On the algebraic functional equation of the eigenspaces of mixed signed Selmer groups of elliptic curves with good reduction at primes above $p$

TL;DR

This paper proves a reflection principle relating eigenspaces of mixed signed Selmer groups of elliptic curves with good reduction at primes above p, showing their pseudo-isomorphism and triviality conditions.

Contribution

It establishes a novel algebraic functional equation connecting eigenspaces of Selmer groups, extending understanding of their structure in Iwasawa theory.

Findings

Eigenspaces are pseudo-isomorphic under certain conditions.

Triviality of one eigenspace implies triviality of the reflected eigenspace.

Provides a reflection principle linking two Iwasawa modules.

Abstract

Let $p$ be an odd prime number, and let $E$ be an elliptic curve defined over a number field which has good reduction at every prime above $p$. Under suitable assumptions, we prove that the $\eta$-eigenspace and the $\bar{\eta}$-eigenspace of mixed signed Selmer group of the elliptic curve are pseudo-isomorphic. As a corollary, we show that the $\eta$-eigenspace is trivial if and only if the $\bar{\eta}$-eigenspace is trivial. Our results can be thought as a reflection principle which relate an Iwasawa module in a given eigenspace with another Iwasawa module in a "reflected" eigenspace.