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

Autor: Ashay Burungale, Francesc Castella, Christopher Skinner, Ye Tian
Rok vydání: 2022
Předmět:
Zdroj: Annales mathématiques du Québec. 46:325-346
ISSN: 2195-4763
2195-4755
DOI: 10.1007/s40316-022-00203-y
Popis: 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].
Databáze: OpenAIRE
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