Hostname: page-component-cd9895bd7-7cvxr Total loading time: 0 Render date: 2024-12-26T06:37:49.178Z Has data issue: false hasContentIssue false
  • English
  • Français

p-Torsion points on elliptic curves defined over quadratic fields

Published online by Cambridge University Press:  22 January 2016

Fumiyuki Momose
Fumiyuki Momose*
Affiliation:
Department of Mathematics, Faculty of Science, University of Tokyo, Hongo, Tokyo 113, Japan
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Let p be a prime number and k an algebraic number field of finite degree d. Manin [14] showed that there exists an integer n = n(k,p) (≧0) which satisfies the condition

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 96 , December 1984 , pp. 139 - 165
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1984

References

[ 1 ] Borevich, Z.I. Shafarevich, I. R., Number theory, Academic Press, New York and London (1966).Google Scholar
[ 2 ] Deligne, P. Rapoport, M., Les schémas de modules de courbes elliptiques, Proceedings of the International Summer School on Modular functions of one variable, vol. II, Lecture Notes in Math., 349, Springer-Verlag, Berlin-Heiderberg-New York (1973).Google Scholar
[ 3 ] Gross, B.H., Arithmetic on elliptic curves with complex multiplication, Lecture Notes in Math., 776, Springer-Verlag, Berlin-Heiderberg-New York (1980).Google Scholar
[ 4 ] Grothendieck, A., Fondements de la géométrie algébrique, Sém. Bourbaki, 1957-1962.Google Scholar
[ 5 ] Hartshorne, R., Algebraic Geometry, Springer-Verlag, New York (1977).Google Scholar
[ 6 ] Kenku, M.A., Certain torsion points on elliptic curves defined over quadratic fields, J. London Math. Soc, (2) 19 (1979), 233–240.Google Scholar
[ 7 ] Kenku, M.A., The modular curve X0(169) and rational isogeny, J. London Math. Soc, (2) 22 (1980), 239–244.Google Scholar
[ 8 ] Kenku, M.A., On the modular curves X0(125), X1(25) and X1(49), J. London Math. Soc, (2) 23 (1981), 415–427.Google Scholar
[ 9 ] Kenku, M.A., Rational torsion points on elliptic curves defined over quadratic fields, to appear.Google Scholar
[10] Kubert, D., Universal bounds on the torsion of elliptic curves, Proc. London Math. Soc, (3) 33 (1976), 193–237.Google Scholar
[11] Kubert, D. Lang, S., Units in the modular function fields I, II, III, Math. Ann., 218 (1975), 67-96, 175-189, 273–285.Google Scholar
[12] Lang, S., Elliptic Functions, Addison-Wesley, Reading Math. (1973).Google Scholar
[13] Lang, S., Cyclotomic Fields, Addison-Wesley, Reading Math. (1973).Google Scholar
[14] Manin, Yu. I., The p-torsion of elliptic curves is uniformly bounded, Math. USSRIzv., 3 (1969), 433–438.Google Scholar
[15] Manin, Yu. I., Parabolic points and zeta functions of modular curves, Math. USSR-Izv., 6 (1972), 19–64.Google Scholar
[16] Mazur, B., Rational points on modular curves, Proceedings of Conference on Modular Functions held in Bonn, Lecture Notes in Math., 601, Springer-Verlag, Berlin-Heiderberg-New York (1977).Google Scholar
[17] Mazur, B., Modular curves and the Eisenstein ideal, I.H. E.S. Pubi. Math., 47 (1977), 33–186.Google Scholar
[18] Mazur, B., Rational isogenies of prime degree, Invent. Math., 44 (1978), 129–162.Google Scholar
[19] Mazur, B. Swinnerton-Dyer, P., Arithmetic of Weil curves, Invent. Math., 25 (1974), 1–61.CrossRefGoogle Scholar
[20] Mazur, B. Tate, J., Points of order 13 on elliptic curves, Invent. Math., 22 (1973), 41–49.Google Scholar
[21] Mestre, J.F., Points rationnels de la courbe modulaire X0(169), Ann. Inst. Fourier, 30, 2 (1980), 17–27.Google Scholar
[22] Ogg, A., Rational points on certain elliptic modular curves, Proc. Symposia in Pure Math. XXIV, AMS (1973), 221–231.Google Scholar
[23] Ogg, A., Hyperelliptic modular curves, Bull. Soc. Math. France, 102 (1974), 449–462.Google Scholar
[24] Ogg, A., Diophantine equation and modular forms, Bull. Amer. Math. Soc, 81 (1975), 14–27.Google Scholar
[25] F. Oort Tate, J., Group schemes of prime order, Ann. Sci. Ecole Norm. Sup. série 4, 3 (1970), 1–21.Google Scholar
[26] Raynaud, M., Schémas en groupes de type (p,…,p), Bull. Soc. Math. France, 102 (1974), 241–280.Google Scholar
[27] Ribet, K.A., Endomorphisms of semi-stable abelian varieties over number fields, Ann. of Math., 101 (1975), 555–562.Google Scholar
[28] Serre, J.P., p-torsion des courbes elliptiques (d’àpres Y. Manin), Sém. Bourbaki 1969/70 pp. 281-294, Lecture Notes in Math., 180, Springer-Verlag, BerlinHeiderberg-New York (1971).Google Scholar
[29] Serre, J.P., Corps Locaux, Publ. Inst, de Math, de Univ. de Nancago Hermann, Paris (1968).Google Scholar
[30] Shimura, G., On elliptic curves with complex multiplication as factors of jacobians of modular function fields, Nagoya Math. J., 43 (1971), 199–208.Google Scholar
[31] Shimura, G., Introduction to the Arithmetic theory of Automorphic Functions, Publ. Math. Soc. Japan 11, Iwanami Shoten, Tokyo-Princeton Univ. Press, Princeton, N.J. Google Scholar
[32] Wada, H., A table of Hecke operators II, Proc. Japan Acad., 49 (1973) 380–384.Google Scholar
[33] Weil, A., Adèles and Algebraic Groups, Lecture Notes, Inst, for Advanced Study, Princeton, N.J. Google Scholar
[34] Éléments de Géométrie Algébrique III (par A. Grothendieck), I.H.E.S. Publ. Math., 17 (1963).Google Scholar
[35] Modular Functions of One Variable IV (Ed. By B. J. Birch and W. Kuyk), Lecture Notes in Math., 476, Springer-Verlag, Berlin-Heiderberg-New York (1975).Google Scholar