  We present a new technique to study Jacobian variety decompositions using
subgroups of the automorphism group of the curve and the corresponding
intermediate covers. In particular, this new method allows us to produce many
new examples of genera for which there is a curve with completely decomposable
Jacobian. These examples greatly extend the list given by Ekedahl and Serre of
genera containing such curves, and provide more evidence for a positive answer
to two questions they asked. Additionally, we produce new examples of families
of curves, all of which have completely decomposable Jacobian varieties. These
families relate to questions about special subvarieties in the moduli space of
principally polarized abelian varieties.
