Rational points on Atkin-Lehner quotients of geometrically hyperelliptic Shimura curves

Guo and Yang give defining equations for all geometrically hyperelliptic Shimura curves X_0(D,N). In this talk, I will describe how we created a database containing all their Atkin-Lehner quotients and how we computed their sets of Q-rational points when these sets are finite. We also determine which rational points are CM for many of these curves. This is joint work with Ciaran Schembri.