# Fundamentals of Diophantine Geometry

Textbook

1. Front Matter
Pages i-xviii
2. Serge Lang
Pages 1-17
3. Serge Lang
Pages 18-49
4. Serge Lang
Pages 50-75
5. Serge Lang
Pages 76-94
6. Serge Lang
Pages 95-137
7. Serge Lang
Pages 138-157
8. Serge Lang
Pages 158-187
9. Serge Lang
Pages 188-224
10. Serge Lang
Pages 225-246
11. Serge Lang
Pages 247-265
12. Serge Lang
Pages 266-295
13. Serge Lang
Pages 296-323
14. Serge Lang
Pages 324-345
15. Back Matter
Pages 347-370

### Introduction

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.

### Keywords

Abelian varieties Diophantische Geometrie Geometry algebra algebraic geometry diophantine equation field finite field

#### Authors and affiliations

1. 1.Department of MathematicsYale UniversityNew HavenUSA

### Bibliographic information

• Book Title Fundamentals of Diophantine Geometry
• Authors S. Lang
• DOI https://doi.org/10.1007/978-1-4757-1810-2
• Copyright Information Springer-Verlag New York 1983
• Publisher Name Springer, New York, NY
• eBook Packages
• Hardcover ISBN 978-0-387-90837-3
• Softcover ISBN 978-1-4419-2818-4
• eBook ISBN 978-1-4757-1810-2
• Edition Number 1
• Number of Pages XVIII, 370
• Number of Illustrations 0 b/w illustrations, 0 illustrations in colour