A remark on the characteristic elements of anticyclotomic Selmer groups of elliptic curves with complex multiplication at supersingular primes

Let $p\ge 5$ be a prime number. Let $E\mathrm{/}\mathbb{Q}$ be an elliptic curve with complex multiplication by an imaginary quadratic field $K$ such that $p$ is inert in $K$ and that $E$ has good reduction at $p$. Let $K_{\mathrm{\infty }}$ be the anticyclotomic ${\mathbb{Z}}_p$-extension of $K$. Agboola–Howard defined Kobayashi-type signed Selmer groups of $E$ over $K_{\mathrm{\infty }}$ and showed that exactly one of them is cotorsion over the corresponding Iwasawa algebra. In this short note, we discuss a link between the characteristic ideals of the cotorsion signed Selmer group and the fine Selmer group building on a recent breakthrough of Burungale–Kobayashi–Ota on the structure of local points.