Rational torsion points on Jacobians of Shimura curves

Let p and q be distinct primes. Consider the Shimura curveX p qassociated to the indefinite quaternion algebra of discriminant p q over Q . Let J p qbe the Jacobian variety ofX p q , which is an abelian variety over Q . For an odd prime ℓ , we provide sufficient conditions for the non‐existence of rational points of order ℓ on J p q . As an application, we find some non‐trivial subgroups of the kernel of an isogeny from the new quotientJ 0( p q ) newofJ 0 ( p q )to J p q .