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

TL;DR

This paper investigates the rational points on quotients of Shimura curves of discriminant pq by Atkin-Lehner involutions, providing a method to verify conditions under which these quotients lack rational points, extending previous criteria.

Contribution

It introduces a general verification method for Parent and Yafaev's criterion, applying explicit congruence conditions to determine the absence of rational points on certain Shimura curve quotients.

Findings

Under specific congruence conditions, the quotient has no rational points.

The method applies when p is sufficiently large relative to q.

It extends known non-rational point results to a broader class of Shimura curves.

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.