Algebraic structure and characteristic ideals of fine Mordell--Weil groups and plus/minus Mordell--Weil groups

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.