Ulteriori informazioni
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.
Sommario
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.
Info autore
Prof. Dr. Gisbert Wüstholz ist Professor für Mathematik an der ETH Zürich.
