2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/71740We 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 attached to the indefinite rational quaternion algebra of discriminant 23*107 by the Atkin-Lehner involution w107. The quadratic twist of X by Q(sqrt{-23}) with respect to this involution is also a counter-example to the Hasse principle over Q.To appear in Moscow Math. Journal. Completely revised and improved versionNumber TheoryAlgebraic Geometry11G18 4G35text