with the proofs of selected inequalities, at least strong enough to obtain the finiteness of integral points.


Algebra Arithmetic Curves Diophantische Approximation Diophantische Ungleichung Elliptische Kurve equation function theorem

Authors and affiliations

  1. 1.Department of MathematicsYale UniversityNew HavenUSA

Bibliographic information

  • Book Title Elliptic Curves
  • Book Subtitle Diophantine Analysis
  • Authors S. Lang
  • Series Title Grundlehren der mathematischen Wissenschaften
  • DOI
  • Copyright Information Springer-Verlag Berlin Heidelberg 1978
  • Publisher Name Springer, Berlin, Heidelberg
  • eBook Packages Springer Book Archive
  • Hardcover ISBN 978-3-540-08489-1
  • Softcover ISBN 978-3-642-05717-5
  • eBook ISBN 978-3-662-07010-9
  • Series ISSN 0072-7830
  • Edition Number 1
  • Number of Pages XI, 264
  • Number of Illustrations 0 b/w illustrations, 0 illustrations in colour
  • Topics Analysis
  • Buy this book on publisher's site
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