  We study a natural question in the Iwasawa theory of algebraic curves of
genus $>1$. Fix a prime number $p$. Let $X$ be a smooth, projective,
geometrically irreducible curve defined over a number field $K$ of genus $g>1$,
such that the Jacobian of $X$ has good ordinary reduction at the primes above
$p$. Fix an odd prime $p$ and for any integer $n>1$, let $K_n^{(p)}$ denote the
degree-$p^n$ extension of $K$ contained in $K(\mu_{p^{\infty}})$. We prove
explicit results for the growth of $\#X(K_n^{(p)})$ as $n\rightarrow \infty$.
When the Jacobian of $X$ has rank zero and the associated adelic Galois
representation has big image, we prove an explicit condition under which
$X(K_{n}^{(p)})=X(K)$ for all $n$. This condition is illustrated through
examples. We also prove a generalization of Imai's theorem that applies to
abelian varieties over arbitrary pro-$p$ extensions.
