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Let p be an odd prime and let E be an elliptic curve defined over a quadraticimaginary field where p splits completely. Suppose E has supersingularreduction at primes above p. We define and study the fine double-signedresidual Selmer groups in these settings for Z_p^2-extensions. We prove thatfor 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 theSelmer group and the double-signed Selmer groups have no non-trivialpseudo-null submodules for these extensions.

Residual Supersingular Iwasawa Theory over quadratic imaginary fields