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

Upper bounds for the growth of Mordell-Weil ranks in pro-p towers of Jacobians