The Shimura covering of a Shimura curve: automorphisms and étale subcoverings

Let XDXD be the Shimura curve associated with an indefinite rational quaternion algebra of discriminant D, and let p be a prime dividing D  . In their investigations on the arithmetic of XDXD, Jordan and Skorobogatov introduced a covering XD,pXD,p of XDXD whose maximal étale quotient is referred to as the Shimura covering of XDXD at p  . The goal of this note is to describe the group of modular automorphisms of the curve XD,pXD,p and its quotients. As an application, we construct cyclic étale Galois coverings of Atkin–Lehner quotients of XDXD.