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Elliptic curves with abelian division fields
Let E be an elliptic curve over Q , and let n ≥ 1 . The central object of study of this article is the division field Q ( E [ n ] ) that results by adjoining to Q the coordinates of all n -torsion points on E ( Q ¯ ) . In particular, we classify all curves E / Q such that Q ( E [ n ] ) is as small a...
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| Published in: | Mathematische Zeitschrift 2016-08, Vol.283 (3-4), p.835-859 |
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| Main Authors: | , |
| Format: | Article |
| Language: | English |
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| cites | cdi_FETCH-LOGICAL-c288t-6804a0aa44471a4d6eeed3b9eef5e5c4dce4e799072b820fea7b0f0e911c24163 |
| container_end_page | 859 |
| container_issue | 3-4 |
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| container_title | Mathematische Zeitschrift |
| container_volume | 283 |
| creator | González–Jiménez, Enrique Lozano-Robledo, Álvaro |
| description | Let
E
be an elliptic curve over
Q
, and let
n
≥
1
. The central object of study of this article is the division field
Q
(
E
[
n
]
)
that results by adjoining to
Q
the coordinates of all
n
-torsion points on
E
(
Q
¯
)
. In particular, we classify all curves
E
/
Q
such that
Q
(
E
[
n
]
)
is as small as possible, that is, when
Q
(
E
[
n
]
)
=
Q
(
ζ
n
)
, and we prove that this is only possible for
n
=
2
,
3
,
4
, or 5. More generally, we classify all curves such that
Q
(
E
[
n
]
)
is contained in a cyclotomic extension of
Q
or, equivalently (by the Kronecker–Weber theorem), when
Q
(
E
[
n
]
)
/
Q
is an abelian extension. In particular, we prove that this only happens for
n
=
2
,
3
,
4
,
5
,
6
, or 8, and we classify the possible Galois groups that occur for each value of
n
. |
| doi_str_mv | 10.1007/s00209-016-1623-z |
| format | article |
| fullrecord | <record><control><sourceid>crossref_sprin</sourceid><recordid>TN_cdi_crossref_primary_10_1007_s00209_016_1623_z</recordid><sourceformat>XML</sourceformat><sourcesystem>PC</sourcesystem><sourcerecordid>10_1007_s00209_016_1623_z</sourcerecordid><originalsourceid>FETCH-LOGICAL-c288t-6804a0aa44471a4d6eeed3b9eef5e5c4dce4e799072b820fea7b0f0e911c24163</originalsourceid><addsrcrecordid>eNp9j0FOwzAQRS0EEqFwAHY5AIYZx4mdJaoKRarEBtaW44zBVUgqOy2ipydVWLOaxZ_39R9jtwj3CKAeEoCAmgNWHCtR8OMZy1AWgqMWxTnLprjkpVbykl2ltAWYQiUzdrfqurAbg8vdPh4o5d9h_MxtQ12wfd6GQ0hh6HMfqGvTNbvwtkt083cX7P1p9bZc883r88vyccOd0HrklQZpwVoppUIr24qI2qKpiXxJpZOtI0mqrkGJRgvwZFUDHqhGdEJiVSwYzr0uDilF8mYXw5eNPwbBnHTNrGsmXXPSNceJETOTpt_-g6LZDvvYTzP_gX4Bni5Yjw</addsrcrecordid><sourcetype>Aggregation Database</sourcetype><iscdi>true</iscdi><recordtype>article</recordtype></control><display><type>article</type><title>Elliptic curves with abelian division fields</title><source>Springer Nature</source><creator>González–Jiménez, Enrique ; Lozano-Robledo, Álvaro</creator><creatorcontrib>González–Jiménez, Enrique ; Lozano-Robledo, Álvaro</creatorcontrib><description>Let
E
be an elliptic curve over
Q
, and let
n
≥
1
. The central object of study of this article is the division field
Q
(
E
[
n
]
)
that results by adjoining to
Q
the coordinates of all
n
-torsion points on
E
(
Q
¯
)
. In particular, we classify all curves
E
/
Q
such that
Q
(
E
[
n
]
)
is as small as possible, that is, when
Q
(
E
[
n
]
)
=
Q
(
ζ
n
)
, and we prove that this is only possible for
n
=
2
,
3
,
4
, or 5. More generally, we classify all curves such that
Q
(
E
[
n
]
)
is contained in a cyclotomic extension of
Q
or, equivalently (by the Kronecker–Weber theorem), when
Q
(
E
[
n
]
)
/
Q
is an abelian extension. In particular, we prove that this only happens for
n
=
2
,
3
,
4
,
5
,
6
, or 8, and we classify the possible Galois groups that occur for each value of
n
.</description><identifier>ISSN: 0025-5874</identifier><identifier>EISSN: 1432-1823</identifier><identifier>DOI: 10.1007/s00209-016-1623-z</identifier><language>eng</language><publisher>Berlin/Heidelberg: Springer Berlin Heidelberg</publisher><subject>Mathematics ; Mathematics and Statistics</subject><ispartof>Mathematische Zeitschrift, 2016-08, Vol.283 (3-4), p.835-859</ispartof><rights>Springer-Verlag Berlin Heidelberg 2016</rights><lds50>peer_reviewed</lds50><woscitedreferencessubscribed>false</woscitedreferencessubscribed><citedby>FETCH-LOGICAL-c288t-6804a0aa44471a4d6eeed3b9eef5e5c4dce4e799072b820fea7b0f0e911c24163</citedby><cites>FETCH-LOGICAL-c288t-6804a0aa44471a4d6eeed3b9eef5e5c4dce4e799072b820fea7b0f0e911c24163</cites></display><links><openurl>$$Topenurl_article</openurl><openurlfulltext>$$Topenurlfull_article</openurlfulltext><thumbnail>$$Tsyndetics_thumb_exl</thumbnail><link.rule.ids>314,780,784,27922,27923</link.rule.ids></links><search><creatorcontrib>González–Jiménez, Enrique</creatorcontrib><creatorcontrib>Lozano-Robledo, Álvaro</creatorcontrib><title>Elliptic curves with abelian division fields</title><title>Mathematische Zeitschrift</title><addtitle>Math. Z</addtitle><description>Let
E
be an elliptic curve over
Q
, and let
n
≥
1
. The central object of study of this article is the division field
Q
(
E
[
n
]
)
that results by adjoining to
Q
the coordinates of all
n
-torsion points on
E
(
Q
¯
)
. In particular, we classify all curves
E
/
Q
such that
Q
(
E
[
n
]
)
is as small as possible, that is, when
Q
(
E
[
n
]
)
=
Q
(
ζ
n
)
, and we prove that this is only possible for
n
=
2
,
3
,
4
, or 5. More generally, we classify all curves such that
Q
(
E
[
n
]
)
is contained in a cyclotomic extension of
Q
or, equivalently (by the Kronecker–Weber theorem), when
Q
(
E
[
n
]
)
/
Q
is an abelian extension. In particular, we prove that this only happens for
n
=
2
,
3
,
4
,
5
,
6
, or 8, and we classify the possible Galois groups that occur for each value of
n
.</description><subject>Mathematics</subject><subject>Mathematics and Statistics</subject><issn>0025-5874</issn><issn>1432-1823</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2016</creationdate><recordtype>article</recordtype><recordid>eNp9j0FOwzAQRS0EEqFwAHY5AIYZx4mdJaoKRarEBtaW44zBVUgqOy2ipydVWLOaxZ_39R9jtwj3CKAeEoCAmgNWHCtR8OMZy1AWgqMWxTnLprjkpVbykl2ltAWYQiUzdrfqurAbg8vdPh4o5d9h_MxtQ12wfd6GQ0hh6HMfqGvTNbvwtkt083cX7P1p9bZc883r88vyccOd0HrklQZpwVoppUIr24qI2qKpiXxJpZOtI0mqrkGJRgvwZFUDHqhGdEJiVSwYzr0uDilF8mYXw5eNPwbBnHTNrGsmXXPSNceJETOTpt_-g6LZDvvYTzP_gX4Bni5Yjw</recordid><startdate>20160801</startdate><enddate>20160801</enddate><creator>González–Jiménez, Enrique</creator><creator>Lozano-Robledo, Álvaro</creator><general>Springer Berlin Heidelberg</general><scope>AAYXX</scope><scope>CITATION</scope></search><sort><creationdate>20160801</creationdate><title>Elliptic curves with abelian division fields</title><author>González–Jiménez, Enrique ; Lozano-Robledo, Álvaro</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-c288t-6804a0aa44471a4d6eeed3b9eef5e5c4dce4e799072b820fea7b0f0e911c24163</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2016</creationdate><topic>Mathematics</topic><topic>Mathematics and Statistics</topic><toplevel>peer_reviewed</toplevel><toplevel>online_resources</toplevel><creatorcontrib>González–Jiménez, Enrique</creatorcontrib><creatorcontrib>Lozano-Robledo, Álvaro</creatorcontrib><collection>CrossRef</collection><jtitle>Mathematische Zeitschrift</jtitle></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext</fulltext></delivery><addata><au>González–Jiménez, Enrique</au><au>Lozano-Robledo, Álvaro</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>Elliptic curves with abelian division fields</atitle><jtitle>Mathematische Zeitschrift</jtitle><stitle>Math. Z</stitle><date>2016-08-01</date><risdate>2016</risdate><volume>283</volume><issue>3-4</issue><spage>835</spage><epage>859</epage><pages>835-859</pages><issn>0025-5874</issn><eissn>1432-1823</eissn><abstract>Let
E
be an elliptic curve over
Q
, and let
n
≥
1
. The central object of study of this article is the division field
Q
(
E
[
n
]
)
that results by adjoining to
Q
the coordinates of all
n
-torsion points on
E
(
Q
¯
)
. In particular, we classify all curves
E
/
Q
such that
Q
(
E
[
n
]
)
is as small as possible, that is, when
Q
(
E
[
n
]
)
=
Q
(
ζ
n
)
, and we prove that this is only possible for
n
=
2
,
3
,
4
, or 5. More generally, we classify all curves such that
Q
(
E
[
n
]
)
is contained in a cyclotomic extension of
Q
or, equivalently (by the Kronecker–Weber theorem), when
Q
(
E
[
n
]
)
/
Q
is an abelian extension. In particular, we prove that this only happens for
n
=
2
,
3
,
4
,
5
,
6
, or 8, and we classify the possible Galois groups that occur for each value of
n
.</abstract><cop>Berlin/Heidelberg</cop><pub>Springer Berlin Heidelberg</pub><doi>10.1007/s00209-016-1623-z</doi><tpages>25</tpages></addata></record> |
| fulltext | fulltext |
| identifier | ISSN: 0025-5874 |
| ispartof | Mathematische Zeitschrift, 2016-08, Vol.283 (3-4), p.835-859 |
| issn | 0025-5874 1432-1823 |
| language | eng |
| recordid | cdi_crossref_primary_10_1007_s00209_016_1623_z |
| source | Springer Nature |
| subjects | Mathematics Mathematics and Statistics |
| title | Elliptic curves with abelian division fields |
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