P-converse to a theorem of Gross–Zagier, Kolyvagin and Rubin

Abstract

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).