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ON A SHIMURA CURVE THAT IS A COUNTEREXAMPLE TO THE HASSE PRINCIPLE

Published online by Cambridge University Press:  12 May 2003

SAMIR SIKSEK  and
ALEXEI SKOROBOGATOV
SAMIR SIKSEK
Affiliation:
Department of Mathematics, Faculty of Science, Sultan Qaboos University PO Box 36, Al-Khod 123, [email protected]
ALEXEI SKOROBOGATOV
Affiliation:
Department of Mathematics, The Huxley Building, Imperial College, 180 Queen's Gate, London, SW7 2BZ [email protected]
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Abstract

Let $X$ be the Shimura curve corresponding to the quaternion algebra over $\Q$ ramified only at 3 and 13. B. Jordan showed that $X_{\Q(\sqrt{-13})}$ is a counterexample to the Hasse principle. Using an equation of $X$ found by A. Kurihara, it is shown here, by elementary means, that $X$ has no $\Q(\sqrt{-13})$-rational divisor classes of odd degree. A corollary of this is the fact that this counterexample is explained by the Manin obstruction.

Keywords

11G1811G0511G30
Type
Research Article
Information
Bulletin of the London Mathematical Society , Volume 35 , Issue 3 , May 2003 , pp. 409 - 414
Copyright
© The London Mathematical Society 2003

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