I: Moduli Spaces -- ยง 1 Introduction -- ยง 2 Generalities about moduli spaces -- ยง 3 Examples -- ยง 4 Metrics with logarithmic singularities -- ยง 5 The minimal compactification of Ag/? -- ยง 8 The toroidal compactification -- II: Heights -- ยง 1 The definition -- ยง 2 Nรฉron-Tate heights -- ยง 3 Heights on the moduli space -- ยง 4 Applications -- III: Some Facts from the Theory of Group Schemes -- ยง 0 Introduction -- ยง 1 Generalities on group schemes -- ยง 2 Finite group schemes -- ยง 3 p-divisible groups -- ยง 4 A theorem of Raynaud -- ยง 5 A theorem of Tate -- IV: Tateโs Conjecture on the Endomorphisms of Abelian Varieties -- ยง 1 Statements -- ยง 2 Reductions -- ยง 3 Heights -- ยง 4 Variants -- V: The Finiteness Theorems of Faltings -- ยง 1 Introduction -- ยง 2 The finiteness theorem for isogeny classes -- ยง 3 The finiteness theorem for isomorphism classes -- ยง 4 Proof of Mordellโs conjecture -- ยง 5 Siegelโs Theorem on integer points -- VI: Complements to Mordell -- ยง 1 Introduction -- ยง 2 Preliminaries -- ยง 3 The Tate conjecture -- ยง 4 The Shafarevich conjecture -- ยง 5 Endomorphisms -- ยง 6 Effectivity -- VII: Intersection Theory on Arithmetic Surfaces -- ยง 0 Introduction -- ยง 1 Hermitian line bundles -- ยง 2 Arakelov divisors and intersection theory -- ยง 3 Volume forms on IR?(X, ?) -- ยง 4 Riemann Roch -- ยง 5 The Hodge index theorem -- Appendix: New Developments in Diophantine and Arithmetic Algebraic Geometry (Gisbert Wรผstholz) -- ยง 2 The transcendental approach -- ยง 3 Vojtaโs approach -- ยง 4 Arithmetic Riemann-Roch Theorem -- ยง 5 Applications in Arithmetic -- ยง 6 Small sections -- ยง 7 Vojtaโs proof in the number field case -- ยง 8 Langโs conjecture -- ยง 9 Proof of Faltingsโ theorem -- ยง 10 An elementary proof of Mordellโs conjecture -- ยง 11 ?-adic representations attached to abelian varieties

1. Mathematics
2. Difference equations
3. Functional equations
4. Geometry
5. Number theory
6. Mathematics
7. Number Theory
8. Geometry
9. Difference and Functional Equations