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

TL;DR

This paper investigates the algebraic structure of fine Mordell--Weil groups over number fields, establishing control theorems and connecting their characteristic ideals with predictions from Iwasawa theory, especially in supersingular cases.

Contribution

It generalizes existing results on Mordell--Weil groups to fine and plus/minus variants, linking their algebraic invariants to conjectural $p$-adic $L$-function properties.

Findings

Proves a control theorem for fine Mordell--Weil groups over $Z_p$-extensions.

Shows the characteristic ideal matches Greenberg's prediction in the rational case.

Establishes the relation between plus/minus Mordell--Weil groups and plus/minus $p$-adic $L$-functions.

Abstract

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.