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Failure of the Hasse principle for Atkin-Lehner quotients of Shimura curves over \Q

V. Rotger·A. Skorobogatov·A. Yafaev
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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 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.

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V. Rotger, A. Skorobogatov, A. Yafaev. 2005-02-22. Failure of the Hasse principle for Atkin-Lehner quotients of Shimura curves over \Q. https://arxiv.org/abs/math/0406625

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