RESIDUAL SUPERSINGULAR IWASAWA THEORY OVER QUADRATIC IMAGINARY FIELDS
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Let $ p $ be an odd prime. 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 $. Under appropriate hypotheses, we extend the results of \cite{SujFil} to $ \bZ_p^{2} $-extensions. We define and study the fine double-signed residual Selmer groups in these settings. 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.
Abstract
Let $ p $ be an odd prime. 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 $. Under appropriate hypotheses, we extend the results of \cite{SujFil} to $ \bZ_p^{2} $-extensions. We define and study the fine double-signed residual Selmer groups in these settings. 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.
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