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

Details Points rationnels sur les quotients d'Atkin-Lehner de courbes de Shimura de discriminant $pq$ Points rationnels sur les quotients d'Atkin-Lehner de courbes de Shimura de discriminant $pq$

2010-12-15

arxiv.org/abs/1012.3414v1

Mathematics

Number Theory

32 pages, in french

Scientific paper

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.

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