Intrinsic Characterization of Optimality

There is a notion of an optimal quotient $A$ such that $J_0(N) \rightarrow A$ is minimal in its isogeny class. One derives different optimal quotients of $J_0(N)$ , $J_1(N)$ , and from the Jacobians of Shimura curves. It would be very interesting to have algorithms for computing structural isomorphism invariants which distinguish these quotients. See work of Glenn Stevens [Ste89] for a conjectural answer in the case of elliptic curves (and recent work of Nike Vatsal (and Stein-Watkins [SW04]) for proofs of Stevens' conjectures.

Remark 8.7.2 (From Mark Watkins)   This only makes sense (in general) over $\mathbb{Q}$ , as else you can have that neither an isogeny or its dual is étale.