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

Ashay Burungale; Francesc Castella; Christopher Skinner; Ye Tian

doi:10.1007/s40316-022-00203-y

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

Ashay Burungale, Francesc Castella, Christopher Skinner, Ye Tian

Mathematics

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

1 Scopus citations

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)
Pages (from-to)	325-346
Number of pages	22
Journal	Annales Mathematiques du Quebec
Volume	46
Issue number	2
DOIs	https://doi.org/10.1007/s40316-022-00203-y
State	Published - Oct 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 Francesc Castella 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).",

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AU - Burungale, Ashay

AU - Castella, Francesc

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 - 2022/10

Y1 - 2022/10

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

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