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.