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, pp. 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.

DOI: 10.5802/aif.2810
Classification: 10X99,  14A12,  11L05
Keywords: Shimura curves, rational points, Gross vectors, Atkin-Lehner involutions
Gillibert, Florence 1

1 IMB Bordeaux I 351, cours de la Libération 33405 Talence (France)
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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. http://archive.numdam.org/articles/10.5802/aif.2810/

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