  Given an elliptic curve defined over a number field $F$, we study the
algebraic structure and prove a control theorem for Wuthrich's fine
Mordell--Weil groups over a $\mathbb{Z}_p$-extension of $F$, generalizing
results of Lee on the usual Mordell--Weil groups. In the case where
$F=\mathbb{Q}$, we show that the characteristic ideal of the Pontryagin dual of
the fine Mordell--Weil group over the cyclotomic $\mathbb{Z}_p$-extension
coincides with Greenberg's prediction for the characteristic ideal of the dual
fine Selmer group. If furthermore $E$ has good supersingular reduction at $p$
with $a_p(E)=0$, we generalize Wuthrich's fine Mordell--Weil groups to define
"plus and minus Mordell--Weil groups". We show that the greatest common divisor
of the characteristic ideals of the Pontryagin duals of these groups coincides
with Kurihara--Pollack's prediction for the greatest common divisor of the plus
and minus $p$-adic $L$-functions.
