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AuthorHindry, Marc. author
TitleDiophantine Geometry [electronic resource] : An Introduction / by Marc Hindry, Joseph H. Silverman
ImprintNew York, NY : Springer New York : Imprint: Springer, 2000
Connect tohttp://dx.doi.org/10.1007/978-1-4612-1210-2
Descript XIII, 561 p. online resource

SUMMARY

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


CONTENT

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 Mordellโ{128}{148}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 Tateโ{128}{148}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 Rothโ{128}{153}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 (Rothโ{128}{153}s Lemma) -- D.7 Completion of the Proof of Rothโ{128}{153}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 Vojtaโ{128}{153}s Geometric Inequality and Faltingsโ{128}{153} Theorem -- E.2 Pinning Down Some Height Functions -- E.3 An Outline of the Proof of Vojtaโ{128}{153}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 Riemannโ{128}{148}Roch -- E.7 Constructing Sections of Small Height II: Applying Siegelโ{128}{153}s Lemma -- E.8 Lower Bound for h?(z,w) at Admissible Version I -- E.9 Eisensteinโ{128}{153}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 Vojtaโ{128}{153}s Inequality -- Exercises -- F Further Results and Open Problems -- F.1 Curves and Abelian Varieties -- F.2 Discreteness of Algebraic Points -- F.3 Height Bounds and Height Conjectures -- F.4 The Search for Effectivity -- F.5 Geometry Governs Arithmetic -- Exercises -- References -- List of Notation


Mathematics Algebraic geometry Number theory Mathematics Algebraic Geometry Number Theory



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