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

Abstract

Let $p\ge 5$ 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.

To appear in Q. J. Math

References in corpus (2)

• On fine Selmer groups and the greatest common divisor of signed and chromatic $p$-adic $L$-functions
• On fine Selmer groups and signed Selmer groups of elliptic modular forms

Cited by in corpus (1)

• On fine Mordell-Weil groups over ${\mathbb{Z}}_p$-extensions of an imaginary quadratic field