Points rationnels de la courbe modulaire X 0 ( 169 )

Jean-François Mestre

Annales de l'institut Fourier (1980)

  • Volume: 30, Issue: 2, page 17-27
  • ISSN: 0373-0956

Abstract

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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 .

How to cite

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Mestre, Jean-François. "Points rationnels de la courbe modulaire $X_0(169)$." Annales de l'institut Fourier 30.2 (1980): 17-27. <http://eudml.org/doc/74447>.

@article{Mestre1980,
abstract = {On démontre que les seuls points rationnels sur $\{\bf Q\}$ de la courbe $X_0(169)$ sont les pointes.En conséquence, il n’existe pas de courbe elliptique définie sur $\{\bf Q\}$ possédant un sous-groupe cyclique rationnel d’ordre $13^2$.},
author = {Mestre, Jean-François},
journal = {Annales de l'institut Fourier},
keywords = {modular curve; X0(169); rational point; cusp},
language = {fre},
number = {2},
pages = {17-27},
publisher = {Association des Annales de l'Institut Fourier},
title = {Points rationnels de la courbe modulaire $X_0(169)$},
url = {http://eudml.org/doc/74447},
volume = {30},
year = {1980},
}

TY - JOUR
AU - Mestre, Jean-François
TI - Points rationnels de la courbe modulaire $X_0(169)$
JO - Annales de l'institut Fourier
PY - 1980
PB - Association des Annales de l'Institut Fourier
VL - 30
IS - 2
SP - 17
EP - 27
AB - On démontre que les seuls points rationnels sur ${\bf Q}$ de la courbe $X_0(169)$ sont les pointes.En conséquence, il n’existe pas de courbe elliptique définie sur ${\bf Q}$ possédant un sous-groupe cyclique rationnel d’ordre $13^2$.
LA - fre
KW - modular curve; X0(169); rational point; cusp
UR - http://eudml.org/doc/74447
ER -

References

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  1. [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. Zbl0385.14007
  2. [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. Zbl0281.14010
  3. [3] R. FRICKE, Die elliptischen Funktionen und ihre Anwendungen, II, Leipzig-Berlin, Teubner, 1922. JFM48.0432.01
  4. [4] M. A. KENKU, The modular curve X0(39) and rational isogeny, Math. Proc. Cambridge Philo. Soc., 85, (1979), 21-23. Zbl0392.14011MR80g:14023
  5. [5] Y. MANIN, Parabolic points and zeta functions of modular forms (Russian), Isv. Acad. Nauk., (1972), 19-66. Zbl0243.14008
  6. [6] B. MAZUR, Rational isogenies of prime degree, Inventiones Mathematicae, 44 (1978), 129-163. Zbl0386.14009MR80h:14022
  7. [7] A. OGG, Rational points on certain elliptic modular curves, Proc. Symp. Pure Math., A.M.S., Providence, 24 (1973), 221-231. Zbl0273.14008MR49 #2743
  8. [8] F. OORT, J. TATE, Group schemes of prime order, Ann. Scient. Ec. Norm. Sup., série 4,3 (1970), 1-21. Zbl0195.50801MR42 #278

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