Residual Supersingular Iwasawa Theory over quadratic imaginary fields

TL;DR

This paper investigates the properties of residual Selmer groups for supersingular elliptic curves over quadratic imaginary fields, establishing relations between invariants of isomorphic curves and structural properties of these groups.

Contribution

It introduces and analyzes fine double-signed residual Selmer groups in Z_p^2-extensions for supersingular elliptic curves over quadratic imaginary fields, linking invariants of isomorphic curves.

Findings

Vanishing of signed μ-invariants for one curve implies the same for an isomorphic curve.

Pontryagin duals of Selmer groups lack non-trivial pseudo-null submodules.

Structural properties of residual Selmer groups are established in supersingular settings.

Abstract

Let p be an odd prime and let E be an elliptic curve defined over a quadratic imaginary field where p splits completely. Suppose E has supersingular reduction at primes above p. We define and study the fine double-signed residual Selmer groups in these settings for Z_p^2-extensions. We prove that for two residually isomorphic elliptic curves, the vanishing of the signed {\mu}-invariants of one elliptic curve implies the vanishing of the signed {\mu}-invariants of the other. Finally, we show that the Pontryagin dual of the Selmer group and the double-signed Selmer groups have no non-trivial pseudo-null submodules for these extensions.