Number Theory III

1. Front Matter

   Pages i-xiii

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2. Some Qualitative Diophantine Statements

   • R. V. Gamkrelidze

   Pages 1-42

3. Heights and Rational Points

   • R. V. Gamkrelidze

   Pages 43-67

4. Abelian Varieties

   • R. V. Gamkrelidze

   Pages 68-100

5. Faltings’ Finiteness Theorems on Abelian Varieties and Curves

   • R. V. Gamkrelidze

   Pages 101-122

6. Modular Curves Over Q

   • R. V. Gamkrelidze

   Pages 123-142

7. The Geometric Case of Mordell’s Conjecture

   • R. V. Gamkrelidze

   Pages 143-162

8. Arakelov Theory

   • R. V. Gamkrelidze

   Pages 163-175

9. Diophantine Problems and Complex Geometry

   • R. V. Gamkrelidze

   Pages 176-204

10. Weil Functions, Integral Points and Diophantine Approximations

    • R. V. Gamkrelidze

    Pages 205-243

11. Existence of (Many) Rational Points

    • R. V. Gamkrelidze

    Pages 244-262

12. Back Matter

    Pages 263-296

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In 1988 Shafarevich asked me to write a volume for the Encyclopaedia of Mathematical Sciences on Diophantine Geometry. I said yes, and here is the volume. By definition, diophantine problems concern the solutions of equations in integers, or rational numbers, or various generalizations, such as finitely generated rings over Z or finitely generated fields over Q. The word Geometry is tacked on to suggest geometric methods. This means that the present volume is not elementary. For a survey of some basic problems with a much more elementary approach, see [La 9Oc]. The field of diophantine geometry is now moving quite rapidly. Out­ standing conjectures ranging from decades back are being proved. I have tried to give the book some sort of coherence and permanence by em­ phasizing structural conjectures as much as results, so that one has a clear picture of the field. On the whole, I omit proofs, according to the boundary conditions of the encyclopedia. On some occasions I do give some ideas for the proofs when these are especially important. In any case, a lengthy bibliography refers to papers and books where proofs may be found. I have also followed Shafarevich's suggestion to give examples, and I have especially chosen these examples which show how some classical problems do or do not get solved by contemporary in­ sights. Fermat's last theorem occupies an intermediate position. Al­ though it is not proved, it is not an isolated problem any more.

• Abelian varieties
• Abelian variety
• Dimension
• Diophantine approximation
• Divisor
• elliptic curve
• Isogenie
• number theory
• sheaves

From the reviews: "Between number theory and geometry there have been several stimulating influences, and this book records these enterprises. This author, who has been at the centre of such research for many years, is one of the best guides a reader can hope for. The book is full of beautiful results, open questions, stimulating conjectures and suggestions where to look for future developments. This volume bears witness of the broad scope of knowledge of the author, and the influence of several people who have commented on the manuscript before publication... Although in the series of number theory, this volume is on diophantine geometry, the reader will notice that algebraic geometry is present in every chapter. ...The style of the book is clear. Ideas are well explained, and the author helps the reader to pass by several technicalities. Mededelingen van het wiskundig genootschap

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

  Serge Lang