Fundamentals of Diophantine Geometry

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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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.

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

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

  Serge Lang

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