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

Abstract

R\'esum\'eLet $$E/{\mathbb {Q}}$$ E / Q be a CM elliptic curve and p a prime of good ordinary reduction for E. We show that if $$\text {Sel}_{p^\infty }(E/{\mathbb {Q}})$$ Sel p ∞ ( E / Q ) has $${\mathbb {Z}}_p$$ Z p -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 $${{\,\mathrm{ord}\,}}_{s=1}L(E,s)=1$$ 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 $$p>3$$ 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].

p-ADIC L-FUNCTIONS AND RATIONAL POINTS ON CM ELLIPTIC CURVES AT INERT PRIMES

“…(ii) Theorem B.3 is a tool in the proof of yet another CM p-converse (cf. [15]), and in turn, a result towards the cube sum problem (cf. [1]).…”

“…(ii) One may seek a refined p-converse: [56], [18], [16], [10], [11]). While it may be possible to approach Theorem 1.5 via the p-adic Gross-Zagier formula [32], with a view to (1.6), our approach instead employs Theorem 1.1.…”

“…The first general results for non-CM curves were independently due to Skinner [52] and Zhang [56] a few years back. One may seek a refined p -converse: (cf. [56], [18], [16],[10], [11]). While it may be possible to approach Theorem 1.5 via the p -adic Gross–Zagier formula [32], with a view to (1.6), our approach instead employs Theorem 1.1. …”