Points on Shimura curves over fields of even degree (Q584332)

\textit{G. Shimura} [Math. Ann. 215, 135-164 (1975; Zbl 0394.14007)] proved that there were no real points of any Shimura curve. In particular, there are no points defined over any field of odd degree. In this note we study points defined over fields of even degree on certain Shimura curves. We prove that these curves have only finitely many points rational over the totality of all quadratic fields and certain infinite families of quartic fields. The proof depends upon a well known result of \textit{K. Ribet} [C. R. Acad. Sci., Paris, Sér. A 291, 121-123 (1980; Zbl 0442.14014)], and a slight abstraction of a very beautiful idea of \textit{G. Frey} [Compos. Math. 58, 133-134 (1986; Zbl 0596.14021)].