  We study the variation of Mordell-Weil ranks in the Jacobians of curves in a
pro-p tower over a fixed number field. In particular, we show that under mild
conditions the Mordell-Weil rank of a Jacobian in the tower is bounded above by
a constant multiple of its dimension. In the case of the tower of Fermat
curves, we show that the constant can be taken arbitrarily close to 1. The main
result is used in the forthcoming paper of Guillermo Mantilla-Soler on the
Mordell-Weil rank of the modular Jacobian J(Np^m).
