Pub. Date:
Birkhäuser Basel
Rational Points on Algebraic Varieties: Zweite, aktualisierte und erweiterte Auflage / Edition 1

Rational Points on Algebraic Varieties: Zweite, aktualisierte und erweiterte Auflage / Edition 1

by Emmanuel Peyre, Yuri Tschinkel


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Product Details

ISBN-13: 9783764366124
Publisher: Birkhäuser Basel
Publication date: 01/18/2002
Series: Progress in Mathematics , #199
Edition description: 2001
Pages: 446
Product dimensions: 6.10(w) x 9.25(h) x 0.04(d)

Table of Contents

Diagonal cubic equations in four variables with prime coefficients.- References.- Rational points on cubic surfaces.- 1. Notations and preliminaries.- 2. Ternary quadratic forms.- 3. Proof of the main theorem.- References.- Torseurs arithmétiques et espaces fibrés.- Notations et conventions.- 1. Torseurs arithmétiques.- 2. Espaces fibrés.- Références.- Fonctions zêta des hauteurs des espaces fibrés.- Notationset conventions.- 3. Fonctions holomorphes dans un tube.- 4. Variétés toriques.- 5. Application aux fibrations en variétés toriques.- Appendice A. Un théorème taubérien.- Appendice B. Démonstration de quelques inégalités.- Références.- Hasse principle for pencils of curves of genus one whose Jacobians have a rational 2-division point, close variation on a paper of Bender and Swinnerton-Dyer.- Statement of the Theorems.- 1. Selmer groups associated to a degree 2 isogeny.- 2. Proof of Theorem A.- 3. Proof of Theorem B.- References.- Enriques surfaces with a dense set of rational points, Appendix to the paper by J.-L. Colliot-Thélène.- References.- Density of integral points on algebraic varieties.- 1. Generalities.- 2. Geometry.- 3. The fibration method and nondegenerate multisections.- 4. Approximation techniques.- 5. Conic bundles and integral points.- 6. Potential density for log K3 surfaces.- References.- Composition of points and the Mordell–Weil problem for cubic surfaces.- 1. Introduction.- 2. Cardinality of generators of subgroups in a reflection group.- 3. Structure of universal equivalence.- 4. A group–theoretic description of universal equivalence.- 5. Birationally trivial cubic surfaces: a finiteness theorem.- References.- Torseurs universels et méthode du cercle.- 1. Une version raffinée d’une conjecture de Manin.- 2. Passage au torseur universel.- 3. Intersections complètes.- 4. Conclusion.- Références.- Tamagawa numbers of diagonal cubic surfaces of higher rank.- 1. Description of the conjectural constant.- 2. The Galois module Pic($$\bar{V}$$).- 3. Euler product for the good places.- 4. Density at the bad places.- 5. The constant a(V).- 6. Some statistical formulae.- 7. Presentation of the results.- References.- The Hasse principle for complete intersections in projective space.- References.- Une construction de courbes k-rationnelles sur les surfaces de Kummer d’un produit de courbes de genre 1..- 1. Relèvement des courbes de P1,k × P1,k sur la surface de Kummer.- 2. Exemples.- Références.- Arithmetic Stratifications and Partial Eisenstein Series.- 1. The fibre bundles: geometric-arithmetic preliminaries.- 2. Height zeta functions.- 3. Arithmetic stratification.- References.- Weak Approximation and R-equivalence on Cubic Surfaces.- 1. Introduction.- 2. Geometric background.- 3. Approximation at an infinite prime.- 4. Approximation at a finite prime.- 5. The lifting process.- 6. The dense lifting process.- 7. Adelic results.- 8. Surfaces X13 + X23 + X33 ? dX03 = 0.- References.- Hua’s lemma and exponential sums over binary forms.- 1. Introduction.- 2. Preliminary reductions.- 3. Integral points on affine plane curves.- 4. The inductive step.- 5. The completion of the proof of Theorem 1.1.- References.

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