Under Iwasawa-main-conjecture hypotheses, a rank-one elliptic curve over a real quadratic field with finite Tate-Shafarevich group has analytic rank one.
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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.