<p>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 <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\mathbb {Z}_p$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="double-struck">Z</mi> <mi>p</mi> </msub> </math></EquationSource> </InlineEquation>-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 <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\Lambda (\mathcal {G})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="normal">Λ</mi> <mo stretchy="false">(</mo> <mi mathvariant="script">G</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>-module with <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\mathcal {G}\simeq \mathbb {Z}_p\times \textsf{G}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="script">G</mi> <mo>≃</mo> <msub> <mi mathvariant="double-struck">Z</mi> <mi>p</mi> </msub> <mo>×</mo> <mi mathvariant="sans-serif">G</mi> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\textsf{G}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="sans-serif">G</mi> </math></EquationSource> </InlineEquation> a finite abelian group. We prove refinements of previously known results over <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\mathbb {Q}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="double-struck">Q</mi> </math></EquationSource> </InlineEquation> 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.</p>

Structure of (fine) Mordell–Weil groups

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发表日期：

2026年6月25日

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Rusiru Gambheera,
Debanjana Kundu

摘要

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 $\mathbb {Z}_p$ $Z_{p}$ -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 $\Lambda (\mathcal {G})$ $Λ (G)$ -module with $\mathcal {G}\simeq \mathbb {Z}_p\times \textsf{G}$ $G ≃ Z_{p} \times G$ and $\textsf{G}$ $G$ a finite abelian group. We prove refinements of previously known results over $\mathbb {Q}$ $Q$ 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.