On maps between modular Jacobians and Jacobians of Shimura curves

Fix a squarefree integer N , divisible by an even number of primes, and let Γ be a congruence subgroup of level M , where M is prime to N . For each D dividing N and divisible by an even number of primes, the Shimura curve X(Γ0(N/D)∩Γ ) associated to the indefinite quaternion algebra of discriminant D and Γ0(N/D) ∩ Γ -level structure is well defined, and we can consider its Jacobian J(Γ0(N/D) ∩ Γ ). Let J denote the ND -new subvariety of this Jacobian. By the Jacquet-Langlands correspondence [J-L] and Faltings’ isogeny theorem [Fa], there are Heckeequivariant among the various varieties J defined above. However, since the isomorphism of JacquetLanglands is noncanonical, this perspective gives no information about the isogenies so obtained beyond their existence. In this paper, we study maps between the varieties J 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 classify the possible kernels of maps from J to J ′ , for D dividing D, up to support on a small finite set of maximal ideals of the Hecke algebra. This allows us to compute the Tate modules TmJ D of J 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.