Which quotients of Shimura curves are hyperelliptic?

A Shimura curve is built from an indefinite quaternion algebra over the rationals: the units of an Eichler order act on the upper half-plane, and the quotient is an algebraic curve. When the quaternion algebra is the algebra of two-by-two matrices, these are the classical modular curves. The normalizer of the order acts on the curve through a finite group of involutions, the Atkin—Lehner involutions, and we determine which of the quotients by these involutions are hyperelliptic, generalizing works of Ogg (trivial quotients) and Furumoto—Hasegawa (modular curves). This is joint work with Sachi Hashimoto.