Rational Points on Atkin-Lehner Quotients of Shimura Curves

We study three families of Atkin-Lehner quotients of quaternionic Shimura curves: X, X 0 (N), and X D+ 1 (N), which serve as moduli spaces of abelian surfaces with potential quaternionic multiplication (PQM) and level N structure. The arithmetic geometry of these curves is similar to, but even richer than, that of the classical modular curves. Two important differences are the existence of a nontrivial obstruction to an abelian surface being defined over its field of moduli and the lack of cusps, due to which there may fail to be any points rational over a given field. We study the existence of points on these curves rational over both local and global fields, and consider applications to the existence of PQM surfaces over Q.