Failure of the Hasse Principle for Atkin–Lehner Quotients of Shimura Curves over Q

Abstract

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 X23∙107 attached to the indefinite rational quaternion algebra of discriminant 23∙107 by the Atkin–Lehner involution ω107. The quadratic twist of X23∙107 by Q( √ −23) with respect to this involution is also a counter-example to the Hasse principle over Q.