Points rationnels sur les quotients d’Atkin-Lehner de courbes de Shimura de discriminant pq  [ Rational points on Atkin-Lehner quotients of Shimura curves of discriminant pq ]
Annales de l'Institut Fourier, Volume 63 (2013) no. 4, p. 1613-1649

Let p and q be two distinct prime numbers, and X pq /w q be the quotient of the Shimura curve of discriminant pq by the Atkin-Lehner involution w q . We describe a way to verify in wide generality a criterion of Parent and Yafaev to prove that if p and q satisfy some explicite congruence conditions, known as the conditions of the non ramified case of Ogg, and if p is large enough compared to q, then the quotient X pq /w q has no rational point, except possibly special points.

Soient p et q deux nombres premiers distincts et X pq /w q le quotient de la courbe de Shimura de discriminant pq par l’involution d’Atkin-Lehner w q . Nous décrivons un moyen permettant de vérifier un critère de Parent et Yafaev en grande généralité pour prouver que si p et q satisfont des conditions de congruence explicites, connues comme les conditions du cas non ramifié de Ogg, et si p est assez grand par rapport à q, alors le quotient X pq /w q n’a pas de point rationnel non spécial.

Received : 2011-01-31
Revised : 2012-02-26
Accepted : 2012-05-22
DOI : https://doi.org/10.5802/aif.2810
Classification:  10X99,  14A12,  11L05
Keywords: Shimura curves, rational points, Gross vectors, Atkin-Lehner involutions 
@article{AIF_2013__63_4_1613_0,
     author = {Gillibert, Florence},
     title = {Points rationnels sur les quotients d'Atkin-Lehner de courbes de Shimura de discriminant $pq$},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {63},
     number = {4},
     year = {2013},
     pages = {1613-1649},
     doi = {10.5802/aif.2810},
     zbl = {06359596},
     mrnumber = {3137362},
     language = {fr},
     url = {aif.centre-mersenne.org/item/AIF_2013__63_4_1613_0}
}

Gillibert, Florence. Points rationnels sur les quotients d’Atkin-Lehner de courbes de Shimura de discriminant $pq$. Annales de l'Institut Fourier, Volume 63 (2013) no. 4, pp. 1613-1649. doi : 10.5802/aif.2810. https://aif.centre-mersenne.org/item/AIF_2013__63_4_1613_0/

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