Structure of (Fine) Mordell--Weil Groups

In this article we study the algebraic structure of fine Mordell–Weil groups, plus/minus Mordell–Weil groups, Selmer groups, and plus/minus Selmer groups in the cyclotomic ${\mathbb{Z}}_p$-extensions of abelian number fields. As a first, we prove theorems on the equivariant structure of fine Mordell–Weil groups and plus/minus Mordell–Weil groups. In other words, we study the explicit shape of the fine, plus/minus objects as a $\mathrm{\Lambda }(\mathcal{G})$-module with $\mathcal{G}\simeq {\mathbb{Z}}_p\times G$ and $G$ a finite abelian group. We prove refinements of previously known results over $\mathbb{Q}$ for the classical Selmer group and the plus/minus Selmer group, and subsequently also the Shafarevich–Tate group, and the plus/minus Shafarevich–Tate group. This gives new evidence towards an affirmative answer for the Kurihara–Pollack problem.