Jacobians of Shimura curves and Jacquet-Langlands correspondences

Fix a squarefree integer N, with an even number of prime divisors, and let Gamma' be a congruence subgroup of level M, where M is prime to N. For each D dividing N with an even number of prime divisors, we consider the Shimura curve X D(Gamma0(N/D) ∩ Gamma' ) associated to the indefinite quaternion algebra of discriminant D and Gamma0(N/D) ∩ Gamma '-level structure, and its Jacobian JD(Gamma 0(N/D) ∩ Gamma'). Let JD denote the ND -new subvariety of this Jacobian. (This depends on N and Gamma', but as we will be considering the varieties JD for a fixed N and Gamma we suppress this from the notation.); The Jacquet-Langlands correspondence [JaLa] and Faltings' isogeny theorem [Fa] imply that there are Hecke-equivariant isogenies among the various varieties JD defined above. However, since the isomorphism of Jacquet-Langlands is noncanonical, this perspective gives no information about the isogenies so obtained beyond their existence. In this paper, we study maps between the varieties JD in terms of the maps they induce on the character groups of the tori corresponding to the mod p reductions of these varieties for p dividing N. Our characterization of such maps in these terms allows us to describe the possible kernels of maps from JD to JD', for D dividing D', up to support on a small finite set of maximal ideals of the Hecke algebra. These descriptions are in terms of modules over a Hecke algebra which are explicitly computable. Among other things, this allows us to compute the Tate modules Tm JD of JD at all non-Eisenstein m of residue characteristic l > 3. These computations have implications for the multiplicities of irreducible Galois representations in the torsion of Jacobians of Shimura curves; one such consequence is a "multiplicity one" result for Jacobians of Shimura curves. As a corollary of this result, we are able to prove that certain quotients of Hecke algebras are Gorenstein, a fact which has applications to proving the modularity of Galois representations [Kh].