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

Abstract

R\'esum\'e

Let \(E/{\mathbb {Q}}\) be a CM elliptic curve and p a prime of good ordinary reduction for E. We show that if \(\text {Sel}_{p^\infty }(E/{\mathbb {Q}})\) has \({\mathbb {Z}}_p\)-corank one, then \(E({\mathbb {Q}})\) has a point of infinite order. The non-torsion point arises from a Heegner point, and thus \({{\,\mathrm{ord}\,}}_{s=1}L(E,s)=1\), 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].