Rational points on Atkin-Lehner quotients of Shimura curves of discriminant \(p q\)

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Abstract: Let

p

and

q

be two distinct prime numbers, and

X^{p q} / w_{q}

be the quotient of the Shimura curve of discriminant

p q

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^{p q} / w_{q}

has no rational point, except possibly special points.

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