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

Ashay Burungale; Francisco Castella Cabello; Christopher Skinner; Ye Tian

doi:10.1007/s40316-022-00203-y

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

Ashay Burungale, Francisco Castella Cabello, Christopher Skinner, Ye Tian

Mathematics

Research output: Contribution to journal › Article › peer-review

Abstract

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].

Original language	English (US)
Journal	Annales Mathematiques du Quebec
DOIs	https://doi.org/10.1007/s40316-022-00203-y
State	Accepted/In press - 2022

All Science Journal Classification (ASJC) codes

Mathematics(all)

Keywords

Elliptic waves
Heegen points
L-functions
P-adic
Selmen groups

Access to Document

10.1007/s40316-022-00203-y

Cite this

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title = "p∞ -Selmer groups and rational points on CM elliptic curves",

abstract = "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 Zp-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].",

keywords = "Elliptic waves, Heegen points, L-functions, P-adic, Selmen groups",

author = "Ashay Burungale and {Castella Cabello}, Francisco and Christopher Skinner and Ye Tian",

note = "Funding Information: We thank Matthias Flach, Jacob Ressler and Qiyao Yu for helpful discussions. We also thank Henri Darmon and Antonio Lei for giving us the opportunity to contribute to this special issue, and the anonymous referee for a detailed reading. During the preparation of this paper, A.B. was partially supported by the NSF grant DMS-2001409; F.C. was partially supported by the NSF grants DMS-1946136 and DMS-2101458; C.S. was partially supported by the Simons Investigator Grant #376203 from the Simons Foundation and by the NSF Grant DMS-1901985; Y.T. was partially supported by the NSFC grants #11688101 and #11531008. Publisher Copyright: {\textcopyright} 2022, The Author(s).",

year = "2022",

doi = "10.1007/s40316-022-00203-y",

language = "English (US)",

journal = "Annales Mathematiques du Quebec",

issn = "2195-4755",

publisher = "Springer International Publishing AG",

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TY - JOUR

T1 - p∞ -Selmer groups and rational points on CM elliptic curves

AU - Burungale, Ashay

AU - Castella Cabello, Francisco

AU - Skinner, Christopher

AU - Tian, Ye

N1 - Funding Information: We thank Matthias Flach, Jacob Ressler and Qiyao Yu for helpful discussions. We also thank Henri Darmon and Antonio Lei for giving us the opportunity to contribute to this special issue, and the anonymous referee for a detailed reading. During the preparation of this paper, A.B. was partially supported by the NSF grant DMS-2001409; F.C. was partially supported by the NSF grants DMS-1946136 and DMS-2101458; C.S. was partially supported by the Simons Investigator Grant #376203 from the Simons Foundation and by the NSF Grant DMS-1901985; Y.T. was partially supported by the NSFC grants #11688101 and #11531008. Publisher Copyright: © 2022, The Author(s).

PY - 2022

Y1 - 2022

N2 - 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 Zp-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].

AB - 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 Zp-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].

KW - Elliptic waves

KW - Heegen points

KW - L-functions

KW - P-adic

KW - Selmen groups

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