no. 2
Points rationnels de la courbe modulaire X 0 (169)
Annales de l'Institut Fourier, Tome 30 (1980) no. 2, pp. 17-27.

On démontre que les seuls points rationnels sur Q de la courbe X 0 (169) sont les pointes.

En conséquence, il n’existe pas de courbe elliptique définie sur Q possédant un sous-groupe cyclique rationnel d’ordre 13 2 .

We prove that the only rational point of the curve X 0 (169) are the cusps.

Consequently, there does not exist any elliptic curve defined over Q which possesses a rational cyclic subgroup of order 13 2 .

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     title = {Points rationnels de la courbe modulaire $X_0(169)$},
     journal = {Annales de l'Institut Fourier},
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Mestre, Jean-François. Points rationnels de la courbe modulaire $X_0(169)$. Annales de l'Institut Fourier, Tome 30 (1980) no. 2, pp. 17-27. doi : 10.5802/aif.782. http://archive.numdam.org/articles/10.5802/aif.782/

[1] V. G. Berkovič, The rational points on the Jacobian of modular curves, Mat. Sbornik, 101 (143) (1976) ; traduction anglaise, Math. U.S.S.R. Sbornik, 30, 4 (1976), 478-500. | Zbl

[2] P. Deligne, M. Rapoport, 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. | Zbl

[3] R. Fricke, Die elliptischen Funktionen und ihre Anwendungen, II, Leipzig-Berlin, Teubner, 1922. | JFM

[4] M. A. Kenku, The modular curve X0(39) and rational isogeny, Math. Proc. Cambridge Philo. Soc., 85, (1979), 21-23. | MR | Zbl

[5] Y. Manin, Parabolic points and zeta functions of modular forms (Russian), Isv. Acad. Nauk., (1972), 19-66. | Zbl

[6] B. Mazur, Rational isogenies of prime degree, Inventiones Mathematicae, 44 (1978), 129-163. | MR | Zbl

[7] A. Ogg, Rational points on certain elliptic modular curves, Proc. Symp. Pure Math., A.M.S., Providence, 24 (1973), 221-231. | MR | Zbl

[8] F. Oort, J. Tate, Group schemes of prime order, Ann. Scient. Ec. Norm. Sup., série 4,3 (1970), 1-21. | Numdam | MR | Zbl

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