Elliptic curves with surjective adelic galois representations

Aaron Greicius

doi:10.1080/10586458.2010.10390639

Elliptic curves with surjective adelic galois representations

Aaron Greicius^*

^*Corresponding author for this work

Mathematics

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

12 Scopus citations

Abstract

Let K be a number field. The Gal(K̄/K)-action on the torsion of an elliptic curve E/K gives rise to an adelic representation ρE :Gal(K̄/K) → GL₂(ℤ^̂). From an analysis of maximal closed subgroups of GL₂(ℤ^̂) we derive useful necessary and sufficient conditions for ρE to be surjective. Using these conditions, we compute an example of a number field K and an elliptic curve E/K that admits a surjective adelic Galois representation.

Original language	English (US)
Pages (from-to)	495-507
Number of pages	13
Journal	Experimental Mathematics
Volume	19
Issue number	4
DOIs	https://doi.org/10.1080/10586458.2010.10390639
State	Published - 2010

Keywords

Adelic
Elliptic curves
Galois representations
Maximal subgroups
Profinite
Torsion
ℓ-adic

ASJC Scopus subject areas

Mathematics(all)

Access to Document

10.1080/10586458.2010.10390639

Cite this

@article{18eac6d3483b486497b5a8885cacc791,

title = "Elliptic curves with surjective adelic galois representations",

abstract = "Let K be a number field. The Gal({\=K}/K)-action on the torsion of an elliptic curve E/K gives rise to an adelic representation ρE :Gal({\=K}/K) → GL2({\^ℤ}). From an analysis of maximal closed subgroups of GL2({\^ℤ}) we derive useful necessary and sufficient conditions for ρE to be surjective. Using these conditions, we compute an example of a number field K and an elliptic curve E/K that admits a surjective adelic Galois representation.",

keywords = "Adelic, Elliptic curves, Galois representations, Maximal subgroups, Profinite, Torsion, ℓ-adic",

author = "Aaron Greicius",

note = "Funding Information: Thanks are due to Bjorn Poonen for suggesting this problem to me, to David Zywina for useful discussions, and to the referee, who suggested phrasing Proposition 3.5 and its corollaries in their present, more general, form. This work was partially supported by German Research Foundation (DFG) grant GR 3448/2-1.",

year = "2010",

doi = "10.1080/10586458.2010.10390639",

language = "English (US)",

volume = "19",

pages = "495--507",

journal = "Experimental Mathematics",

issn = "1058-6458",

publisher = "A K Peters",

number = "4",

}

TY - JOUR

T1 - Elliptic curves with surjective adelic galois representations

AU - Greicius, Aaron

N1 - Funding Information: Thanks are due to Bjorn Poonen for suggesting this problem to me, to David Zywina for useful discussions, and to the referee, who suggested phrasing Proposition 3.5 and its corollaries in their present, more general, form. This work was partially supported by German Research Foundation (DFG) grant GR 3448/2-1.

PY - 2010

Y1 - 2010

N2 - Let K be a number field. The Gal(K̄/K)-action on the torsion of an elliptic curve E/K gives rise to an adelic representation ρE :Gal(K̄/K) → GL2(ℤ̂). From an analysis of maximal closed subgroups of GL2(ℤ̂) we derive useful necessary and sufficient conditions for ρE to be surjective. Using these conditions, we compute an example of a number field K and an elliptic curve E/K that admits a surjective adelic Galois representation.

AB - Let K be a number field. The Gal(K̄/K)-action on the torsion of an elliptic curve E/K gives rise to an adelic representation ρE :Gal(K̄/K) → GL2(ℤ̂). From an analysis of maximal closed subgroups of GL2(ℤ̂) we derive useful necessary and sufficient conditions for ρE to be surjective. Using these conditions, we compute an example of a number field K and an elliptic curve E/K that admits a surjective adelic Galois representation.

KW - Adelic

KW - Elliptic curves

KW - Galois representations

KW - Maximal subgroups

KW - Profinite

KW - Torsion

KW - ℓ-adic

UR - http://www.scopus.com/inward/record.url?scp=79952140087&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=79952140087&partnerID=8YFLogxK

U2 - 10.1080/10586458.2010.10390639

DO - 10.1080/10586458.2010.10390639

M3 - Article

AN - SCOPUS:79952140087

SN - 1058-6458

VL - 19

SP - 495

EP - 507

JO - Experimental Mathematics

JF - Experimental Mathematics

IS - 4

ER -

Elliptic curves with surjective adelic galois representations

Abstract

Keywords

ASJC Scopus subject areas

Access to Document

Other files and links

Fingerprint

Cite this