This article is cited in 16 papers

Failure of the Hasse principle for Atkin–Lehner quotients of Shimura curves over $\mathbb Q$

V. Rotger^a, A. D. Skorobogatov^b, A. Yafaev<sup>c

a</sup> Escola Politècnica Superior d'Enginyeria de Vilanova i la Geltrú
^b Imperial College, Department of Mathematics
^c Department of Mathematics, University College London

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 $X_{23\cdot 107}$ attached to the indefinite rational quaternion algebra of discriminant $23\cdot 107$ by the Atkin–Lehner involution $\omega_{107}$. The quadratic twist of $X_{23\cdot 107}$ by $\mathbb Q(\sqrt{-23})$ with respect to this involution is also a counter-example to the Hasse principle over $\mathbb Q$.

Key words and phrases: Shimura curves, rational points, Hasse principle, descent.

MSC: 11G18, 14G35

Received: July 9, 2004

Language: English

DOI: 10.17323/1609-4514-2005-5-2-463-476

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