Let p ≥ 5 be a prime and E/ℚ be an elliptic curve of conductor N that is ordinary at p. Let K/ℚ be an imaginary quadratic field. This thesis is concerned with the dual Selmer group of E over the anticyclotomic extension of K, especially when p is split in K and K satisfies the Heegner hypothesis for E: every prime ℓ dividing N is split in K/ℚ. A fundamental result in this setting is the non-existence of non-zero finite submodules. Using purely algebraic methods, we extend this result to new cases under verifiable hypotheses concerning the Heegner point of E over K. Under similar hypotheses, we establish the vanishing of the μ-invariant using simpler techniques than the literature. As an application, we study the variation of λ-invariants for p-residually isomorphic elliptic curves. This type of congruence question is also explored on the analytic side. Namely, we look at the Bertolini-Darmon-Prasanna (BDP) p-adic L-functions for p-congruent modular forms.
Dac-Nhan-Tam Nguyen (Thu,) studied this question.