Under Iwasawa-main-conjecture hypotheses, a rank-one elliptic curve over a real quadratic field with finite Tate-Shafarevich group has analytic rank one.
Burungale, Christopher Skinner, and Ye Tian, The B irch and S winnerton- D yer conjecture: a brief survey , Nine mathematical challenges---an elucidation, Proc
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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.