On Selmer groups in the supersingular reduction case

TL;DR

This paper investigates the structure and properties of Selmer groups associated with elliptic curves with supersingular reduction over cyclotomic extensions, providing conditions for their non-trivial submodules and calculating Euler characteristics in certain cases.

Contribution

It offers new criteria for the non-existence of finite index submodules in Selmer groups and computes their Euler characteristics when the prime splits completely.

Findings

Selmer groups lack non-trivial finite index submodules under certain conditions.

Euler characteristics of plus/minus Selmer groups are explicitly calculated when $p$ splits in $F$.

Provides insights into the structure of Selmer groups in supersingular reduction scenarios.

Abstract

Let $p$ be a fixed odd prime. Let $E$ be an elliptic curve defined over a number field $F$ with good supersingular reduction at all primes above $p$. We study both the classical and plus/minus Selmer groups over the cyclotomic $\mathbb{Z}_p$-extension of $F$. In particular, we give sufficient conditions for these Selmer groups to not contain a non-trivial sub-module of finite index. Furthermore, when $p$ splits completely in $F$, we calculate the Euler characteristics of the plus/minus Selmer groups over the compositum of all $\mathbb{Z}_p$-extensions of $F$ when they are defined.