This book consists of the notes from the seminar Bonn/ Wuppertal 1983/ 84 on Arithmetic Geometry. It contains a proof for the Mordell conjecture and may be useful as an introduction to Arakelov's point of view in diophantine geometry. The third edition includes an appendix in which a detailed survey on the spectacular recent developments in arithmetic algebraic geometry is given. These beautiful new results have their roots in the material covered by this book.
This book consists of the notes from the seminar Bonn/ Wuppertal 1983/ 84 on Arithmetic Geometry. It contains a proof for the Mordell conjecture and may be useful as an introduction to Arakelov's point of view in diophantine geometry. The third edition includes an appendix in which a detailed survey on the spectacular recent developments in arithmetic algebraic geometry is given. These beautiful new results have their roots in the material covered by this book.
Prof. Dr. Gisbert Wüstholz ist Professor für Mathematik an der ETH Zürich.
Inhaltsangabe
I: Moduli Spaces.- 1 Introduction.- 2 Generalities about moduli spaces.- 3 Examples.- 4 Metrics with logarithmic singularities.- 5 The minimal compactification of Ag/?.- 8 The toroidal compactification.- II: Heights.- 1 The definition.- 2 Néron-Tate heights.- 3 Heights on the moduli space.- 4 Applications.- III: Some Facts from the Theory of Group Schemes.- 0 Introduction.- 1 Generalities on group schemes.- 2 Finite group schemes.- 3 p-divisible groups.- 4 A theorem of Raynaud.- 5 A theorem of Tate.- IV: Tate's Conjecture on the Endomorphisms of Abelian Varieties.- 1 Statements.- 2 Reductions.- 3 Heights.- 4 Variants.- V: The Finiteness Theorems of Faltings.- 1 Introduction.- 2 The finiteness theorem for isogeny classes.- 3 The finiteness theorem for isomorphism classes.- 4 Proof of Mordell's conjecture.- 5 Siegel's Theorem on integer points.- VI: Complements to Mordell.- 1 Introduction.- 2 Preliminaries.- 3 The Tate conjecture.- 4 The Shafarevich conjecture.- 5 Endomorphisms.- 6 Effectivity.- VII: Intersection Theory on Arithmetic Surfaces.- 0 Introduction.- 1 Hermitian line bundles.- 2 Arakelov divisors and intersection theory.- 3 Volume forms on IR?(X, ?).- 4 Riemann Roch.- 5 The Hodge index theorem.- Appendix: New Developments in Diophantine and Arithmetic Algebraic Geometry (Gisbert Wüstholz).- 2 The transcendental approach.- 3 Vojta's approach.- 4 Arithmetic Riemann-Roch Theorem.- 5 Applications in Arithmetic.- 6 Small sections.- 7 Vojta's proof in the number field case.- 8 Lang's conjecture.- 9 Proof of Faltings' theorem.- 10 An elementary proof of Mordell's conjecture.- 11 ?-adic representations attached to abelian varieties.
I: Moduli Spaces.- 1 Introduction.- 2 Generalities about moduli spaces.- 3 Examples.- 4 Metrics with logarithmic singularities.- 5 The minimal compactification of Ag/?.- 8 The toroidal compactification.- II: Heights.- 1 The definition.- 2 Néron-Tate heights.- 3 Heights on the moduli space.- 4 Applications.- III: Some Facts from the Theory of Group Schemes.- 0 Introduction.- 1 Generalities on group schemes.- 2 Finite group schemes.- 3 p-divisible groups.- 4 A theorem of Raynaud.- 5 A theorem of Tate.- IV: Tate's Conjecture on the Endomorphisms of Abelian Varieties.- 1 Statements.- 2 Reductions.- 3 Heights.- 4 Variants.- V: The Finiteness Theorems of Faltings.- 1 Introduction.- 2 The finiteness theorem for isogeny classes.- 3 The finiteness theorem for isomorphism classes.- 4 Proof of Mordell's conjecture.- 5 Siegel's Theorem on integer points.- VI: Complements to Mordell.- 1 Introduction.- 2 Preliminaries.- 3 The Tate conjecture.- 4 The Shafarevich conjecture.- 5 Endomorphisms.- 6 Effectivity.- VII: Intersection Theory on Arithmetic Surfaces.- 0 Introduction.- 1 Hermitian line bundles.- 2 Arakelov divisors and intersection theory.- 3 Volume forms on IR?(X, ?).- 4 Riemann Roch.- 5 The Hodge index theorem.- Appendix: New Developments in Diophantine and Arithmetic Algebraic Geometry (Gisbert Wüstholz).- 2 The transcendental approach.- 3 Vojta's approach.- 4 Arithmetic Riemann-Roch Theorem.- 5 Applications in Arithmetic.- 6 Small sections.- 7 Vojta's proof in the number field case.- 8 Lang's conjecture.- 9 Proof of Faltings' theorem.- 10 An elementary proof of Mordell's conjecture.- 11 ?-adic representations attached to abelian varieties.
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