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

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 Zp-extension of F , generalizing results of Lee on the usual Mordell-Weil groups. In the case where F " Q, we show that the characteristic ideal of the Pontryagin dual of the fine Mordell-Weil group over the cyclotomic Zp-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 appEq " 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.