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

3 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

General Mathematics

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].",

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

AU - Burungale, Ashay

AU - Castella, Francesc

AU - Skinner, Christopher

AU - Tian, Ye

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].

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KW - Heegen points

KW - L-functions

KW - P-adic

KW - Selmen groups

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