Shimura subgroups of Jacobians of Shimura curves

Abstract:

Given an indefinite quaternion algebra of reduced discriminant $D$ and an integer $N$ relatively prime to $D$, one can construct Shimura curves ${\operatorname {Sh} _0}(N,D)$ and ${\operatorname {Sh} _1}(N,D)$, which are analogues of ${X_0}(N)$ and ${X_1}(N)$. The natural morphism ${\operatorname {Sh} _1}(N,D) \to {\operatorname {Sh} _0}(N,D)$ induces a morphism ${J_0}(N,D) \to {J_1}(N,D)$ between the Jacobians. We compute the kernel $\sum (N,D)$ of this latter map, which is finite.