Failure of the Hasse principle for Atkin–Lehner quotients of Shimura curves over $\mathbb Q$

We show how to construct counter-examples to the Hasse principle over the field of rational numbers on Atkin–Lehner quotients of Shimura curves and on twisted forms of Shimura curves by Atkin–Lehner involutions. A particular example is the quotient of the Shimura curve $X_{23\cdot 107}$ attached to the indefinite rational quaternion algebra of discriminant $23\cdot 107$ by the Atkin–Lehner involution $\omega_{107}$. The quadratic twist of $X_{23\cdot 107}$ by $\mathbb Q(\sqrt{-23})$ with respect to this involution is also a counter-example to the Hasse principle over $\mathbb Q$.