Offical version here. Preprint version: arXiv:1505.07141 [math.NT].

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.