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

TL;DR

This paper proves that the plus/minus Selmer group over the anticyclotomic $Z_p$-extension of an inert prime in an imaginary quadratic field has no non-trivial finite index $Z_p$-submodules, extending previous results.

Contribution

It establishes the absence of non-trivial finite index $Z_p$-submodules in the plus/minus Selmer group over the anticyclotomic extension, generalizing prior work to inert primes.

Findings

No non-trivial $Z_p$-submodules of finite index in the Selmer group.

Extension of Greenberg and Kim's results to inert primes.

Constructed explicit examples satisfying the theorem's assumptions.

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 $\Lambda$-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. B\"uy\"ukboduk--A. Lei, we can also construct examples satisfying the assumptions of our theorem.