Heights in Diophantine Geometry: (Series Number 4 New Mathematical Monographs)

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.

Table of Contents:
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.

About the Author :
Professor Enrico Bombieri is a Professor of Mathematics at the Institute for Advanced Study. Dr Walter Gubler is a Lecturer in Mathematics at the University of Dortmund.

Review :
'This monograph is a bridge between the classical theory and a modern approach via arithmetic geometry. The authors aim to 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.' L'enseignement mathematique 'The quality of exposition is exemplary, which is not surprising, given the brilliant expository style of the elder author.' Yuri Bilu, Mathematical Review 'Bombieri and Gubler have written an excellent introduction to some exciting mathematics ... written with an excellent combination of clarity and rigor, with the authors highlighting which parts can be skipped on a first reading and which parts are particularly important for later material. The book also contains a glossary of notation, a good index, and a nice bibliography collecting many of the primary sources in this field.' MAA Reviews