Heights in Diophantine Geometry

This monograph is a bridge between the classical theory and modern approach via arithmetic geometry.

Enrico Bombieri (Author), Walter Gubler (Author)

9780521712293, Cambridge University Press

Paperback, published 6 September 2007

670 pages
22.7 x 15.5 x 3.2 cm, 0.88 kg

'… remarkable …' European Mathematical Society Newsletter

Diophantine geometry has been studied by number theorists for thousands of years, since the time of Pythagoras, and has continued to be a rich area of ideas such as Fermat's Last Theorem, and most recently the ABC conjecture. This monograph is a bridge between the classical theory and modern approach via arithmetic geometry. The authors provide a clear path through the subject for graduate students and researchers. They have re-examined many results and much of the literature, and give a thorough account of several topics at a level not seen before in book form. The treatment is largely self-contained, with proofs given in full detail. Many results appear here for the first time. The book concludes with a comprehensive bibliography. It is destined to be a definitive reference on modern diophantine geometry, bringing a new standard of rigor and elegance to the field.

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. Neron-Tate heights
10. The Mordell-Weil theorem
11. Faltings theorem
12. The ABC-conjecture
13. Nevanlinna theory
14. The Vojta conjectures
Appendix A. Algebraic geometry
Appendix B. Ramification
Appendix C. Geometry of numbers
Bibliography
Glossary of notation
Index.

Subject Areas: Algebraic geometry [PBMW], Complex analysis, complex variables [PBKD], Real analysis, real variables [PBKB], Number theory [PBH]