Structure of (Fine) Mordell--Weil Groups

Mathematics > Number Theory [Submitted on 27 Jul 2025 (v1), last revised 16 Jun 2026 (this version, v3)] Title:Structure of (Fine) Mordell--Weil Groups View PDF HTML (experimental)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. Submission history From: Rusiru Gambheera [view email][v1] Sun, 27 Jul 2025 16:14:36 UTC (27 KB) [v2] Thu, 25 Sep 2025 00:56:55 UTC (27 KB) [v3] Tue, 16 Jun 2026 17:26:55 UTC (31 KB) References & Citations Loading... Bibliographic and Citation Tools Bibliographic Explorer (What is the Explorer?) Connected Papers (What is Connected Papers?) Litmaps (What is Litmaps?) scite Smart Citations (What are Smart Citations?) Code, Data and Media Associated with this Article alphaXiv (What is alphaXiv?) CatalyzeX Code Finder for Papers (What is CatalyzeX?) DagsHub (What is DagsHub?) Gotit.pub (What is GotitPub?) Hugging Face (What is Huggingface?) ScienceCast (What is ScienceCast?) Demos Recommenders and Search Tools Influence Flower (What are Influence Flowers?) CORE Recommender (What is CORE?) arXivLabs: experimental projects with community collaborators arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website. Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them. Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs.