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.
Inhaltsverzeichnis
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.
