Local points on Shimura coverings of Shimura curves at bad reduction primes

Let $X_D$ be the Shimura curve associated with an indefinite rational quaternion algebra of reduced discriminant $D>1$. For each prime $\ell\mid D$, there is a natural cyclic Galois covering of Shimura curves $X_{D,\ell}\to X_D$ constructed by adding certain level structure at $\ell$. The main goal of this note is to study the existence of local points at primes $p\neq\ell$ of bad reduction on the intermediate curves of these coverings and their Atkin–Lehner quotients.