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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.

William Stein 2006-10-20