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

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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.
To appear in Moscow Math. Journal. Completely revised and improved version

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Number Theory, Algebraic Geometry, 11G18 4G35

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http://salesiana.dossiersoluciones.com/handle/123456789/71740

Consulte el texto completo en el siguiente enlace:

https://arxiv.org/abs/math/0406625
 http://arxiv.org/abs/math/0406625

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arXiv