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

Abstract

Let X be the Shimura curve corresponding to the quaternion algebra over ramified only at 3 and 13. B. Jordan showed that X ( √ −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 ( √ −13)-rational divisor classes of odd degree. A corollary of this is the fact that this counterexample is explained by the Manin obstruction.