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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
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.
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- © The London Mathematical Society 2003
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