Points rationnels sur les quotients d’Atkin-Lehner de courbes de Shimura de discriminant pq

Abstract

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. RésuméSoient 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.

Galois Representations Over Fields of Moduli and Rational Points on Shimura Curves

“…Pairs (p, m) such that X (m) pm violates the Hasse principle over Q. As opposed to the approach taken in [PY07] and [Gil10], observe that if we consider discriminants of the form D = pm with p, m different odd primes, the conditions on the above theorem place us in Ogg's "ramifié" case (see [Ogg85]). Therefore, our results are complementary to those in [PY07,Gil10].…”

“…In [PY07], explicit conditions for rational points on these quotients to be "trivial" (arising from CM points) are given, and they also find an infinite family of such quotients satisfying them. This work has recently been taken a step further by Gillibert in [Gil10], where Parent-Yafaev conditions are made explicit.…”