Download PDF - Double fibres and double covers: paucity of rational pointsAll rights reserved. Copyright exceptions and limitations apply

ArticleOriginal scientific text

Title

Double fibres and double covers: paucity of rational points

Authors 1, 2, 3, 4

Affiliations

  1. C.N.R.S., URA D0752, Mathématiques, Bâtiment 425, Université de Paris-Sud, F-91405 Orsay, France
  2. Institute for Problems of Information Transmission, Russian Academy of Sciences, 19, Bolshoi Karetnyi, Moscow 101447, Russia
  3. Laboratoire de mathématiques discrètes, C.N.R.S., UPR 9016, Equipe "Arithmétique et théorie de l'information", Luminy Case 930, F-13288 Marseille Cédex 9, France
  4. Isaac Newton Institute, 20 Clarkson Road, Cambridge CB3 0EH, Great Britain

Bibliography

  1. [A] D. Abramovich, Lang maps and Harris's conjecture, Israel J. Math., to appear.
  2. [BPV] W. Barth, C. Peters and A. Van de Ven, Compact Complex Surfaces, Ergeb. Math. Grenzgeb. (3) 4, Springer, 1984.
  3. [B] A. Beauville, Surfaces algébriques complexes, Astérisque 54 (1978).
  4. [Ca.1] J. W. S. Cassels, The rational solutions of the diophantine equation Y²=X³-D, Acta Math. 82 (1950), 243-273.
  5. [Ca.2] J. W. S. Cassels, Diophantine equations with special reference to elliptic curves, J. London Math. Soc. 41 (1966), 193-291.
  6. [CT/S.1] J.-L. Colliot-Thélène et J.-J. Sansuc, La descente sur les variétés rationnelles, in: Journées de géométrie algébrique d'Angers, A. Beauville (éd.), Sijthoff and Noordhoff, Alphen aan den Rijn, 1980, 223-237.
  7. [CT/S.2] J.-L. Colliot-Thélène et J.-J. Sansuc, La descente sur les variétés rationnelles, II, Duke Math. J. 54 (1987), 375-492.
  8. [CT/Sk/SwD.1] J.-L. Colliot-Thélène, A. N. Skorobogatov and Sir Peter Swinnerton-Dyer, Hasse principle for pencils of curves of genus one whose Jacobians have rational 2-division points, in preparation.
  9. [CT/Sk/SwD.2] J.-L. Colliot-Thélène, A. N. Skorobogatov and Sir Peter Swinnerton-Dyer, Rational points and zero-cycles on fibred varieties: Schinzel's hypothesis and Salberger's device, in preparation.
  10. [CT/SwD] J.-L. Colliot-Thélène and Sir Peter Swinnerton-Dyer, Hasse principle and weak approximation for pencils of Severi-Brauer and similar varieties, J. Reine Angew. Math. 453 (1994), 49-112.
  11. [F] G. Faltings, Complements to Mordell, in: Rational Points, Seminar Bonn/Wuppertal 1983/1984, G. Faltings, G. Wüstholz et al. (eds.), Aspekte der Math. E6, Vieweg, 1984, 203-227.
  12. [K] A. W. Knapp, Elliptic Curves, Princeton University Press, 1992.
  13. [MD] M. Martin-Deschamps, La construction de Kodaira-Parshin, Astérisque 127 (1985), 256-272.
  14. [Maz.1] B. Mazur, The topology of rational points, Experiment. Math. 1 (1992), 35-45.
  15. [Maz.2] B. Mazur, Speculations about the topology of rational points: an up-date, Astérisque 228 (1995), 165-181.
  16. [R] D. E. Rohrlich, Variation of the root number in families of elliptic curves, Compositio Math. 87 (1993), 119-151.
  17. [Sh] I. R. Shafarevich et al., Algebraic surfaces, Proc. Steklov Inst. Math. 75 (1967).
  18. [S] J. H. Silverman, The Arithmetic of Elliptic Curves, Grad. Texts in Math. 106, Springer, 1986.
  19. [Sk] A. N. Skorobogatov, Descent on fibrations over the projective line, Amer. J. Math. 118 (1996), 905-923.
  20. [SwD] Sir Peter Swinnerton-Dyer, Rational points on certain intersections of two quadrics, in: Abelian Varieties, Proc. Conf. Egloffstein, W. Barth, K. Hulek and H. Lange (eds.), de Gruyter, Berlin, 1995, 273-292.
  21. [W.1] A. Weil, Arithmétique et géométrie sur les variétés algébriques, in: Actualités Sci. Indust. 206, Hermann, Paris, 1935, 3-16; reprinted in: Oeuvres Scientifiques, Vol. I, Springer, 1980, 87-100.
  22. [W.2] A. Weil, Arithmetic on algebraic varieties, Ann. of Math. 53 (1951), 412-444.
Acta Arithmetica
1997/79/2
Export to PBN, Opens in new tab
Pages:
113-135
Main language of publication
English
Received
1996-06-28
Published
1997
Exact and natural sciences