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

TL;DR

This paper explores the relationship between characteristic ideals of cotorsion signed Selmer groups and fine Selmer groups for CM elliptic curves at supersingular primes, building on recent structural breakthroughs.

Contribution

It establishes a link between the characteristic ideals of signed Selmer groups and the fine Selmer group for CM elliptic curves at supersingular primes.

Findings

Identifies a connection between characteristic ideals of Selmer groups and fine Selmer groups.

Builds on recent advances in the structure of local points by Burungale--Kobayashi--Ota.

Provides insights into the structure of anticyclotomic Selmer groups for CM elliptic curves.

Abstract

Let $p\ge5$ be a prime number. Let $E/\mathbb{Q}$ 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 $K_\infty$ be the anticyclotomic $\mathbb{Z}_p$-extension of $K$. Agboola--Howard defined Kobayashi-type signed Selmer groups of $E$ over $K_\infty$ 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.