Book contents
- Frontmatter
- Contents
- Preface
- Terminology
- 1 Heights
- 2 Weil heights
- 3 Linear tori
- 4 Small points
- 5 The unit equation
- 6 Roth's theorem
- 7 The subspace theorem
- 8 Abelian varieties
- 9 Néron–Tate heights
- 10 The Mordell–Weil theorem
- 11 Faltings's theorem
- 12 The abc-conjecture
- 13 Nevanlinna theory
- 14 The Vojta conjectures
- Appendix A Algebraic geometry
- Appendix B Ramification
- Appendix C Geometry of numbers
- References
- Glossary of notation
- Index
Preface
Published online by Cambridge University Press: 14 August 2009
- Frontmatter
- Contents
- Preface
- Terminology
- 1 Heights
- 2 Weil heights
- 3 Linear tori
- 4 Small points
- 5 The unit equation
- 6 Roth's theorem
- 7 The subspace theorem
- 8 Abelian varieties
- 9 Néron–Tate heights
- 10 The Mordell–Weil theorem
- 11 Faltings's theorem
- 12 The abc-conjecture
- 13 Nevanlinna theory
- 14 The Vojta conjectures
- Appendix A Algebraic geometry
- Appendix B Ramification
- Appendix C Geometry of numbers
- References
- Glossary of notation
- Index
Summary
Diophantine geometry, the study of equations in integer and rational numbers, is one of the oldest subjects of mathematics and possibly the most popular part of number theory, for the professional mathematician and the amateur alike. Certainly, one of its main attractions is that, far from being a disconnected assembly of isolated results, it provides glimpses of a view which hints at a well-organized underlying structure.
Diophantine equations are of course determined by the underlying algebraic equations and therefore their associated algebraic geometry, obtained by dropping the condition that the solutions must be integers or rational numbers, plays a big role in their study. However, algebraic geometry is already not an easy subject. A pioneer and one of the founding fathers of algebraic geometry, the German mathematician Max Noether, after seeing the theory of algebraic curves with its elegance, simplicity, and also depth of results, and comparing it with the collection of the existing examples of algebraic surfaces at the time, for which nothing comparable could be found, used to say that algebraic curves were created by God and algebraic surfaces by the Devil. Only later, with the development of new tools, in particular the introduction of cohomological and topological methods, the theory of surfaces and higher-dimensional varieties over a field found a satisfactory status.
Of special importance for arithmetic was the development of algebraic geometry over fields of positive characteristic and p-adic fields, since the study of polynomial congruences leads very naturally to such problems.
- Type
- Chapter
- Information
- Heights in Diophantine Geometry , pp. xi - xivPublisher: Cambridge University PressPrint publication year: 2006