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

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 .

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 .

@article{AIF_1980__30_2_17_0,
     author = {Mestre, Jean-Fran\c{c}ois},
     title = {Points rationnels de la courbe modulaire $X_0(169)$},
     journal = {Annales de l'Institut Fourier},
     pages = {17--27},
     publisher = {Institut Fourier},
     address = {Grenoble},
     volume = {30},
     number = {2},
     year = {1980},
     doi = {10.5802/aif.782},
     mrnumber = {81h:10036},
     zbl = {0432.14017},
     language = {fr},
     url = {http://www.numdam.org/articles/10.5802/aif.782/}
}
TY  - JOUR
AU  - Mestre, Jean-François
TI  - Points rationnels de la courbe modulaire $X_0(169)$
JO  - Annales de l'Institut Fourier
PY  - 1980
SP  - 17
EP  - 27
VL  - 30
IS  - 2
PB  - Institut Fourier
PP  - Grenoble
UR  - http://www.numdam.org/articles/10.5802/aif.782/
DO  - 10.5802/aif.782
LA  - fr
ID  - AIF_1980__30_2_17_0
ER  - 
%0 Journal Article
%A Mestre, Jean-François
%T Points rationnels de la courbe modulaire $X_0(169)$
%J Annales de l'Institut Fourier
%D 1980
%P 17-27
%V 30
%N 2
%I Institut Fourier
%C Grenoble
%U http://www.numdam.org/articles/10.5802/aif.782/
%R 10.5802/aif.782
%G fr
%F AIF_1980__30_2_17_0
Mestre, Jean-François. Points rationnels de la courbe modulaire $X_0(169)$. Annales de l'Institut Fourier, Volume 30 (1980) no. 2, pp. 17-27. doi : 10.5802/aif.782. http://www.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

Cited by Sources: