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

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. Agoboola--B. Howard or A. Burungale--K. B\"uy\"ukboduk--A. Lei, we can also construct examples satisfying the assumptions of our theorem.