Algebraic functional equations and completely faithful Selmer groups

TL;DR

This paper develops a pairing on the Selmer group of a non-CM elliptic curve over certain p-adic Lie extensions, leading to an algebraic functional equation for the associated p-adic L-function and constructing faithful Selmer groups.

Contribution

It introduces a new pairing on Selmer groups over p-adic Lie extensions and establishes an algebraic functional equation under torsion hypotheses, with applications to faithful Selmer groups.

Findings

Constructed a pairing on the dual p-infinity Selmer group.

Derived an algebraic functional equation for the p-adic L-function.

Built faithful Selmer groups in specific p-adic Lie extensions.

Abstract

Let $E$ be an elliptic curve---defined over a number field $K$---without complex multiplication and with good ordinary reduction at all the primes above a rational prime $p \geq 5$. We construct a pairing on the dual $p^\infty$-Selmer group of $E$ over any strongly admissible $p$-adic Lie extension $K_\infty/K$ under the assumption that it is a torsion module over the Iwasawa algebra of the Galois group $G=\operatorname{Gal}(K_\infty/K)$. Under some mild additional hypotheses this gives an algebraic functional equation of the conjectured $p$-adic L-function. As an application we construct completely faithful Selmer groups in case the $p$-adic Lie extension is obtained by adjoining the $p$-power division points of another non-CM elliptic curve $A$.