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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 the primes above p. The main purpose of this thesis is to study the signed μ-invariants of the dual signed Selmer groups over Zₚ²-extensions of an imaginary quadratic field, as well as the signed μ-invariants of the dual signed Selmer groups over Zₚ-cyclotomic extensions. We give an overview of some of the important results proven for the fine Selmer group and the signed Selmer groups over cyclotomic and Zₚ²-extensions of an imaginary quadratic field. Under appropriate hypotheses, we define and study the fine double-signed residual Selmer groups and extend the results of [35] to Zₚ²-extensions in these settings. We prove that for two residually isomorphic elliptic curves, the vanishing of the signed μ-invariants of one elliptic curve implies the vanishing of the signed μ-invariants of the other. Moreover, we show that the μ-invariant of the classical Selmer groups is bounded by the μ-invariant of the signed Selmer groups. Finally, we show that the Pontryagin duals of the Selmer group and the double-signed Selmer groups have no non-trivial pseudo-null submodules for these extensions, with purely algebraic methods.