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Some New Hasse Principles For Conic Bundle Surfaces

  • Per Salberger
Part of the Progress in Mathematics book series (PM, volume 81)

Abstract

Let k be a number field and let X be a smooth projective geometrically integral variety defined over k. If K is an overfield of k, denote by X(K) the set of K—points on X.

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Bibliography

  1. [Co]
    J-L. Colliot-Thélène — Quelques propriétés arithmttiques des surfaces rationnelles (d’aprés Manin), Séminaire de Théorie des Nombres, Bordeaux 1972–72, Exp. 13, Lab. Theorie des Nombres, C.N.R.S., Talence 1972. Google Scholar
  2. [CC]
    J.-L. Colliot-Thélenè, D. Coray.— L’équivalence rationnelle sur les points fermés des surfaces rationnelles fibrées en coniques, Compositio Math. 39 (1979), 301–332. Google Scholar
  3. [CCS]
    J.-L. Colliot-Thélène, D. Coray, J.-J. Sansuc.- Descente et principe de Hasse pour certaines variétés rationnelles, J. reine angew. Math., 320 (1980), 150–191. Google Scholar
  4. [CKS]
    J.-L. Colliot-Thelene, D. Kanevsky, J.-J. Sansuc.— Arithmétique des surfaces cubiques diagonales, in Diophantine Approximation and Transcendence Theory, G. Wüstholz ed., Springer Lecture Notes in Mathematics 1290 (1987), 1–108. Google Scholar
  5. [CSi]
    J.-L. Colliot-Thelene, J-J. Sansuc.— On the Chow group of certain rational surfaces: a sequel to a paper of S. Bloch, Duke Math. J. 48 (1981), 421–447. Google Scholar
  6. [CS2]
    J.-L. Colliot-Thélène, J.-J. Sansuc — La descente sur les surfaces rationnelles fibr&es en coniques, C.R. Acad. Sci. Paris 303, Serie I 1986, 303–306. Google Scholar
  7. [CS3]
    J.-L. Colliot-Thélène, J.-J. Sansuc.— La descente sur les varieties rationnelles, II, Duke Math. J. 54 (1987), 375–492. Google Scholar
  8. [CSS]
    J.-L. Colliot-Thelene, J-J. Sansuc, Sir Peter Swinnerton-Dyer - Intersections of two quadrics and Ch&telet surfaces, J. reine angew. Math. 373 (1987), 37–107 et 374 (1987), 72–168. Google Scholar
  9. [Is]
    V.A. Iskovskih.— Minimal models of rational surfaces over arbitrary fields, Izv. Ak. Nauk. SSSR Ser. Mat. 43 (1979), 19–43 (engl. transl.: Math. USSR-Izv. 14 (1980), 17–39). Google Scholar
  10. [La]
    S. Lang - Algebraic number theory, Addison-Wesley, Reading 1970. Google Scholar
  11. [Ma1]
    Yu.I. Manin — Rational surfaces over perfect fields (Russian), Inst.des Hautes Etudes Sci., Publ. Math. 30 (1966), 55–113 (engl. transl.: Translations AMS (2) 84 (1969) 137–186Google Scholar
  12. [Ma2]
    Yu.I. Manin.— Le groupe de Brauer-Grothendieck en géomttrie diophantienne, in Actes du congrès intern, math, Nice 1,Google Scholar
  13. [MT]
    Yu.I. Manin, M.A. Tsfasman— Rational varieties: Algebra, geometry and arithmetic, Uspekhi Mat. Nauk 41 (1986), 43–94 (engl. transl.: Russian Math. Surveys 41 (1986), 51–116). Google Scholar
  14. [Sai]
    S. Saito.— Some observations on motivic cohomologies of arithmetical schemes, preprint.Google Scholar
  15. [Sa1]
    P. Salberger— K-theory of orders and their Brauer-Severi schemes, Thesis, Department of Mathematics, University of Goteborg 1985. Google Scholar
  16. [Sa2]
    P. Salberger — Sur l’arithmétique de certaines surfaces de del Pezzo, C.R. Acad. Sci. Paris 303, série I (1986), 273–276. Google Scholar
  17. [Sa3]
    P. Salberger.— On the arithmetic of conic bundle surfaces, in Séminaire de Théorie des Nombres Paris 1985–86, Progr. Math. 71, Birkhaüser, Basel Boston 1987, 175–197 (cf. also the Errata in this volume). Google Scholar
  18. [Sa4]
    P. Salberger — Zero-cycles on rational surfaces over number fields, Invent. Math. 91 (1988), 505–524. MathSciNetMATHCrossRefGoogle Scholar
  19. [San1]
    J J.-J. Sansuc.— Descente et principe de Hasse pour certaines varieties rationnelles, in Séminaire de Théorie des Nombres, Paris 1980–81, Progr. Math. 22, Birkhäuser, Basel Boston 1982, 253–271. Google Scholar
  20. [San2]
    J.-J. Sansuc.— A propos d’une conjecture arithmttique sur le groupe de Chow d’une surface rationnelle, Séminaire de Théorie des Nombres, Bordeaux 1981–81, Exp. 33 Lab. Theorie des Nombres, C.N.R.S., Talence 1972. Google Scholar
  21. [Sc]
    W.M. Schmidt. — The density of integer points on homogeneous varieties, Acta Math. 154 (1985), 243–296. MathSciNetMATHCrossRefGoogle Scholar
  22. [Se]
    J-P. Serre.— Corps locaux, deuxiéme éd, Herman, Paris 1968Google Scholar
  23. [Si]
    J. Silverman - The Arithmetic of Elliptic Curves, Graduate Texts in Mathematics, 106, Springer-Verlag, Berlin Heidelberg New York 1986. Google Scholar
  24. [Sp]
    T.A. Springer.— Sur les formes quadratiques d’indice zero, C.R. Acad. Sci. 234, 1517–1519 (1952). Google Scholar
  25. [Sw]
    H.P.F. Swinnerton—Dyer — Rational points on del Pezzo surfaces of degree 5, in Algebraic geometry Oslo,Google Scholar

Copyright information

© Birkhäuser Boston 1990

Authors and Affiliations

  • Per Salberger
    • 1
  1. 1.C.N.R.S. Salberger URA D0752Université de Paris—SudOrsay CedexFrance

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