Rational points on the modular curves X+0(N)

Fumiyuki MOMOSE

doi:10.2969/jmsj/03920269

Rational points on the modular curves X⁺₀(N)

Fumiyuki MOMOSE

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JOURNAL FREE ACCESS

1987 Volume 39 Issue 2 Pages 269-286

DOI https://doi.org/10.2969/jmsj/03920269

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Published: 1987 Received: January 05, 1984 Available on J-STAGE: October 20, 2006 Accepted: - Advance online publication: - Revised: -
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Date of correction: October 20, 2006 Reason for correction: - Correction: CITATION Details: Wrong : 1) A. O. L. Atkin and J. Lehner, Hecke operators on Γ₀(m), Math. Ann., 185 (1970), 134-160.
2) V. G. Berkovic, The rational points on the jacobians of modular curves, Math. USSR Sbornik, 30-4 (1976), 485-500.
3) P. Deligne and M. Rapoport, Schémas de modules des courbes elliptiques, Proc. International Summer School on Modular Functions, Antwerp 1972, Vol. II, Lecture Notes in Math., 349, Springer, 1973.
4) N. Ishii and F. Momose, Hyperelliptic modular curves, to appear.
5) M. A. Kenku, The modular curve X₀(39) and rational isogeny, Math. Proc. Cambridge Philos. Soc., 85 (1979), 21-23.
6) M. A. Kenku, The modular curves X₀(65) and X₀(91) and rational isogeny, Math. Cambridge Philos. Soc., 87 (1980), 15-20.
7) M. A. Kenku, The modular curve X₀(169) and rational isogeny, J. London Math. Soc. (2), 22 (1980), 239-244.
8) M. A. Kenku, On the modular curves X₀(125), X₁(25) and X₁(49), J. London Math. Soc. (2), 23 (1981), 415-427.
9) B. Mazur, Modular curves and the Eisenstein ideals, Publ. Math. I.H.E.S., 47, 1977.
10) B. Mazur, Rational isogenies of prime degree, Invent. Math., 44 (1978), 129-162.
11) B. Mazur and H. P. F. Swinnerton-Deyer, Arithmetic of Weil curves, Invent. Math., 25 (1974), 1-61.
12) J. F. Mestre, Points rationnels de la courbe modulaire X₀(169), Ann. Inst. Fourier (Grenoble), 30-2 (1980), 17-27.
13) F. Momose, Rational points on the modular curves X_split(p), Compositio Math., 52 (1984), 115-137.
14) F. Momose, Rational points on the modular curves X⁺₀(p^r), to appear.
15) A. Ogg, Über die Automorphismengruppe von X₀(N), Math. Ann., 228 (1977), 279-292.
16) A. Ogg, Hyperelliptic modular curves, Bull. Soc. Math. France, 102 (1974), 449-462.
17) F. Oort and J. Tate, Group schemes of prime order, Ann. Sci. École Norm. Sup. (4), 3(1970), 1-21.
18) M. Raynaud, Spécialisation du foncteur de Picard, Publ. Math. I.H.E.S., 38, 1970, pp. 27-76.
19) M. Raynaud, Schémas en groupes de type (p, ..., p), Bull. Soc. Math. France, 102 (1974), 241-280.
20) K. A. Ribet, Endomorphisms of semi-stable abelian varieties over number fields, Ann. of Math., 101 (1975), 555-562.
21) J. P. Serre, Propriétés galoissiennes des points d'ordre fini des courbes elliptiques, Invent. Math., 15 (1972), 259-331.
22) B. J. Birch and W. Kuyk ed., Modular functions of one variable IV, Lecture Notes in Math., 476, Springer, 1975.

Right : [1] A. O. L. Atkin and J. Lehner, Hecke operators on Γ₀(m), Math. Ann., 185 (1970), 134-160.
[2] V. G. Berkovic, The rational points on the jacobians of modular curves, Math. USSR Sbornik, 30-4 (1976), 485-500.
[3] P. Deligne and M. Rapoport, Schémas de modules des courbes elliptiques, Proc. International Summer School on Modular Functions, Antwerp 1972, Vol. II, Lecture Notes in Math., 349, Springer, 1973.
[4] N. Ishii and F. Momose, Hyperelliptic modular curves, to appear.
[5] M. A. Kenku, The modular curve X₀(39) and rational isogeny, Math. Proc. Cambridge Philos. Soc., 85 (1979), 21-23.
[6] M. A. Kenku, The modular curves X₀(65) and X₀(91) and rational isogeny, Math. Cambridge Philos. Soc., 87 (1980), 15-20.
[7] M. A. Kenku, The modular curve X₀(169) and rational isogeny, J. London Math. Soc. (2), 22 (1980), 239-244.
[8] M. A. Kenku, On the modular curves X₀(125), X₁(25) and X₁(49), J. London Math. Soc. (2), 23 (1981), 415-427.
[9] B. Mazur, Modular curves and the Eisenstein ideals, Publ. Math. I. H. E. S., 47, 1977.
[10] B. Mazur, Rational isogenies of prime degree, Invent. Math., 44 (1978), 129-162.
[11] B. Mazur and H. P. F. Swinnerton-Deyer, Arithmetic of Weil curves, Invent. Math., 25 (1974), 1-61.
[12] J. F. Mestre, Points rationnels de la courbe modulaire X₀(169), Ann. Inst. Fourier (Grenoble), 30-2 (1980), 17-27.
[13] F. Momose, Rational points on the modular curves X_split(p), Compositio Math., 52 (1984), 115-137.
[14] F. Momose, Rational points on the modular curves X⁺₀(p^r), to appear.
[15] A. Ogg, Über die Automorphismengruppe von X₀(N), Math. Ann., 228 (1977), 279-292.
[16] A. Ogg, Hyperelliptic modular curves, Bull. Soc. Math. France, 102 (1974), 449-462.
[17] F. Oort and J. Tate, Group schemes of prime order, Ann. Sci. École Norm. Sup. (4), 3 (1970), 1-21.
[18] M. Raynaud, Spécialisation du foncteur de Picard, Publ. Math. I. H. E. S., 38, 1970, pp. 27-76.
[19] M. Raynaud, Schémas en groupes de type (p,…,p), Bull. Soc. Math. France, 102 (1974), 241-280.
[20] K. A. Ribet, Endomorphisms of semi-stable abelian varieties over number fields, Ann. of Math., 101 (1975), 555-562.
[21] J. P. Serre, Propriétés galoissiennes des points d'ordre fini des courbes elliptiques, Invent. Math., 15 (1972), 259-331.
[22] B. J. Birch and W. Kuyk ed., Modular functions of one variable IV, Lecture Notes in Math., 476, Springer, 1975.

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