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.
Springer Book Archives
"In this excellent 500-page volume, the authors introduce the
reader to four fundamental finiteness theorems in Diophantine
geometry. After reviewing algebraic geometry and the theory of
heights in Parts A and B, the Mordell-Weil theorem (the group of
rational points on an abelian variety is finitely generated) is
presented in Part C, Roth's theorem (an algebraic number has
finitely many approximations of order $2 + \varepsilon$) and
Siegel's theorem (an affine curve of genus $g \ge 1$ has finitely
many integral points) are proved in Part D, and Faltings' theorem
(a curve of genus $g \ge 2$ has finitely many rational points) is
discussed in Part E.
Together, Parts C--E form the core of the book and can be read by
any reader already acquainted with algebraic number theory,
classical (i.e., not scheme-theoretical) algebraic geometry, and
the height machine. The authors write clearly and strive to help
the reader understand this difficult material. They provide
insightful introductions, clear motivations for theorems, and
helpful outlines of complicated proofs.
This volume will not only serve as a very useful reference for the
advanced reader, but it will also be an invaluable tool for
students attempting to study Diophantine geometry. Indeed, such
students usually face the difficult task of having to acquire a
sufficient grasp of algebraic geometry to be able to use
algebraic-geometric tools to study Diophantine applications. Many
beginners feel overwhelmed by the geometry before they read any of
the beautiful arithmetic results. To help such students, the
authors have devoted about a third of the volume, Part A, to a
lengthy introduction to algebraic geometry, and suggest that the
reader begin by skimming Part A, possibly reading more closely any
material that covers gaps in the reader's knowledge. Then Part A
should be used as a reference source for geometric facts as they
are needed while reading the rest of the book.
The first arithmetic portion of the book is Part B, which deals
with the theory of height functions, functions which measure the
"size" of a point on an algebraic variety. These objects are a key
tool for the Diophantine study in Parts C--E, and the authors, in
their characteristically clear and insightful style, fully prove in
Part B most results on heights later used in the book.
The book concludes in Part F with a survey of further results and
open problems, such as the generalization of Mordell's conjecture
to higher-dimensional subvarieties of abelian varieties and
questions of quantitative and effective results on the solutions of
Diophantine problems.
This book is a most welcome addition to the literature. It is well
written and renders accessible to students of Diophantine geometry
some of the most elegant and beautiful arithmetical results of the
20th century." (Dino J. Lorenzini, Mathematical Reviews)
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