Structure of (Fine) Mordell--Weil Groups

TL;DR

This paper investigates the algebraic structure of fine Mordell--Weil groups and related Selmer groups in cyclotomic extensions, providing new structural theorems and refinements that support conjectures in number theory.

Contribution

It offers explicit descriptions of the modules' structures over Iwasawa algebras and refines known results for Selmer and Shafarevich--Tate groups over number fields.

Findings

Proved equivariant structure theorems for fine and plus/minus Mordell--Weil groups.

Refined the structure of Selmer and Shafarevich--Tate groups over cyclotomic extensions.

Provided evidence supporting the Kurihara--Pollack conjecture.

Abstract

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 $\Lambda(\mathcal{G})$-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.