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

Suman Ahmed; Meng Fai Lim

doi:10.1007/s10114-020-9556-1

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arXiv · 1912.09023

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

Suman Ahmed·Meng Fai Lim

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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.

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Suman Ahmed, Meng Fai Lim. 2019-12-19. On the algebraic functional equation of the eigenspaces of mixed signed Selmer groups of elliptic curves with good reduction at primes above $p$. https://doi.org/10.1007/s10114-020-9556-1

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