Diophantine geometry is the study of integral and rational solutions to systems of polynomial equations using ideas and techniques from algebraic number theory and algebraic geometry. The ultimate goal is to describe the solutions in terms of geometric invariants of the underlying algebraic variety. This book contains complete proofs of four of the fundamental finiteness theorems in Diophantine geometry:
• The Mordell-Weil theorem - The group of rational points on an Abelian variety is finitely generated
• Roth's theorem - An algebraic number has finitely many approximations of order 2+r
• Siegel's theorem - An affine curve of genus at least one has finitely many integral points
• Faltings' theorem (Mordell conjecture) - A curve of genus at least two has finitely many rational points
Also included are a lengthy overview (with sketched or omitted proofs) of algebraic geometry, a detailed development of the theory of height functions, a discussion of further results and open problems, numerous exercises, and a comprehensive index.