http://www.loc.gov/mods/v3 http://www.loc.gov/standards/mods/v3/mods-3-2.xsd
1 online resource (xiii, 558 pages) : illustrations.
"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."--BOOK JACKET.
Part A Geometry of Curves and Abelian Varieties 6 -- A.1 Algebraic Varieties 8 -- A.2 Divisors 34 -- A.3 Linear Systems 49 -- A.4 Algebraic Curves 67 -- A.5 Abelian Varieties over C 91 -- A.6 Jacobians over C 110 -- A.7 Abelian Varieties over Arbitrary Fields 119 -- A.8 Jacobians over Arbitrary Fields 134 -- A.9 Schemes 151 -- Part B Height Functions 168 -- B.1 Absolute Values 170 -- B.2 Heights on Projective Space 174 -- B.3 Heights on Varieties 183 -- B.4 Canonical Height Functions 195 -- B.5 Canonical Heights on Abelian Varieties 199 -- B.6 Counting Rational Points on Varieties 210 -- B.7 Heights and Polynomials 224 -- B.8 Local Height Functions 237 -- B.9 Canonical Local Heights on Abelian Varieties 241 -- B.10 Introduction to Arakelov Theory 243 -- Part C Rational Points on Abelian Varieties 257 -- C.1 Weak Mordell-Weil Theorem 260 -- C.2 Kernel of Reduction Modulo p 267 -- C.3 Appendix Finiteness Theorems in Algebraic Number Theory 273 -- C.4 Appendix Selmer and Tate-Shafarevich Groups 279 -- C.5 Appendix Galois Cohomology and Homogeneous Spaces 283 -- Part D Diophantine Approximation and Integral Points on Curves 299 -- D.1 Two Elementary Results on Diophantine Approximation 300 -- D.2 Roth's Theorem 304 -- D.3 Preliminary Results 307 -- D.4 Construction of the Auxiliary Polynomial 316 -- D.5 Index Is Large 323 -- D.6 Index Is Small (Roth's Lemma) 329 -- D.7 Completion of the Proof of Roth's Theorem 341 -- D.8 Application: The Unit Equation U + V = 1 345 -- D.9 Application: Integer Points on Curves 353 -- Part E Rational Points on Curves of Genus at Least 2 367 -- E.1 Vojta's Geometric Inequality and Faltings' Theorem 369 -- E.2 Pinning Down Some Height Functions 373 -- E.3 An Outline of the Proof of Vojta's Inequality 379 -- E.4 An Upper Bound for h[subscript Omega](z, w) 381 -- E.5 A Lower Bound for h[subscript Omega](z, w) for Nonvanishing Sections 385 -- E.6 Constructing Sections of Small Height I: Applying Riemann-Roch 389 -- E.7 Constructing Sections of Small Height II: Applying Siegel's Lemma 393 -- E.8 Lower Bound for h[subscript Omega](z, w) at Admissible (i*[subscript 1], i*[subscript 2]): Version I 401 -- E.9 Eisenstein's Estimate for the Derivatives of an Algebraic Function 408 -- E.10 Lower Bound for h[subscript Omega](z, w) at Admissible (i*[subscript 1], i*[subscript 2]): Version II 412 -- E.11 A Nonvanishing Derivative of Small Order 418 -- E.12 Completion of the Proof of Vojta's Inequality 421 -- Part F Further Results and Open Problems 433 -- F.1 Curves and Abelian Varieties 434 -- F.1.1 Rational Points on Subvarieties of Abelian Varieties 434 -- F.1.2 Application to Points of Bounded Degree on Curves 439 -- F.2 Discreteness of Algebraic Points 443 -- F.2.1 Bogomolov's Conjecture 444 -- F.2.2 Height of a Variety 445 -- F.3 Height Bounds and Height Conjectures 451 -- F.4 Search for Effectivity 456 -- F.4.1 Effective Computation of the Mordell-Weil Group A([kappa]) 457 -- F.4.2 Effective Computation of Rational Points on Curves 465 -- F.4.3 Quantitative Bounds for Rational Points 472 -- F.5 Geometry Governs Arithmetic 474 -- F.5.1 Kodaira Dimension 475 -- F.5.2 Bombieri-Lang Conjecture 479 -- F.5.3 Vojta's Conjecture 482 -- F.5.4 Varieties Whose Rational Points Are Dense 487 -- Part A Geometry of Curves and Abelian Varieties 6 -- A.1 Algebraic Varieties 8 -- A.1.1 Affine and Projective Varieties 9 -- A.1.2 Algebraic Maps and Local Rings 15 -- A.1.3 Dimension 22 -- A.1.4 Tangent Spaces and Differentials 24 -- A.2 Divisors 34 -- A.2.1 Weil Divisors 34 -- A.2.2 Cartier Divisors 37 -- A.2.3 Intersection Numbers 44 -- A.3 Linear Systems 49 -- A.3.1 Linear Systems and Maps 49 -- A.3.2 Ampleness and the Enriques-Severi-Zariski Lemma 52 -- A.3.3 Line Bundles and Sheavese 56 -- A.4 Algebraic Curves 67 -- A.4.1 Birational Models of Curves 68 -- A.4.2 Genus of a Curve and the Riemann-Roch Theorem 70 -- A.4.3 Curves of Genus 0 74 -- A.4.4 Curves of Genus 1 76 -- A.4.5 Curves of Genus at Least 2 81 -- A.4.6 Algebraic Surfaces 84 -- A.5 Abelian Varieties over C 91 -- A.5.1 Complex Tori 93 -- A.5.2 Divisors, Theta Functions, and Riemann Forms 97 -- A.5.3 Riemann-Roch for Abelian Varieties 103 -- A.6 Jacobians over C 110 -- A.6.1 Abelian Integrals 110 -- A.6.2 Periods of Riemann Surfaces 111 -- A.6.3 Jacobian of a Riemann Surface 113 -- A.6.4 Albanese Varieties 116 -- A.7 Abelian Varieties over Arbitrary Fields 119 -- A.7.1 Generalities 119 -- A.7.2 Divisors and the Theorem of the Cube 121 -- A.7.3 Dual Abelian Varieties and Poincare Divisors 128 -- A.8 Jacobians over Arbitrary Fields 134 -- A.8.1 Construction and Properties 134 -- A.8.2 Divisor [Theta] 138 -- A.8.3 Appendix Families of Subvarieties 142 -- A.9 Schemes 151 -- A.9.1 Varieties over Z 151 -- A.9.2 Analogies Between Number Fields and Function Fields 159 -- A.9.3 Minimal Model of a Curve 160 -- A.9.4 Neron Model of an Abelian Variety 162.
Includes bibliographical references (pages [504]-519) and index.