# Elliptic Curves

## Diophantine Analysis

• Serge Lang
Book

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

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 193-217
5. Serge Lang
Pages 218-233
6. Serge Lang
Pages 234-252
4. Back Matter
Pages 253-264

### Introduction

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.

### Keywords

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

#### Authors and affiliations

• Serge Lang
• 1
1. 1.Department of MathematicsYale UniversityNew HavenUSA

### Bibliographic information

• DOI https://doi.org/10.1007/978-3-662-07010-9
• Copyright Information Springer-Verlag Berlin Heidelberg 1978
• Publisher Name Springer, Berlin, Heidelberg
• eBook Packages
• Print ISBN 978-3-642-05717-5
• Online ISBN 978-3-662-07010-9
• Series Print ISSN 0072-7830