Rational Points on Algebraic Varieties -

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Résumé :
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Sommaire:
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 arithm?tiques et espaces fibr?s.- Notations et conventions.- 1. Torseurs arithm?tiques.- 2. Espaces fibr?s.- R?f?rences.- Fonctions z?ta des hauteurs des espaces fibr?s.- Notationset conventions.- 3. Fonctions holomorphes dans un tube.- 4. Vari?t?s toriques.- 5. Application aux fibrations en vari?t?s toriques.- Appendice A. Un th?or?me taub?rien.- Appendice B. D?monstration de quelques in?galit?s.- R?f?rences.- 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-Th?l?ne.- 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 m?thode du cercle.- 1. Une version raffin?e d'une conjecture de Manin.- 2. Passageau torseur universel.- 3. Intersections compl?tes.- 4. Conclusion.- R?f?rences.- 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. Rel?vement des courbes de P1,k ? P1,k sur la surface de Kummer.- 2. Exemples.- R?f?rences.- 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....