$$p^\infty $$-Selmer groups and rational points on CM elliptic curves

Abstract

R\'esum\'eLet$$E/{\mathbb {Q}}$$E/Qbe a CM elliptic curve andpa prime of good ordinary reduction forE. We show that if$$\text {Sel}_{p^\infty }(E/{\mathbb {Q}})$$Selp∞(E/Q)has$${\mathbb {Z}}_p$$Zp-corank one, then$$E({\mathbb {Q}})$$E(Q)has a point of infinite order. The non-torsion point arises from a Heegner point, and thus$${{\,\mathrm{ord}\,}}_{s=1}L(E,s)=1$$ords=1L(E,s)=1, yielding ap-converse to a theorem of Gross–Zagier, Kolyvagin, and Rubin in the spirit of [49, 54]. For$$p>3$$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].