Tamagawa measures on universal torsors and points of bounded height on Fano varieties
Nombre et répartition de points de hauteur bornée, Astérisque, no. 251 (1998), pp. 91-258.
@incollection{AST_1998__251__91_0,
     author = {Salberger, Per},
     title = {Tamagawa measures on universal torsors and points of bounded height on {Fano} varieties},
     booktitle = {Nombre et r\'epartition de points de hauteur born\'ee},
     editor = {Peyre Emmanuel},
     series = {Ast\'erisque},
     publisher = {Soci\'et\'e math\'ematique de France},
     number = {251},
     year = {1998},
     zbl = {0959.14007},
     mrnumber = {1679841},
     language = {en},
     url = {http://www.numdam.org/item/AST_1998__251__91_0/}
}
TY  - CHAP
AU  - Salberger, Per
TI  - Tamagawa measures on universal torsors and points of bounded height on Fano varieties
BT  - Nombre et répartition de points de hauteur bornée
ED  - Peyre Emmanuel
T3  - Astérisque
PY  - 1998
DA  - 1998///
IS  - 251
PB  - Société mathématique de France
UR  - http://www.numdam.org/item/AST_1998__251__91_0/
UR  - https://zbmath.org/?q=an%3A0959.14007
UR  - https://www.ams.org/mathscinet-getitem?mr=1679841
LA  - en
ID  - AST_1998__251__91_0
ER  - 
%0 Book Section
%A Salberger, Per
%T Tamagawa measures on universal torsors and points of bounded height on Fano varieties
%B Nombre et répartition de points de hauteur bornée
%E Peyre Emmanuel
%S Astérisque
%D 1998
%N 251
%I Société mathématique de France
%G en
%F AST_1998__251__91_0
Salberger, Per. Tamagawa measures on universal torsors and points of bounded height on Fano varieties, in Nombre et répartition de points de hauteur bornée, Astérisque, no. 251 (1998), pp. 91-258. http://www.numdam.org/item/AST_1998__251__91_0/

[1] M. Artin, A. Grothendieck and J.-L. Verdier - Théorie des topos et cohomologie étale des schémas, Lecture notes in Math., vol. 269, 270, 305, Springer-Verlag, 1972-1973. | MR | Zbl

[2] M. Artin and D. Mumford - Some elementary examples of unirational varieties which are not rational, Proc. London Math. Soc. 25 (1972), p. 75-95. | DOI | MR | Zbl

[3] V. Batyrev and Y. I. Manin - Sur les nombres des points rationnels de hauteur bornée des variétés algébriques, Math. Ann. 286 (1980), p. 27-43. | DOI | EuDML | MR | Zbl

[4] V. Batyrev and Y. Tschinkel - Manin's conjecture for toric varieties, to appear in J. Alg. Geometry. | MR | Zbl

[5] V. Batyrev and Y. Tschinkel, Rational points on some fano cubic bundles, Preprint. | MR | Zbl

[6] V. Batyrev and Y. Tschinkel, Saturated Varieties, this volume.

[7] V. Batyrev and Y. Tschinkel, Rational points of bounded height on compactifications of anisotropic tori, Int. Math. Research Notes 12 (1995), p. 591-635. | DOI | MR | Zbl

[8] S. Bloch - A note on height pairings, Tamagawa numbers and the Birch-Swinnerton-Dyer conjecture, Invent. Math. 58 (1980), p. 65-76. | DOI | EuDML | MR | Zbl

[9] A. Borel - Linear algebraic groups, 2nd ed., Graduate texts in Math., vol. 126, Springer-Verlag, Berlin, 1991. | MR | Zbl

[10] S. Bosch, W. Lüthkebohmert and M. Raynaud - Néron models, Springer-Verlag, 1990. | DOI | MR | Zbl

[11] N. Bourbaki - Intégration, Hermann, Paris, Chap I-VII. | MR | Zbl

[12] N. Bourbaki, Variétés différentielles et analytiques, Paris, 1971, Fascicule de résultats, paragraphes 8 à 15. | Zbl

[13] R. De La Bretèche - Sur le nombre de points de hauteur bornée d'une certaine surface cubique singulière, this volume. | MR | Zbl

[14] J.-L. Colliot-Thélène and J.-J. Sansuc - La R-équivalence sur les tores, Ann. Sci. École Norm. Sup. (4) 10 (1977), p. 175-229. | DOI | EuDML | Numdam | MR | Zbl

[15] J.-L. Colliot-Thélène and J.-J. Sansuc, La descente sur les variétés rationnelles II, Duke Math. J. 54 (1987), p. 375-492. | DOI | MR | Zbl

[16] D. A. Cox - The homogenous coordinate ring of a toric variety, J. Alg. Geom. 4 (1995), p. 17-50. | MR | Zbl

[17] H. Delange - Généralisation du théorème de Ikehara, Ann. Sci. École Norm. Sup. (4) 3 (1971), p. 213-242. | EuDML | Numdam | MR | Zbl

[18] P. Deligne - Cohomologie étale, Springer-Verlag, Berlin, 1977, Séminaire de Géométrie Algébrique du Bois-Marie SGA 4 1 2, Avec la collaboration de J. F. Boutot, A. Grothendieck, L. Illusie et J. L. Verdier, Lecture Notes in Mathematics, Vol. 569. | MR | Zbl

[19] P. Deligne - La conjecture de Weil II, Inst. Hautes Études Sci. Publ. Math. 52 (1981), p. 313-428. | Numdam | MR | Zbl

[20] M. Demazure - Sous-groupes algébriques de rang maximum du groupe de Cremona, Ann. Sci. École Norm. Sup. (4) (1970). | DOI | EuDML | Numdam | MR | Zbl

[21] M. Demazure and A. Grothendieck - Schémas en groupes, Lecture notes in Math., vol. 151, Springer-Verlag, 1970. | MR | Zbl

[21] M. Demazure and A. Grothendieck - Schémas en groupes, Lecture notes in Math., vol. 152, Springer-Verlag, 1970. | Zbl

[21] M. Demazure and A. Grothendieck - Schémas en groupes, Lecture notes in Math., vol. 153, Springer-Verlag, 1970. | MR | Zbl

[22] E. Fouvry - Sur la hauteur des points d'une certaine surface cubique singulière, this volume. | Zbl

[23] J. Fraenke, Y. I. Manin and Y. Tschinkel - Rational points of bounded height of Fano varieties, Invent. Math. 95 (1989), p. 425-35. | EuDML | MR | Zbl

[24] W. Fulton - Introduction to toric varieties, Ann. of Math. Studies, vol. 131, Princeton University Press, 1993. | MR | Zbl

[25] P. Griffiths and J. Harris - Principles of algebraic geometry, Wiley Interscience, 1978. | MR | Zbl

[26] A. Grothendieck - Classes de Chern et représentations linéaires des groupes discrets, Adv. Stud. Pure Math., vol. 3, Adv. Stud. Pure Math., North-Holland, Amsterdam, 1968. | MR | Zbl

[27] A. Grothendieck, Le groupe de Brauer I-III, Dix exposés sur la cohomologie des schémas, Adv. Stud. Pure Math., vol. 3, North-Holland, Amsterdam, 1968. | MR | Zbl

[28] A. Grothendieck, Séminaire de Géométrie Algébrique : Revêtements étales et groupe fondamental, Lecture notes in Math., vol. 224, Springer-Verlag, Heidelberg, 1971. | MR

[29] A. Grothendieck and J. Dieudonné - Éléments de Géométrie Algébrique, Inst. Hautes Études Sci. Publ. Math. 8 (1961). | Zbl

[30] A. Grothendieck and J. Dieudonné, Éléments de Géométrie Algébrique, Inst. Hautes Études Sci. Publ. Math. 20, 24, 28, 32 (1964). | DOI | Zbl

A. Grothendieck and J. Dieudonné, Éléments de Géométrie Algébrique, Inst. Hautes Études Sci. Publ. Math. 20, 24, 28, 32 (1966).

[31] A. Grothendieck and J. Dieudonné, Éléments de Géométrie Algébrique I, Springer, 1971. | MR | Zbl

[32] R. Hartshorne - Algebraic geometry, Graduate texts in Math., vol. 52, Springer-Verlag, Berlin, 1991. | Zbl

[33] D. R. Heath-Brown - The density of zeros of forms for which weak approximation fails, Math. Comp. 59 (1992), p. 613-623. | DOI | MR | Zbl

[34] N. Katz - Review of -adic cohomology, Motives, vol. Volume 55, Symp. in pure Mathematics, no. 1, American Mathematical Society. | MR | Zbl

[35] J. Kollar - Rational curves on algebraic varieties, Ergebnisse der Mathematik 3 Folge Band 32, Springer-Verlag, 1996. | MR | Zbl

[36] S. Lang - Algebraic number theory, Addison-Wesley, 1970. | MR | Zbl

[37] S. Lang, Fundamentals of Diophantine Geometry, Springer-Verlag, 1983. | DOI | MR | Zbl

[38] S. Lang, Real and Functional Analysis, 3 ed., Springer-Verlag, 1993. | MR | Zbl

[39] S. Lang, Differential and Riemannian Manifolds, Springer-Verlag, 1995. | DOI | MR | Zbl

[40] Y. I. Manin - Le groupe de Brauer-Grothendieck en géométrie diophantienne, Actes du congrès intern. Math. Nice 1 (1970), Gauthiers-Villars, Paris, 1971, p. 401-411. | MR | Zbl

[41] Y. I. Manin, Cubic forms : Algebra, geometry and arithmetic, 2nd ed., North-Holland, Amsterdam, 1974. | MR | Zbl

Y. I. Manin, Cubic forms : Algebra, geometry and arithmetic, 2nd ed., North-Holland, Amsterdam, 1986. | MR | Zbl

[42] Y. I. Manin, Notes on arithmetic on fano three-folds, Compositio Math. 85 (1993), p. 37-55. | EuDML | Numdam | MR | Zbl

[43] Y. I. Manin, Problems on rational points and rational curves on algebraic varieties, Surveys in Differential Geometry, vol. II, International Press, 1995. | MR | Zbl

[44] Y. I. Manin and M. A. Tsfasman - Rational varieties : Algebra, geometry and arithmetic, Russian mathematical surveys 41 (1986), no. 2. | DOI | MR | Zbl

[45] J. S. Milne - Etale Cohomology, Princeton University Press, Princeton, 1980. | MR | Zbl

[46] T. Oda - Convex Bodies and Algebraic Geometry, Springer-Verlag, Berlin, 1988. | EuDML | MR | Zbl

[47] J. Oesterlé - Construction de hauteurs archimédienne et p-adiques suivant la méthode de Bloch, Séminaire de théorie des nombres 1980-81, Birkhäuser, Boston, 1980. | MR | Zbl

[48] J. Oesterlé, Réduction modulo p n des sous-ensembles analytiques fermés de NP, Invent. Math. 66 (1982). | DOI | EuDML | MR | Zbl

[49] T. Ono - Arithmetic of algebraic tori, Ann. of Math. 74 (1961), p. 101-139. | DOI | MR | Zbl

[50] T. Ono, On the Tamagawa number of algebraic tori, Ann. of Math. 78 (1963), p. 47-73. | DOI | MR | Zbl

[51] E. Peyre - Terme principal de la fonction zêta des hauteurs et torseurs universels, these proceedings. | Zbl

[52] E. Peyre, Hauteurs et mesures de Tamagawa sur les variétés de Fano, Duke Math. J. 79 (1995), p. 101-218. | MR | Zbl

[53] V. Platonov and A. Rapinchuk - Algebraic Groups and Number Theory, Academic Press, 1994. | MR | Zbl

[54] I. Reiner - Maximal Orders, Academic Press, 1975. | MR | Zbl

[55] P. Salberger - Manuscript on universal torsors, 1993.

[56] S. Schanuel - Heights in number fields, Bull. Soc. Math. France 107 (1979), p. 433-449. | DOI | EuDML | Numdam | MR | Zbl

[57] J.-P. Serre - Quelques applications du théorème du densité de Chebotarev, Inst. Hautes Études Sci. Publ. Math. 54, p. 123-201. | DOI | EuDML | Numdam | Zbl

[58] J.-P. Serre, Classification des variétés analytiques p-adiques compactes, vol. 3, 1965. | MR | Zbl

[59] J.-P. Serre, Lie algebras and Lie groups, Benjamin, New York, 1965. | MR | Zbl

[60] J.-P. Serre, Divisibilité de certaines fonctions arithmétiques, l'Ens. Math. 22 (1976), p. 227-260. | MR | Zbl

[61] J.-P. Serre, Résumé de Cours au Collège de France, 1980.

[62] J.-P. Serre, Lectures on the Mordell-Weil theorem, Viehweg-Verlag, 1989. | MR | Zbl

[63] J. H. Silverman - The Theory of Height Functions, Arithmetic Geometry (G. Cornell and J. Silverman, eds.), Springer-Verlag, Berlin, 1986, p. 151-166. | DOI | MR | Zbl

[64] T. A. Springer - Galois cohomology of linear groups, Algebraic Groups and Discontinuous Subgroups (A. Borel and G. Mostow, eds.), Proc. of Symposia in Pure Math., vol. 9, Amer. Math. Society, 1966, p. 149-158. | DOI | MR | Zbl

[65] J. Tate - Global class field theory, Algebraic number theory (Cassels and Fröhlich, eds.), Academic Press, 1967. | Zbl

[66] A. Weil - Basic number theory, Springer-Verlag, Berlin, 1967. | MR | Zbl

[67] A. Weil, Adeles and algebraic groups, Birkhäuser, Boston, 1982. | MR | Zbl

[68] D. B. Zagier - Zeta Funktionen und quadratische Körper, Springer-Verlag, Berlin, 1991. | MR | Zbl