Let E be a CM elliptic curve over the rationals and $$p>3$$ a good ordinary prime for E. We show that $$\begin{aligned} {\mathrm {corank}}_{{\mathbb {Z}}_{p}} {\mathrm {Sel}}_{p^{\infty }}(E_{/{\mathbb {Q}}})=1 \implies {\mathrm {ord}}_{s=1}L(s,E_{/{\mathbb {Q}}})=1 \end{aligned}$$for the $$p^{\infty }$$-Selmer group $${\mathrm {Sel}}_{p^{\infty }}(E_{/{\mathbb {Q}}})$$ and the complex L-function $$L(s,E_{/{\mathbb {Q}}})$$. In particular, the Tate–Shafarevich group $$\hbox {X}(E_{/{\mathbb {Q}}})$$ is finite whenever $${\mathrm {corank}}_{{\mathbb {Z}}_{p}} {\mathrm {Sel}}_{p^{\infty }}(E_{/{\mathbb {Q}}})=1$$. We also prove an analogous p-converse for CM abelian varieties arising from weight two elliptic CM modular forms with trivial central character. For non-CM elliptic curves over the rationals, first general results towards such a p-converse theorem are independently due to Skinner (A converse to a theorem of Gross, Zagier and Kolyvagin, arXiv:1405.7294, 2014) and Zhang (Camb J Math 2(2):191–253, 2014).

中文翻译：

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与 Gross-Zagier、Kolyvagin 和 Rubin 定理的 p 反演

设 E 是有理数上的 CM 椭圆曲线，$$p>3$$ 是 E 的一个很好的普通素数。我们证明 $$\begin{aligned} {\mathrm {corank}}_{{\mathbb {Z} }_{p}} {\mathrm {Sel}}_{p^{\infty }}(E_{/{\mathbb {Q}}})=1 \implies {\mathrm {ord}}_{s= 1}L(s,E_{/{\mathbb {Q}}})=1 \end{aligned}$$对于 $$p^{\infty }$$-Selmer 组 $${\mathrm {Sel} }_{p^{\infty }}(E_{/{\mathbb {Q}}})$$ 和复杂的 L 函数 $$L(s,E_{/{\mathbb {Q}}})$ $. 特别是，Tate-Shafarevich 群 $$\hbox {X}(E_{/{\mathbb {Q}}})$$ 是有限的，只要 $${\mathrm {corank}}_{{\mathbb {Z} }_{p}} {\mathrm {Sel}}_{p^{\infty }}(E_{/{\mathbb {Q}}})=1$$。我们还证明了 CM 阿贝尔变体的类似 p-逆变换，该变体是由两个具有平凡中心特征的椭圆 CM 模形式产生的。对于有理数上的非 CM 椭圆曲线，