Advertisement

Rational points on modular curves

  • B. Mazur
Conference paper
Part of the Lecture Notes in Mathematics book series (LNM, volume 601)

Keywords

Elliptic Curve Rational Point Elliptic Curf Abelian Variety Group Scheme 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Bibliography

  1. 1.
    Berkovic, V.: On rational points on the jacobians of modular curves [in Russian]. To appear.Google Scholar
  2. 2.
    Brylinski, J.-L.: Torsion des courbes elliptiques (d'après Demjanenko). D.E.A. de Mathématique Pure presented at the Faculté des Sciences de Paris-Sud (1973).Google Scholar
  3. 3.
    Cassels, J. W. S.: Diophantine equations with special reference to elliptic curves. J. London Math. Soc. 41 (193–291) (1966).MathSciNetCrossRefGoogle Scholar
  4. 4.
    Deligne, P., Rapoport, M.: Schémas de modules des courbes elliptiques. Vol. II of the Proceedings of the International Summer School on Modular Functions, Antwerp (1972). Lecture Notes in Mathematics 349. Berlin-Heidelberg-New York: Springer 1973.Google Scholar
  5. 5.
    Demjanenko, V. A.: Torsion of elliptic curves [in Russian], Izv. Akad. Nauk. CCCP, 35, 280–307 (1971) [MR 44, 2755].MathSciNetGoogle Scholar
  6. 6.
    Dörrie, H.: 100 great problems of elementary mathematics; their history and solution. Dover, New York 1965.zbMATHGoogle Scholar
  7. 7.
    Griffiths, P.: Variations on a theme of Abel. Inventiones Math. 35 321–390 (1976).zbMATHCrossRefGoogle Scholar
  8. 8.
    Hellegouarch, Y.: Courbes elliptiques et équation de Fermet. Thèse d'Etat. Faculté des Sciences de Besançon (1972). See also the series of notes in the Computes-Rendus de l'Académie des Sciences de Paris. 260 5989–5992, 6256–6258 (1965); 273 540–543, 1194–1196 (1971).Google Scholar
  9. 9.
    Herbrand, J.: Sur les classes des corps circulaires. Journal de Math. Pures et Appliquées. 9e série II, 417–441 (1932).Google Scholar
  10. 10.
    Kubert, D.: Universal bounds on torsion of elliptic curves. Proc. London Math. Soc. (3) 33 193–237 (1976).MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Kubert, D., Lang, S.: Units in the modular function field. I, II, III Math. Ann. 218, 67–96, 175–189, 273–285 (1975).Google Scholar
  12. 12.
    Lang, S.: Elliptic Functions. Addison Wesley, Reading 1974.Google Scholar
  13. 13.
    Manin, Y.: A uniform bound for p — torsion in elliptic curves [in Russian]. Izv. Akad. Nauk. CCCP, 33 459–465 (1969).MathSciNetzbMATHGoogle Scholar
  14. 14a.
    Mazur, B.: Modular curves and the Eisenstein Ideal. To appear: Publ. Math. I.H.E.S.Google Scholar
  15. 14b.
    Mazur, B.: p — adic analytic number theory of elliptic curves and abelian varieties over Q. Proc. of International Congress of Mathematicians at Vancouver, 1974, vol. I, 369–377, Canadian Math. Soc. (1975).Google Scholar
  16. 15.
    Mazur, B., Messing, W.: Universal extensions and one dimensional crystalline cohomology. Lecture Notes in Mathematics. 370. Berlin-Heidelberg-New York: Springer 1974.zbMATHGoogle Scholar
  17. 16.
    Mazur, B., Serre, J.-P.: Points rationnels des courbes modulaires X0(N). Séminaire Bourbaki no. 469. Lecture Notes in Mathematics. 514 Berlin-Heidelberg-New York: Springer 1976.Google Scholar
  18. 17a.
    Ogg, A.: Rational points on certain elliptic modular curves. Proc. Symp. Pure Math. 24 221–231 (1973) AMS, Providence.MathSciNetCrossRefGoogle Scholar
  19. 17b.
    Ogg, A.: Diophantine equations and modular forms. Bull. AMS 18 14–27 (1975).MathSciNetCrossRefGoogle Scholar
  20. 18.
    Oort, F., Tate, J.: Group schemes of prime order. Ann. Scient. Ec. Norm. Sup. série 4, 3, 1–21 (1970).MathSciNetzbMATHGoogle Scholar
  21. 19.
    Raynaud, M.: Schémas en groupes de type (p, ..., p). Bull. Soc. Math. France. 102 fasc. 3, 241–280 (1974).MathSciNetzbMATHGoogle Scholar
  22. 20a.
    Ribet, K.: Endomorphisms of semi-stable abelian varieties over number fields. Ann. of Math. 101 no. 3, 555–562 (1975).MathSciNetzbMATHCrossRefGoogle Scholar
  23. 20b.
    Ribet, K.: A modular construction of unramified p — extension of Qp). Inventiones Math. 34, 151–162 (1976).MathSciNetzbMATHCrossRefGoogle Scholar
  24. 21.
    Robert, G.: Nombres de Hurwitz et regularité des idéaux premiers d'un corps quadratique imaginaire. Séminaire Delange-Pisot-Poitou. Exposé given April 28, 1975.Google Scholar
  25. 22a.
    Serre, J.-P.: Propriétés galoisiennes des points d'ordre fini des courbes elliptiques. Inventiones math. 15, 259–331 (1972).zbMATHCrossRefGoogle Scholar
  26. 22b.
    Serre, J.-P.: p — torsion des courbes elliptiques (d'après Y. Manin) Séminaire Bourbaki 69/70 no. 380. Lecture Notes in Mathematics. 180. Berlin-Heidelberg-New York: Springer 1971.Google Scholar
  27. 22c.
    Serre, J.-P.: Abelian ℓ — adic representations and elliptic curves. Lectures at McGill University. New York-Amsterdam: W. A. Benjamin Inc., 1968.Google Scholar
  28. 23.
    Serre, J.-P., Tate, J.: Good reduction of abelian varieties. Ann. of Math. 88, 492–517 (1968).MathSciNetzbMATHCrossRefGoogle Scholar
  29. 24.
    Tate, J.: Algorithm for determining the Type of a Singular Fiber in an Elliptic Pencil. 33–52. Modular Functions of one variable IV. Proceedings of the International Summer School, Antwerp RUCA. Lecture Notes in Mathematics 476. Berlin-Heidelberg-New York: Springer 1975.Google Scholar
  30. 25.
    SGA 3: Schémas en groupes I. Lecture Notes in Mathematics. 151. Berlin-Heidelberg-New York: Springer 1970.Google Scholar

Copyright information

© Springer-Verlag 1977

Authors and Affiliations

  • B. Mazur
    • 1
  1. 1.Department of Mathematics Science CenterHarvard UniversityCambridge

Personalised recommendations