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

ISBN-13: 9783034895361 / Angielski / Miękka / 2012 / 446 str.

Emmanuel Peyre;Yuri Tschinkel

Opis

This book is devoted to the study of rational and integral points on higher- dimensional algebraic varieties. It contains research papers addressing the arithmetic geometry of varieties which are not of general type, with an em- phasis on how rational points are distributed with respect to the classical, Zariski and adelic topologies. The book gives a glimpse of the state of the art of this rapidly expanding domain in arithmetic geometry. The techniques involve explicit geometric con- structions, ideas from the minimal model program in algebraic geometry as well as analytic number theory and harmonic analysis on adelic groups. In recent years there has been substantial progress in our understanding of the arithmetic of algebraic surfaces. Five papers are devoted to cubic surfaces: Basile and Fisher study the existence of rational points on certain diagonal cubics, Swinnerton-Dyer considers weak approximation and Broberg proves upper bounds on the number of rational points on the complement to lines on cubic surfaces. Peyre and Tschinkel compare numerical data with conjectures concerning asymptotics of rational points of bounded height on diagonal cubics of rank 2. Kanevsky and Manin investigate the composition of points on cubic surfaces. Satge constructs rational curves on certain Kummer surfaces. Colliot-Thelene studies the Hasse principle for pencils of curves of genus 1. In an appendix to this paper Skorobogatov produces explicit examples of Enriques surfaces with a Zariski dense set of rational points.

Spis treści

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($$\bar$$).- 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.