LAUSR

Let [Formula: see text] be an elliptic curve, [Formula: see text] a prime and [Formula: see text] the anticyclotomic [Formula: see text]-extension of a quadratic imaginary field [Formula: see text] satisfying the Heegner hypothesis. In this paper, we make a conjecture about the fine Selmer group over [Formula: see text]. We also make a conjecture about the structure of the module of Heegner points in [Formula: see text] where [Formula: see text] is the union of the completions of the fields [Formula: see text] at a prime of [Formula: see text] above [Formula: see text]. We prove that these conjectures are equivalent. When [Formula: see text] has supersingular reduction at [Formula: see text] we also show that these conjectures are equivalent to the conjecture in our earlier work. Assuming these conjectures when [Formula: see text] has supersingular reduction at [Formula: see text], we prove various results about the structure of the Selmer group over [Formula: see text].