  By the Jacquet-Langlands correspondence and Faltings' isogeny theorem, it is
known that the abelian variety J_0^{Dpq}(N) (the Jacobian of a Shimura curve of
discriminant Dpq with Gamma_0(N) level structure) is isogenous to the pq-new
subvariety of J_0^D(pqN), whenever p and q are distinct primes not dividing D
or N. However, this approach is entirely nonconstructive, and gives no
information at all about any particular isogeny between the two varieties. In
this paper, we determine the kernels of all possible maps from one variety to
the other up to support on a small finite set of maximal ideals of the Hecke
algebra. We also examine consequences of this computation for the geometry of
Jacobians of Shimura curves. In particular, we obtain a "multiplicity one"
result for certain Galois representations in the torsion of Jacobians of such
curves.
