Under Iwasawa-main-conjecture hypotheses, a rank-one elliptic curve over a real quadratic field with finite Tate-Shafarevich group has analytic rank one.
Iwasawa theory for elliptic curves
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abstract
We study this subject by first proving that the p-primary subgroup of the classical Selmer group for an elliptic curve with good, ordinary reduction at a prime p has a very simple and elegant description which involves just the Galois module of p-power torsion points. We then prove theorems of Mazur, Schneider, and Perrin-Riou on the basis of this description. The final section, which is half of this long paper, contains a number of results and examples including a thorough study of the mu-invariant.
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A $p$-Converse theorem for Real Quadratic Fields
Under Iwasawa-main-conjecture hypotheses, a rank-one elliptic curve over a real quadratic field with finite Tate-Shafarevich group has analytic rank one.