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On non-trivial $Λ$-submodules with finite index of the plus/minus Selmer group over anticyclotomic $\mathbb{Z}_{p}$-extension at inert primes

Ryota Shii

TL;DR

This work studies plus/minus $p$-primary Selmer groups over the inert anticyclotomic $\mathbb{Z}_p$-extension of an imaginary quadratic field, proving that under mild hypotheses the dual Selmer group has no non-trivial $\Lambda$-submodules of finite index when it is $\Lambda$-cotorsion. It leverages Rubin's plus/minus Lubin–Tate formal-group framework (as established by BKO21) and a control theorem to transfer local conditions to the global Selmer module, extending Greenberg–Kim-type results from the cyclotomic/cyclotomic-ordinary setting to the inert anticyclotomic case. The approach yields corank and surjectivity statements via Poitou–Tate duality and a twisting technique, and provides concrete CM and non-CM examples through recent works of Agboola–Howard and Burungale–Kobayashi–Lei. The results have implications for the $p$-part of BSD and deepen understanding of $\Lambda$-adic invariants in non-abelian Iwasawa theory in the inert setting.

Abstract

Let $K$ be an imaginary quadratic field where $p$ is inert. Let $E$ be an elliptic curve defined over $K$ and suppose that $E$ has good supersingular reduction at $p$. In this paper, we prove that the plus/minus Selmer group of $E$ over the anticyclotomic $\mathbb{Z}_{p}$-extension of $K$ has no non-trivial $Λ$-submodules of finite index under mild assumptions for $E$. This is an analogous result to R. Greenberg and B. D. Kim for the anticyclotomic $\mathbb{Z}_{p}$-extension essentially. By applying the results of A. Agboola--B. Howard or A. Burungale--K. Büyükboduk--A. Lei, we can also construct examples satisfying the assumptions of our theorem.

On non-trivial $Λ$-submodules with finite index of the plus/minus Selmer group over anticyclotomic $\mathbb{Z}_{p}$-extension at inert primes

TL;DR

This work studies plus/minus -primary Selmer groups over the inert anticyclotomic -extension of an imaginary quadratic field, proving that under mild hypotheses the dual Selmer group has no non-trivial -submodules of finite index when it is -cotorsion. It leverages Rubin's plus/minus Lubin–Tate formal-group framework (as established by BKO21) and a control theorem to transfer local conditions to the global Selmer module, extending Greenberg–Kim-type results from the cyclotomic/cyclotomic-ordinary setting to the inert anticyclotomic case. The approach yields corank and surjectivity statements via Poitou–Tate duality and a twisting technique, and provides concrete CM and non-CM examples through recent works of Agboola–Howard and Burungale–Kobayashi–Lei. The results have implications for the -part of BSD and deepen understanding of -adic invariants in non-abelian Iwasawa theory in the inert setting.

Abstract

Let be an imaginary quadratic field where is inert. Let be an elliptic curve defined over and suppose that has good supersingular reduction at . In this paper, we prove that the plus/minus Selmer group of over the anticyclotomic -extension of has no non-trivial -submodules of finite index under mild assumptions for . This is an analogous result to R. Greenberg and B. D. Kim for the anticyclotomic -extension essentially. By applying the results of A. Agboola--B. Howard or A. Burungale--K. Büyükboduk--A. Lei, we can also construct examples satisfying the assumptions of our theorem.
Paper Structure (5 sections, 21 theorems, 61 equations)

This paper contains 5 sections, 21 theorems, 61 equations.

Table of Contents

  1. Introduction
  2. Rubin's conjecture
  3. Global settings and Local conditions
  4. Control theorem
  5. Proof of Main Theorem

Key Result

Theorem 1.1

Let $F$ be a finite extension of $K$ and $h_{K}$ the class number of $K$. Assume that $p$ splits completely over $F/K$, and $p \nmid h_{K}[F: K]$. Let $E$ be an elliptic curve defined over $F$ that has supersingular reduction at any prime lying over $p$. Suppose that the formal group $\widehat{E}$ o

Theorems & Definitions (46)

  • Theorem 1.1: Theorem \ref{['主定理']}
  • Remark 1.2
  • Corollary 1.3: CM case
  • proof
  • Corollary 1.4: non-CM case
  • proof
  • Corollary 1.5
  • Theorem 2.1
  • Theorem 2.2: BKOpre
  • Proposition 2.3
  • ...and 36 more