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Open image computations for elliptic curves over number fields
For a non-CM elliptic curve E defined over a number field K , the Galois action on its torsion points gives rise to a Galois representation ρ E : Gal ( K ¯ / K ) → GL 2 ( Z ^ ) that is unique up to isomorphism. A renowned theorem of Serre says that the image of ρ E is an open, and hence finite index...
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| Published in: | Research in number theory 2025-03, Vol.11 (1), Article 1 |
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| container_title | Research in number theory |
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| creator | Zywina, David |
| description | For a non-CM elliptic curve
E
defined over a number field
K
, the Galois action on its torsion points gives rise to a Galois representation
ρ
E
:
Gal
(
K
¯
/
K
)
→
GL
2
(
Z
^
)
that is unique up to isomorphism. A renowned theorem of Serre says that the image of
ρ
E
is an open, and hence finite index, subgroup of
GL
2
(
Z
^
)
. In an earlier work of the author, an algorithm was given that computed the image of
ρ
E
up to conjugacy in
GL
2
(
Z
^
)
in the special case
K
=
Q
. A fundamental ingredient of this earlier work was the Kronecker–Weber theorem whose conclusion fails for number fields
K
≠
Q
. We shall give an overview of an analogous algorithm for a general number field and work out the required group theory. We also give some bounds on the index in Serre’s theorem for a typical elliptic curve over a fixed number field. |
| doi_str_mv | 10.1007/s40993-024-00599-2 |
| format | article |
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E
defined over a number field
K
, the Galois action on its torsion points gives rise to a Galois representation
ρ
E
:
Gal
(
K
¯
/
K
)
→
GL
2
(
Z
^
)
that is unique up to isomorphism. A renowned theorem of Serre says that the image of
ρ
E
is an open, and hence finite index, subgroup of
GL
2
(
Z
^
)
. In an earlier work of the author, an algorithm was given that computed the image of
ρ
E
up to conjugacy in
GL
2
(
Z
^
)
in the special case
K
=
Q
. A fundamental ingredient of this earlier work was the Kronecker–Weber theorem whose conclusion fails for number fields
K
≠
Q
. We shall give an overview of an analogous algorithm for a general number field and work out the required group theory. We also give some bounds on the index in Serre’s theorem for a typical elliptic curve over a fixed number field.</description><identifier>ISSN: 2522-0160</identifier><identifier>EISSN: 2363-9555</identifier><identifier>DOI: 10.1007/s40993-024-00599-2</identifier><language>eng</language><publisher>Cham: Springer International Publishing</publisher><subject>Mathematics ; Mathematics and Statistics ; Number Theory</subject><ispartof>Research in number theory, 2025-03, Vol.11 (1), Article 1</ispartof><rights>The Author(s), under exclusive licence to Springer Nature Switzerland AG 2024 Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.</rights><lds50>peer_reviewed</lds50><woscitedreferencessubscribed>false</woscitedreferencessubscribed><cites>FETCH-LOGICAL-c172t-cdaf231f061b998d6b5cb193578be2ff5ffd096bad8650873056fe39e36eacfd3</cites></display><links><openurl>$$Topenurl_article</openurl><openurlfulltext>$$Topenurlfull_article</openurlfulltext><thumbnail>$$Tsyndetics_thumb_exl</thumbnail><link.rule.ids>314,776,780,27903,27904</link.rule.ids></links><search><creatorcontrib>Zywina, David</creatorcontrib><title>Open image computations for elliptic curves over number fields</title><title>Research in number theory</title><addtitle>Res. number theory</addtitle><description>For a non-CM elliptic curve
E
defined over a number field
K
, the Galois action on its torsion points gives rise to a Galois representation
ρ
E
:
Gal
(
K
¯
/
K
)
→
GL
2
(
Z
^
)
that is unique up to isomorphism. A renowned theorem of Serre says that the image of
ρ
E
is an open, and hence finite index, subgroup of
GL
2
(
Z
^
)
. In an earlier work of the author, an algorithm was given that computed the image of
ρ
E
up to conjugacy in
GL
2
(
Z
^
)
in the special case
K
=
Q
. A fundamental ingredient of this earlier work was the Kronecker–Weber theorem whose conclusion fails for number fields
K
≠
Q
. We shall give an overview of an analogous algorithm for a general number field and work out the required group theory. We also give some bounds on the index in Serre’s theorem for a typical elliptic curve over a fixed number field.</description><subject>Mathematics</subject><subject>Mathematics and Statistics</subject><subject>Number Theory</subject><issn>2522-0160</issn><issn>2363-9555</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2025</creationdate><recordtype>article</recordtype><recordid>eNp9kMtqwzAQRUVpoSHND3SlH1A7GkWytSmU0BcEsmnXwpZHwcGxjWQH-vdVm66zuncxZ7gcxu4lPEiA4jGtwVolANcCQFsr8IotUBklrNb6OneNKEAauGWrlA4Auas1Ii7Y026knrfHak_cD8dxnqqpHfrEwxA5dV07Tq3nfo4nSnw4UeT9fKxzhJa6Jt2xm1B1iVb_uWRfry-fm3ex3b19bJ63wssCJ-GbKqCSAYysrS0bU2tfS6t0UdaEIegQGrCmrprSaCgLBdoEUpaUocqHRi0Znv_6OKQUKbgx5tHx20lwvxLcWYLLEtyfBIcZUmco5eN-T9Edhjn2eecl6gcobWBa</recordid><startdate>20250301</startdate><enddate>20250301</enddate><creator>Zywina, David</creator><general>Springer International Publishing</general><scope>AAYXX</scope><scope>CITATION</scope></search><sort><creationdate>20250301</creationdate><title>Open image computations for elliptic curves over number fields</title><author>Zywina, David</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-c172t-cdaf231f061b998d6b5cb193578be2ff5ffd096bad8650873056fe39e36eacfd3</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2025</creationdate><topic>Mathematics</topic><topic>Mathematics and Statistics</topic><topic>Number Theory</topic><toplevel>peer_reviewed</toplevel><toplevel>online_resources</toplevel><creatorcontrib>Zywina, David</creatorcontrib><collection>CrossRef</collection><jtitle>Research in number theory</jtitle></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext</fulltext></delivery><addata><au>Zywina, David</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>Open image computations for elliptic curves over number fields</atitle><jtitle>Research in number theory</jtitle><stitle>Res. number theory</stitle><date>2025-03-01</date><risdate>2025</risdate><volume>11</volume><issue>1</issue><artnum>1</artnum><issn>2522-0160</issn><eissn>2363-9555</eissn><abstract>For a non-CM elliptic curve
E
defined over a number field
K
, the Galois action on its torsion points gives rise to a Galois representation
ρ
E
:
Gal
(
K
¯
/
K
)
→
GL
2
(
Z
^
)
that is unique up to isomorphism. A renowned theorem of Serre says that the image of
ρ
E
is an open, and hence finite index, subgroup of
GL
2
(
Z
^
)
. In an earlier work of the author, an algorithm was given that computed the image of
ρ
E
up to conjugacy in
GL
2
(
Z
^
)
in the special case
K
=
Q
. A fundamental ingredient of this earlier work was the Kronecker–Weber theorem whose conclusion fails for number fields
K
≠
Q
. We shall give an overview of an analogous algorithm for a general number field and work out the required group theory. We also give some bounds on the index in Serre’s theorem for a typical elliptic curve over a fixed number field.</abstract><cop>Cham</cop><pub>Springer International Publishing</pub><doi>10.1007/s40993-024-00599-2</doi></addata></record> |
| fulltext | fulltext |
| identifier | ISSN: 2522-0160 |
| ispartof | Research in number theory, 2025-03, Vol.11 (1), Article 1 |
| issn | 2522-0160 2363-9555 |
| language | eng |
| recordid | cdi_crossref_primary_10_1007_s40993_024_00599_2 |
| source | Springer Nature |
| subjects | Mathematics Mathematics and Statistics Number Theory |
| title | Open image computations for elliptic curves over number fields |
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