Points on Shimura curves rational over imaginary quadratic fields in the non-split case

Keisuke Arai

Published 2014-11-05Version 1

For an imaginary quadratic field $k$ of class number $>1$, Jordan proved that there are only finitely many isomorphism classes of rational indefinite quaternion division algebras $B$ such that the associated Shimura curve has $k$-rational points and $k$ splits $B$. In this article, we study the case where $k$ does not split $B$, and obtain an analogous result by imposing a certain congruent condition on the discriminant of $B$.