Gerd Faltings, Gisbert Wüstholz

Rational Points (eBook, PDF)

Seminar Bonn/Wuppertal 1983/84

• Format: PDF

Produktbeschreibung

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.

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Produktdetails

• Verlag: Vieweg+Teubner Verlag
• Seitenzahl: 312
• Erscheinungstermin: 6. Dezember 2012
• Englisch
• ISBN-13: 9783322803405
• Artikelnr.: 53145382

Autorenporträt

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.