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Torsion points on elliptic curves defined over quadratic fields

  • M. A. Kenku (a1) and F. Momose (a2)
Abstract

Let k be a quadratic field and E an elliptic curve defined over k. The authors [8, 12, 13] [23] discussed the k-rational points on E of prime power order. For a prime number p, let n = n(k, p) be the least non negative integer such that

for all elliptic curves E defined over a quadratic field k ([15]).

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[1] A. O. L Atkin and J. Lehner , Hecke operators on A(m), Math. Ann., 185 (1972), 134160.

[1] J. Coates and A. Wiles , On the conjecture of Birch and Swinnerton Dyer, Invent. Math., 39 (1977), 223251.

[1] P. Deligne and M. Rapoport , Les schémas de modules de courbes elliptique, Proceedings of the International Summer School on Modular functions of one variable, vol. II, Lecture Notes in Math., 349, Springer-Verlag, Berlin-Heidelberg-New York (1973).

[1] M. A. Kenku , The modular curve X 0(39) and rational isogeny, Math. Proc. Cambridge Philos. Soc, 85 (1979), 2123.

[10] M. A. Kenku , The modular curves X0(65) and X 0(91) and rational isogeny, Math. Proc. Cambridge Philos. Soc, 87 (1980), 1520.

[15] Yu. I. Manin , The p-torsion of elliptic curves is uniformally bounded, Math. USSR-Izvestija, 3 (1969), 433438.

[16] Yu. I. Manin , Parabolic points and zeta-functions of modular curves, Math. USSR-Izvestija, 6 (1972), 1964.

[17] B. Mazur , 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).

[18] B. Mazur , Rational isogenies of prime degree, Invent. Math., 44 (1978), 129162.

[19] B. Mazur and J. Tate , Points of order 13 on elliptic curves, Invent. Math., 22 (1973), 4149.

[20] J. F. Mestre , Points rationnels de la courbe modulaire X0(169), Ann. Inst. Fourier, 30, 2 (1980), 1727.

[21] J. S. Milne , On the arithmetic of abelian varieties, Invent. Math., 17 (1972), 177190.

[26] A. Ogg , Diophantine equations and modular forms, Bull. AMS, 81 (1975), 1427.

[29] K. Rubin , Congruences for special values of L-functions of elliptic curves with complex multiplication, Invent. Math., 71 (1983), 339364.

[30] J. P. Serre , Propriétés galoissiennes des points d’ordre fini des courbes elliptiques, Invent. Math., 15 (1972), 259331.

[33] G. Shimura , On elliptic curves with complex multiplication as factors of the jacobians of modular function fields, Nagoya Math. J., 43 (1971), 199208.

[35] J. Yu , A cuspidal class number formula for the modular curves Xi(N), Math. Ann., 252 (1980), 197216.

[36]Modular functions of one variable IV (ed. by B. J. Birch and W. Kuyk ), Lecture Notes in Math., 476, Springer-Verlag, Berlin-Heidelberg-New York (1975).

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Nagoya Mathematical Journal
  • ISSN: 0027-7630
  • EISSN: 2152-6842
  • URL: /core/journals/nagoya-mathematical-journal
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