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

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

2004-06-30

arxiv.org/abs/math/0406625v2

Mathematics

Number Theory

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

Scientific paper

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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