Let ...-> X_2 -> X_1 -> X be a tower of algebraic curves over number fields. It is natural to ask about the asymptotic growth in n of the Mordell-Weil groups of these curves. The problem has an Iwasawa-theoretic flavor; we make this analogy literal by constructing an Iwasawa module for a certain non-abelian Iwasawa algebra which controls these ranks. This algebra arises in a natural way from the metabelian quotient of the etale fundamental group of X. We will discuss some theorems in the special case X_n = x^{p^n} + y^{p^n} + z^{p^n} = 0 and we will lay out some challenges in non-abelian Iwasawa theory which stand in the way of more general results.