  Answering a question of Ed Schaefer, we show that if J is the Jacobian of a
curve C over a number field, if s is an automorphism of J coming from an
automorphism of C, and if u lies in the subring Z[s] of End J and has connected
kernel, then it is not necessarily the case that u gives a surjective map from
the Mordell-Weil group of J to the Mordell-Weil group of its image.
