On maps between modular Jacobians and Jacobians of Shimura curves

摘要

By the Jacquet-Langlands correspondence and Faltings' isogeny theorem, it isknown that the abelian variety J_0^{Dpq}(N) (the Jacobian of a Shimura curve ofdiscriminant Dpq with Gamma_0(N) level structure) is isogenous to the pq-newsubvariety of J_0^D(pqN), whenever p and q are distinct primes not dividing Dor N. However, this approach is entirely nonconstructive, and gives noinformation at all about any particular isogeny between the two varieties. Inthis paper, we determine the kernels of all possible maps from one variety tothe other up to support on a small finite set of maximal ideals of the Heckealgebra. We also examine consequences of this computation for the geometry ofJacobians of Shimura curves. In particular, we obtain a "multiplicity one"result for certain Galois representations in the torsion of Jacobians of suchcurves.