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Elliptic Curves

Diophantine Analysis

  • Book
  • © 1978

Overview

Authors:
  1. Serge Lang

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

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

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Table of contents (11 chapters)

  1. Front Matter

    Pages i-xi
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  2. General Algebraic Theory

    1. Front Matter

      Pages 1-1
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    2. Elliptic Functions

      • Serge Lang
      Pages 3-32

    3. The Division Equation

      • Serge Lang
      Pages 33-46

    4. p-Adic Addition

      • Serge Lang
      Pages 47-76

    5. Heights

      • Serge Lang
      Pages 77-100

    6. Kummer Theory

      • Serge Lang
      Pages 101-127

    7. Integral Points

      • Serge Lang
      Pages 128-153

  3. Approximation of Logarithms

    1. Front Matter

      Pages 155-158
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    2. Auxiliary Results

      • Serge Lang
      Pages 159-180

    3. The Baker—Feldman Theorem

      • Serge Lang
      Pages 181-192

    4. Linear Combinations of Elliptic Logarithms

      • Serge Lang
      Pages 193-217

    5. The Baker—Tijdeman Theorem

      • Serge Lang
      Pages 218-233

    6. Refined Inequalities

      • Serge Lang
      Pages 234-252

  4. Back Matter

    Pages 253-264
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Keywords

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.

Authors and Affiliations

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

    Serge Lang

Bibliographic Information

  • Book Title: Elliptic Curves

  • Book Subtitle: Diophantine Analysis

  • Authors: Serge Lang

  • Series Title: Grundlehren der mathematischen Wissenschaften

  • DOI: https://doi.org/10.1007/978-3-662-07010-9

  • Publisher: Springer Berlin, Heidelberg

  • eBook Packages: Springer Book Archive

  • Copyright Information: Springer-Verlag Berlin Heidelberg 1978

  • Hardcover ISBN: 978-3-540-08489-1Published: 01 November 1978

  • Softcover ISBN: 978-3-642-05717-5Published: 19 October 2010

  • eBook ISBN: 978-3-662-07010-9Published: 29 June 2013

  • Series ISSN: 0072-7830

  • Series E-ISSN: 2196-9701

  • Edition Number: 1

  • Number of Pages: XI, 264

  • Topics: Analysis

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