Motivating Problem

This paper was motivated by the problem of computing the groups  $\Phi_{A,p}$ attached to quotients $A$ of Jacobians of modular curves $X_0(N)$. When $A$ has semistable reduction, Grothendieck described the component group $\Phi_A$ in terms of a monodromy pairing on certain free abelian groups (see [9, Thm. 11.5]). When $A=J=J_0(N)$ is the Jacobian of $X_0(N)$, this pairing can be explicitly computed, hence the component group $\Phi_J$ can also be computed; this has been done in many cases (see, e.g., [5] and [13]).

Suppose now that $A=A_f$ is an optimal quotient of $J_0(N)$ attached to a newform $f$, so that the kernel of the map $\pi:J\rightarrow A$ is geometrically connected. There is a natural map $\pi_*:\Phi_J\rightarrow \Phi_A$. The results of this paper give an algorithm to compute both the image of $\pi_*$ and the order (but not structure) of the cokernel of $\pi_*$.