p^∞ -Selmer groups and rational points on CM elliptic curves

Let E/ Q be a CM elliptic curve and p a prime of good ordinary reduction for E. We show that if Selp∞(E/Q) has Z_p-corank one, then E(Q) has a point of infinite order. The non-torsion point arises from a Heegner point, and thus ords=1L(E,s)=1, yielding a p-converse to a theorem of Gross–Zagier, Kolyvagin, and Rubin in the spirit of [49, 54]. For p> 3 , this gives a new proof of the main result of [12], which our approach extends to all primes. The approach generalizes to CM elliptic curves over totally real fields [4].