Skip to main content

Variétés presque rationnelles, leurs points rationnels et leurs dégénérescences

  • Chapter
  • First Online:
Arithmetic Geometry

Part of the book series: Lecture Notes in Mathematics ((LNMCIME,volume 2009))

Abstract

Voici une série de résultats classiques. Toute forme quadratique en au moins trois variables sur le corps fini\(\mathbb{F}_p\)(p premier)possède un zéro non trivial (Euler).

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Bibliographie

  1. Araujo, C., Kollár, J.: Rational curves on varieties. In: Higher Dimensional Varieties and Rational Points. Bolyai Society Mathematics Studies, vol. 12, pp. 13–92. Springer, Heidelberg (2003)

    Google Scholar 

  2. Ax, J.: The elementary theory of finite fields. Ann. Math. 88(2), 239–271 (1968)

    Article  MathSciNet  MATH  Google Scholar 

  3. Ax, J., Kochen, S.: Diophantine problems over local fields, I. Amer. J. Math. 87, 605–631 (1965)

    Article  MathSciNet  MATH  Google Scholar 

  4. Borovoi, M., Colliot-Thlène, J.-L., Skorobogatov, A.N.: The elementary obstruction and homogeneous spaces. Duke Math. J. 141, 321–364 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  5. Bosch, S., Lütkebohmert, W., Raynaud, M.: Nron Models, Ergebnisse der Math. und ihrer Grenzg. 3. Folge Band, vol. 21. Springer, Heidelberg

    Google Scholar 

  6. Campana, F., Connexit rationnelle des varits de Fano. Ann. Scient. École Norm. Sup. 25, 539–545 (1992)

    MathSciNet  MATH  Google Scholar 

  7. Chambert-Loir, A.: Points rationnels et groupes fondamentaux: applications de la cohomologie p-adique (d’après P. Berthelot, T. Ekedahl, H. Esnault, etc.), Sminaire Bourbaki 2002–2003, expos 914. Astrisque 294, 125–146 (2004)

    Google Scholar 

  8. Colliot-Thlène, J.-L.: Hilbert’s theorem 90 for K 2, with application to the Chow groups of rational surfaces. Invent. math. 71, 1–20 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  9. Colliot-Thlène, J.-L.: Un thorème de finitude pour le groupe de Chow des zro-cycles d’un groupe algbrique linaire sur un corps p-adique. Invent. math 159, 589–606 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  10. Colliot-Thlène, J.-L.: Fibres spciales des hypersurfaces de petit degr. C. R. Acad. Sc. Paris 346, 63–65 (2008)

    Article  MATH  Google Scholar 

  11. Colliot-Thlène, J.-L., Gille, P.: Remarques sur l’approximation faible sur un corps de fonctions d’une variable. In: Poonen, B., Tschinkel, Yu. (eds.) Arithmetic of Higher Dimensional Arithmetic Varieties, Progress in Mathematics, vol. 226, pp. 121–133. Birkhäuser, Boston (2003)

    Google Scholar 

  12. Colliot-Thlène, J.-L., Gille, P., Parimala, R.: Arithmetic of linear algebraic groups over two-dimensional fields. Duke Math. J. 121, 285–341 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  13. Colliot-Thlène, J.-L., Kunyavskiĭ, B.: Groupe de Picard et groupe de Brauer des compactifications lisses d’espaces homogènes. J. Algebraic Geom. 15, 733–752 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  14. Colliot-Thlène, J.-L., Madore, D.: Surfaces de Del Pezzo sans point rationnel sur un corps de dimension cohomologique un. Journal de l’Institut Mathmatique de Jussieu 3, 1–16 (2004)

    Article  MATH  Google Scholar 

  15. Colliot-Thlène, J.-L., Saito, S.: Zro-cycles sur les varits p-adiques et groupe de Brauer. IMRN 4, 151–160 (1996)

    Article  Google Scholar 

  16. Colliot-Thlène, J.-L., Sansuc, J.-J.: La R-quivalence sur les tores. Ann. Scient. École Norm. Sup. 10, 175–229 (1977)

    MATH  Google Scholar 

  17. Colliot-Thlène, J.-L., Sansuc, J.-J., Sir Peter Swinnerton-Dyer: Intersections of two quadrics and Châtelet surfaces. I, J. für die reine und angew. Math. (Crelle) 373, 37–107 (1987); II, 374, 72–168 (1987)

    Google Scholar 

  18. Colliot-Thlène, J.-L., Skorobogatov, A.N.: R-equivalence on conic bundles of degree 4. Duke Math. J. 54, 671–677 (1987)

    Article  MathSciNet  Google Scholar 

  19. Colliot-Thlène, J.-L., Skorobogatov, A.N.: Groupes de Chow des zro-cycles des fibrs en quadriques. K-Theor. 7, 477–500 (1993)

    Article  MATH  Google Scholar 

  20. Coray, D.F., Tsfasman, M.A.: Arithmetic on singular Del Pezzo surfaces. Proc. Lond. Math. Soc. (3) 57(1), 25–87 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  21. Debarre, O.: Higher-dimensional algebraic geometry, Universitext. Springer, Heidelberg (2001)

    MATH  Google Scholar 

  22. Denef, J., Jarden, M., Lewis, D.J.: On Ax-fields which are C i . Quart. J. Math. Oxford Ser. (2) 34(133), 21–36 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  23. Ducros, A.: Dimension cohomologique et points rationnels sur les courbes. J. Algebra 203, 349–354 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  24. Ducros, A.: Points rationnels sur la fibre spciale d’un schma au-dessus d’un anneau de valuation. Math. Z. 238, 177–185 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  25. Esnault, H.: Varieties over a finite field with trivial Chow group of 0-cycles have a rational point. Invent. math. 151(1), 187–191 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  26. Esnault, H.: Deligne’s integrality theorem in unequal characteristic and rational points over finite fields (with an appendix by H. Esnault and P. Deligne). Ann. Math. 164, 715–730 (2006)

    Article  MathSciNet  Google Scholar 

  27. Esnault, H.: Coniveau over p-adic fields and points over finite fields. C. R. Acad. Sc. Paris Sr. I 345, 73–76 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  28. Esnault, H., Xu, C. Congruence for rational points over finite fields and coniveau over local fields. Trans. Amer. Math. Soc. 361(5), 2679–2688 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  29. Euler, L.: Demonstratio theorematis Fermatiani omnem numerum sive integrum sive fractum esse summam quatuor pauciorumve quadratorum. N. Comm. Ac. Petrop. 5 (1754/5), 1760, pp.13–58. E.242, Opera omnia. vol. I.2, pp.338–372. Birkhäuser, Boston (2003)

    Google Scholar 

  30. Fakhruddin, N., Rajan, C.S.: Congruences for rational points on varieties over finite fields. Math. Ann. 333, 797–809 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  31. Gille, P.: La R-quivalence sur les groupes algbriques rductifs dfinis sur un corps global. Inst. Hautes Études Sci. Publ. Math. 86, 199–235 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  32. Gille, P.: Spcialisation de la R-quivalence pour les groupes rductifs. Trans. Amer. Math. Soc. 356, 4465–4474 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  33. Graber, T., Harris, J., Mazur, B., Starr, J.: Rational connectivity and sections of families over curves. Ann. Scient. École Norm. Sup. 38(4), 671–692 (2005)

    MathSciNet  MATH  Google Scholar 

  34. T. Graber, J. Harris, et J. Starr, Families of rationally connected varieties, J. Amer. Math. Soc. 16 (2003) 57–67

    Article  MathSciNet  MATH  Google Scholar 

  35. Greenberg, M.J.: Rational points in Henselian discrete valuation rings. Inst. Hautes Études Sci. Publ. Math. 31, 59–64 (1966)

    Article  Google Scholar 

  36. Grothendieck, A.: Le groupe de Brauer I, II, III. In: Dix exposs sur la cohomologie des schmas. North-Holland, Amsterdam (1968)

    Google Scholar 

  37. Hassett, B., Tschinkel, Yu.: Weak approximation over function fields. Invent. math. 163(1), 171–190 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  38. Hassett, B., Tschinkel, Yu.: Approximation at places of bad reduction for rationally connected varieties. Pure and Applied Math Quarterly, Bogomolov Festschrift 4(3), 1–24 (2008)

    MathSciNet  Google Scholar 

  39. Hassett, B., Tschinkel, Yu.: Weak approximation for hypersurfaces of low degree. Algebraic geometry – Seattle 2005. Part 2, 937–955. Proc. Sympos. Pure Math. 80, Part 2, Amer. Math. Soc. Providence, RI (2009)

    MathSciNet  Google Scholar 

  40. Hogadi, A., Xu, C.: Degenerations of rationally connected varieties. Trans. Amer. Math. Soc. 361(7), 3931–3949 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  41. de Jong, A.J.: The period-index problem for the Brauer group of an algebraic surface. Duke Math. J. 123, 71–94 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  42. de Jong, A.J., Starr, J.: Every rationally connected variety over the function field of a curve has a rational point. Amer. J. Math. 125, 567–580 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  43. de Jong, A.J., Starr, J.: Low degree complete intersections are rationally simply connected (prpublication)

    Google Scholar 

  44. Kahn, B.: Zeta functions and motives. Pure Appl. Math. Quart. 5(1), à paraître

    Google Scholar 

  45. Kato, K., Kuzumaki, T.: The dimension of fields and algebraic K-theory. J. Number Theor. 24, 229–244 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  46. Kollár, J.: Rational curves on algebraic varieties, Ergebnisse der Mathematik und ihrer Grenzgebiete, 3. Folge, Band, vol. 32. Springer, Heidelberg (1996, rdition avec corrections 1999)

    Google Scholar 

  47. Kollár, J.: Rationally connected varieties over local fields. Ann. Math. (2) 150(1), 357–367 (1999)

    Article  MATH  Google Scholar 

  48. Kollár, J.: Specialization of zero cycles. Publ. Res. Inst. Math. Sci. 40(3), 689–708 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  49. Kollár, J.: A conjecture of Ax and degenerations of Fano varieties. Israel J. Math. 162, 235–252 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  50. Kollár, J.: Looking for rational curves on cubic hypersurfaces. In: Kaledin, D., Tschinkel, Y. (eds.) Higher-Dimensional Geometry over Finite Fields, vol. 16 NATO Science for Peace and Security Series: Information and Communication Security, (2008). Voir aussi: U. Derenthal and J. Kollár, Looking for rational curves on cubic hypersurfaces, matharXivv:0710.5516

    Google Scholar 

  51. Kollár, J., Miyaoka, Y., Mori, S.: Rational connectedness and boundedness of Fano manifolds. J. Diff. Geom. 36(3), 765–779 (1992)

    MATH  Google Scholar 

  52. Kollár, J., Szabó, E.: Rationally connected varieties over finite fields. Duke Math. J. 120(2), 251–267 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  53. Lafon, G.: Une surface d’Enriques sans point sur \(\mathbb{C}((t))\). C.R. Math. Acad. Sci. Paris 138(1), 51–54 (2004)

    Article  MathSciNet  Google Scholar 

  54. Lang, S.: On quasi algebraic closure. Ann. Math. 55(2), 373–390 (1952)

    Article  MATH  Google Scholar 

  55. Madore, D.: Sur la spcialisation de la R-quivalence. In: Hypersurfaces cubiques, R-quivalence et approximation faible, thèse de doctorat. Universit Paris-Sud (2005)

    Google Scholar 

  56. Madore, D.: Équivalence rationnelle sur les hypersurfaces cubiques sur les corps p-adiques. Manuscripta Math. 110, 171–185 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  57. Madore, D.: Approximation faible aux places de bonne rduction sur les surfaces cubiques sur les corps de fonctions. Bull. Soc. Math. France 134(4), 475–485 (2006)

    MathSciNet  MATH  Google Scholar 

  58. Madore, D.: Équivalence rationnelle sur les hypersurfaces cubiques de mauvaise rduction. J. Number Theor. 128, 926–944 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  59. Manin, Yu.I.: Cubic forms. Algebra, geometry, arithmetic. Translated from the Russian by M. Hazewinkel, 2nd edn. North-Holland Mathematical Library, 4. North-Holland, Amsterdam (1986)

    MATH  Google Scholar 

  60. Merkur’ev, A.S.: R-equivalence on three-dimensional tori and zero-cycles. Algebra Number Theor. 2, 69–89 (2008)

    Article  Google Scholar 

  61. Parimala, R. Sujatha, R.: Hasse principle for Witt groups of function fields with special reference to elliptic curves. With an appendix by Colliot-Thlène, J.-L. Duke Math. J. 85(3), 555–582 (1996)

    MathSciNet  MATH  Google Scholar 

  62. Parimala, R., Suresh, V.: Zero-cycles on quadric fibrations: Finiteness theorems and the cycle map. Invent. math. 122, 83–117 (1995)

    Article  MathSciNet  Google Scholar 

  63. Pfister, A.: Quadratic forms with applications to algebraic geometry and topology, London Mathematical Society Lecture Note Series, vol. 217. Cambridge University Press, Cambridge (1995)

    Book  Google Scholar 

  64. Saito, S., Sato, K.: A finiteness theorem for zero-cycles over p-adic fields. à paraître dans Ann. Math

    Google Scholar 

  65. Serre, J.-P.:Cohomologie galoisienne, Cinquième dition, rvise et complte. Springer Lecture Notes in Mathematics, vol. 5 (1994)

    Google Scholar 

  66. Starr, J.: Degenerations of rationally connected varieties and PAC fields, arXiv preprint

    Google Scholar 

  67. Starr, J.M.: Arithmetic over function fields. Arithmetic geometry, 375–418, Clay Math. Proc. 8, Amer. Math. Soc., Providence, RI (2009)

    MathSciNet  Google Scholar 

  68. Tao, D.: A variety associated to an algebra with involution. J. Algebra 168(2), 479–520 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  69. Wittenberg, O.: La connexit rationnelle en arithmtique, notes pour un mini-cours, Session SMF États de la Recherche≪Varits rationnellement connexes≫, Strasbourg, mai (2008)

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Jean-Louis Colliot-Thélène .

Editor information

Editors and Affiliations

Rights and permissions

Reprints and permissions

Copyright information

© 2011 Springer-Verlag Berlin Heidelberg

About this chapter

Cite this chapter

Colliot-Thélène, JL. (2011). Variétés presque rationnelles, leurs points rationnels et leurs dégénérescences. In: Corvaja, P., Gasbarri, C. (eds) Arithmetic Geometry. Lecture Notes in Mathematics(), vol 2009. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-15945-9_1

Download citation

Publish with us

Policies and ethics