Bombieri E., Gubler W. Heights in Diophantine Geometry

Cambridge University Press, 2007 — 670 pp. — (New Mathematical Monographs: 4).
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 provide 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.
The authors 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.
For graduate students and researchers, and is largely self-contained: proofs are given in full detail, and many results appear here for the first time.
Destined to be a definitive reference on modern diophantine geometry, bringing a new standard of rigour and elegance to the field.Heights.
Weil heights.
Linear tori.
Small points.
The unit equation.
Roth's theorem.
The subspace theorem.
Abelian varieties.
Neron-Tate heights.
The Mordell-Weil theorem.
Faltings theorem.
The ABC-conjecture.
Nevanlinna theory.
The Vojta conjectures.
Appendix A. Algebraic geometry.
Appendix B. Ramification.
Appendix C. Geometry of numbers.Glossary of notation.