Diophantine Geometry

This is an introduction to diophantine geometry at the advanced graduate level. The book contains a proof of the Mordell conjecture which will make it quite attractive to graduate students and professional mathematicians. In each part of the book, the reader will find numerous exercises.

This is an introduction to diophantine geometry at the advanced graduate level. The book contains a proof of the Mordell conjecture which will make it quite attractive to graduate students and professional mathematicians. In each part of the book, the reader will find numerous exercises.

A The Geometry of Curves and Abelian Varieties.- A.1 Algebraic Varieties.- A.2 Divisors.- A.3 Linear Systems.- A.4 Algebraic Curves.- A.5 Abelian Varieties over C.- A.6 Jacobians over C.- A.7 Abelian Varieties over Arbitrary Fields.- A.8 Jacobians over Arbitrary Fields.- A.9 Schemes.- B Height Functions.- B.1 Absolute Values.- B.2 Heights on Projective Space.- B.3 Heights on Varieties.- B.4 Canonical Height Functions.- B.5 Canonical Heights on Abelian Varieties.- B.6 Counting Rational Points on Varieties.- B.7 Heights and Polynomials.- B.8 Local Height Functions.- B.9 Canonical Local Heights on Abelian Varieties.- B.10 Introduction to Arakelov Theory.- Exercises.- C Rational Points on Abelian Varieties.- C.1 The Weak Mordell-Weil Theorem.- C.2 The Kernel of Reduction Modulo p.- C.3 Appendix: Finiteness Theorems in Algebraic Number Theory.- C.4 Appendix: The Selmer and Tate-Shafarevich Groups.- C.5 Appendix: Galois Cohomology and Homogeneous Spaces.- Exercises.- D Diophantine Approximation and Integral Points on Curves.- D.1 Two Elementary Results on Diophantine Approximation.- D.2 Roth's Theorem.- D.3 Preliminary Results.- D.4 Construction of the Auxiliary Polynomial.- D.5 The Index Is Large.- D.6 The Index Is Small (Roth's Lemma).- D.7 Completion of the Proof of Roth's Theorem.- D.8 Application: The Unit Equation U + V = 1.- D.9 Application: Integer Points on Curves.- Exercises.- E Rational Points on Curves of Genus at Least 2.- E.I Vojta's Geometric Inequality and Faltings' Theorem.- E.2 Pinning Down Some Height Functions.- E.3 An Outline of the Proof of Vojta's Inequality.- E.4 An Upper Bound for h?(z, w).- E.5 A Lower Bound for h?(z,w) for Nonvanishing Sections.- E.6 Constructing Sections of Small Height I: Applying Riemann-Roch.- E.7 Constructing Sections of Small Height II: Applying Siegel's Lemma.- E.8 Lower Bound for h?(z,w) at Admissible Version I.- E.9 Eisenstein's Estimate for the Derivatives of an Algebraic Function.- E.10 Lower Bound for h?(z,w) at Admissible: Version II.- E.11 A Nonvanishing Derivative of Small Order.- E.12 Completion of the Proof of Vojta's Inequality.- Exercises.- F Further Results and Open Problems.- F.1 Curves and Abelian Varieties.- F.1.1 Rational Points on Subvarieties of Abelian Varieties.- F.1.2 Application to Points of Bounded Degree on Curves.- F.2 Discreteness of Algebraic Points.- F.2.1 Bogomolov's Conjecture.- F.2.2 The Height of a Variety.- F.3 Height Bounds and Height Conjectures.- F.4 The Search for Effectivity.- F.4.1 Effective Computation of the Mordell-Weil Group A(k).- F.4.2 Effective Computation of Rational Points on Curves.- F.4.3 Quantitative Bounds for Rational Points.- F.5 Geometry Governs Arithmetic.- F.5.1 Kodaira Dimension.- F.5.2 The Bombieri-Lang Conjecture.- F.5.3 Vojta's Conjecture.- F.5.4 Varieties Whose Rational Points Are Dense.- Exercises.- References.- List of Notation.