  It is known that in the case of hyperelliptic curves the Shafarevich
conjecture can be made effective, i.e., for any number field k and any finite
set of places S of k, one can effectively compute the set of isomorphism
classes of hyperelliptic curves over k with good reduction outside S. We show
here that an extension of this result to an effective Shafarevich conjecture
for Jacobians of hyperelliptic curves of genus g would imply an effective
version of Siegel's theorem for integral points on hyperelliptic curves of
genus g.

Number Theory