On maps between modular Jacobians and Jacobians of Shimura curves

Abstract

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.