  Consider a function field $K$ with characteristic $p>0$. We investigate the
$\Lambda$-module structure of the Mordell-Weil group of an abelian variety over
$\mathbb{Z}_p$-extensions of $K$, generalizing results due to Lee. Next, we
study the algebraic structure and prove a control theorem for the S-fine
Mordell-Weil groups, the function field analogue for Wuthrich's fine
Mordell-Weil groups, over a $\mathbb{Z}_p$-extension of $K$. In case of
unramified $\mathbb{Z}_p$-extension, $K_\infty$, we compute the characteristic
ideal of the Pontryagin dual of the S-fine Mordell group. This provides an
answer to an analogue of Greenberg's question for the characteristic ideal of
the dual fine Selmer group in the function field setup. In the $\ell\neq p$
case, we prove the triviality of the $\mu$-invariant for the Selmer group (same
as the fine Selmer group in this case) of an elliptic curve over a
non-commutative $GL_2(\mathbb{Z}_\ell)$-extension of $K$ and thus extending
Conjecture A. In the $\ell=p$ case, we compute the change of $\mu$-invariants
of the dual Selmer groups of elliptic curves under isogeny, giving a lower
bound for the $\mu$-invariant.
