Rational points on Shimura varieties classifying abelian varieties with quaternionic multiplication

We study the characters induced by suitable level structures of abelian varieties with quaternionic multiplication following the methods of Mazur, Momose, who studied the characters induced by elliptic curves, and Arai–Momose, who studied the characters induced by abelian surfaces with quaternionic multiplication. Also using results on such characters we show that Shimura varieties parametrizing them have rarely rational points.