An abelian variety in the interior of the Torelli locus is non-decomposable, but it could possibly be non-simple (i.e. isogenous to a product of abelian varieties with lower dimension). For certain families of super elliptic curves, the fibers are generically non-simple. The only examples of such families I've seen in the literature come from super elliptic curves. The set of non-simple abelian varieties should be contained in a countably infinite union of Shimura subvarieties, so in some sense there are very few of them. Is there any expectation that non-simple Jacobians come from a particular set of curves?
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