The T-adic Galois representation is surjective for a positive density of Drinfeld modules
Let F q be the finite field with q ≥ 5 elements, A : = F q [ T ] and F : = F q ( T ) . Assume that q is odd and take | · | to be the absolute value at ∞ that is normalized by | T | = q . Given a pair w = ( g 1 , g 2 ) ∈ A 2 with g 2 ≠ 0 , consider the associated Drinfeld module ϕ w : A → A { τ } of...
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| Veröffentlicht in: | Research in number theory 2024-09, Vol.10 (3), Article 56 |
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| creator | Ray, Anwesh |
| description | Let
F
q
be the finite field with
q
≥
5
elements,
A
:
=
F
q
[
T
]
and
F
:
=
F
q
(
T
)
. Assume that
q
is odd and take
|
·
|
to be the absolute value at
∞
that is normalized by
|
T
|
=
q
. Given a pair
w
=
(
g
1
,
g
2
)
∈
A
2
with
g
2
≠
0
, consider the associated Drinfeld module
ϕ
w
:
A
→
A
{
τ
}
of rank 2 defined by
ϕ
T
w
=
T
+
g
1
τ
+
g
2
τ
2
. Fix integers
c
1
,
c
2
≥
1
and define
|
w
|
:
=
max
{
|
g
1
|
1
c
1
,
|
g
2
|
1
c
2
}
. I show that when ordered by height, there is a positive density of pairs
w
=
(
g
1
,
g
2
)
, such that the
T
-adic Galois representation attached to
ϕ
w
is surjective. |
| doi_str_mv | 10.1007/s40993-024-00541-6 |
| format | Article |
| fullrecord | <record><control><sourceid>proquest_cross</sourceid><recordid>TN_cdi_proquest_journals_3064985685</recordid><sourceformat>XML</sourceformat><sourcesystem>PC</sourcesystem><sourcerecordid>3064985685</sourcerecordid><originalsourceid>FETCH-LOGICAL-c200t-804737d2b1d15617d648ffb02740ce7a07283f1e57aa38257cfa853e2bb8b4373</originalsourceid><addsrcrecordid>eNp9kEtLxDAUhYMoOIzzB1wFXEdv3u1SxicMuBkXrkLaJtqh09SkFebfG6eCO1f33Ms558KH0CWFawqgb5KAsuQEmCAAUlCiTtCCccVJKaU8zVoyRoAqOEerlHYAWXPBGFugt-2Hw1tim7bGj7YLbcLRDdEl1492bEOP8yVNcefqsf1y2IeILR5Cao9r4_qsDjh4fBfb3ruuwfvQTJ1LF-jM2y651e9coteH--36iWxeHp_XtxtSM4CRFCA01w2raEOlorpRovC-AqYF1E5b0KzgnjqpreUFk7r2tpDcsaoqKsE1X6KruXeI4XNyaTS7MMU-vzQclCgLqbJ_idjsqmNIKTpvhtjubTwYCuaHopkpmkzRHCkalUN8DqVs7t9d_Kv-J_UNHMp0MA</addsrcrecordid><sourcetype>Aggregation Database</sourcetype><iscdi>true</iscdi><recordtype>article</recordtype><pqid>3064985685</pqid></control><display><type>article</type><title>The T-adic Galois representation is surjective for a positive density of Drinfeld modules</title><source>SpringerLink Journals - AutoHoldings</source><creator>Ray, Anwesh</creator><creatorcontrib>Ray, Anwesh</creatorcontrib><description>Let
F
q
be the finite field with
q
≥
5
elements,
A
:
=
F
q
[
T
]
and
F
:
=
F
q
(
T
)
. Assume that
q
is odd and take
|
·
|
to be the absolute value at
∞
that is normalized by
|
T
|
=
q
. Given a pair
w
=
(
g
1
,
g
2
)
∈
A
2
with
g
2
≠
0
, consider the associated Drinfeld module
ϕ
w
:
A
→
A
{
τ
}
of rank 2 defined by
ϕ
T
w
=
T
+
g
1
τ
+
g
2
τ
2
. Fix integers
c
1
,
c
2
≥
1
and define
|
w
|
:
=
max
{
|
g
1
|
1
c
1
,
|
g
2
|
1
c
2
}
. I show that when ordered by height, there is a positive density of pairs
w
=
(
g
1
,
g
2
)
, such that the
T
-adic Galois representation attached to
ϕ
w
is surjective.</description><identifier>ISSN: 2522-0160</identifier><identifier>EISSN: 2363-9555</identifier><identifier>DOI: 10.1007/s40993-024-00541-6</identifier><language>eng</language><publisher>Cham: Springer International Publishing</publisher><subject>Density ; Fields (mathematics) ; Mathematics ; Mathematics and Statistics ; Number Theory ; Representations</subject><ispartof>Research in number theory, 2024-09, Vol.10 (3), Article 56</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-c200t-804737d2b1d15617d648ffb02740ce7a07283f1e57aa38257cfa853e2bb8b4373</cites><orcidid>0000-0001-6946-1559</orcidid></display><links><openurl>$$Topenurl_article</openurl><openurlfulltext>$$Topenurlfull_article</openurlfulltext><thumbnail>$$Tsyndetics_thumb_exl</thumbnail><linktopdf>$$Uhttps://link.springer.com/content/pdf/10.1007/s40993-024-00541-6$$EPDF$$P50$$Gspringer$$H</linktopdf><linktohtml>$$Uhttps://link.springer.com/10.1007/s40993-024-00541-6$$EHTML$$P50$$Gspringer$$H</linktohtml><link.rule.ids>314,780,784,27924,27925,41488,42557,51319</link.rule.ids></links><search><creatorcontrib>Ray, Anwesh</creatorcontrib><title>The T-adic Galois representation is surjective for a positive density of Drinfeld modules</title><title>Research in number theory</title><addtitle>Res. number theory</addtitle><description>Let
F
q
be the finite field with
q
≥
5
elements,
A
:
=
F
q
[
T
]
and
F
:
=
F
q
(
T
)
. Assume that
q
is odd and take
|
·
|
to be the absolute value at
∞
that is normalized by
|
T
|
=
q
. Given a pair
w
=
(
g
1
,
g
2
)
∈
A
2
with
g
2
≠
0
, consider the associated Drinfeld module
ϕ
w
:
A
→
A
{
τ
}
of rank 2 defined by
ϕ
T
w
=
T
+
g
1
τ
+
g
2
τ
2
. Fix integers
c
1
,
c
2
≥
1
and define
|
w
|
:
=
max
{
|
g
1
|
1
c
1
,
|
g
2
|
1
c
2
}
. I show that when ordered by height, there is a positive density of pairs
w
=
(
g
1
,
g
2
)
, such that the
T
-adic Galois representation attached to
ϕ
w
is surjective.</description><subject>Density</subject><subject>Fields (mathematics)</subject><subject>Mathematics</subject><subject>Mathematics and Statistics</subject><subject>Number Theory</subject><subject>Representations</subject><issn>2522-0160</issn><issn>2363-9555</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2024</creationdate><recordtype>article</recordtype><recordid>eNp9kEtLxDAUhYMoOIzzB1wFXEdv3u1SxicMuBkXrkLaJtqh09SkFebfG6eCO1f33Ms558KH0CWFawqgb5KAsuQEmCAAUlCiTtCCccVJKaU8zVoyRoAqOEerlHYAWXPBGFugt-2Hw1tim7bGj7YLbcLRDdEl1492bEOP8yVNcefqsf1y2IeILR5Cao9r4_qsDjh4fBfb3ruuwfvQTJ1LF-jM2y651e9coteH--36iWxeHp_XtxtSM4CRFCA01w2raEOlorpRovC-AqYF1E5b0KzgnjqpreUFk7r2tpDcsaoqKsE1X6KruXeI4XNyaTS7MMU-vzQclCgLqbJ_idjsqmNIKTpvhtjubTwYCuaHopkpmkzRHCkalUN8DqVs7t9d_Kv-J_UNHMp0MA</recordid><startdate>20240901</startdate><enddate>20240901</enddate><creator>Ray, Anwesh</creator><general>Springer International Publishing</general><general>Springer Nature B.V</general><scope>AAYXX</scope><scope>CITATION</scope><orcidid>https://orcid.org/0000-0001-6946-1559</orcidid></search><sort><creationdate>20240901</creationdate><title>The T-adic Galois representation is surjective for a positive density of Drinfeld modules</title><author>Ray, Anwesh</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-c200t-804737d2b1d15617d648ffb02740ce7a07283f1e57aa38257cfa853e2bb8b4373</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2024</creationdate><topic>Density</topic><topic>Fields (mathematics)</topic><topic>Mathematics</topic><topic>Mathematics and Statistics</topic><topic>Number Theory</topic><topic>Representations</topic><toplevel>peer_reviewed</toplevel><toplevel>online_resources</toplevel><creatorcontrib>Ray, Anwesh</creatorcontrib><collection>CrossRef</collection><jtitle>Research in number theory</jtitle></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext</fulltext></delivery><addata><au>Ray, Anwesh</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>The T-adic Galois representation is surjective for a positive density of Drinfeld modules</atitle><jtitle>Research in number theory</jtitle><stitle>Res. number theory</stitle><date>2024-09-01</date><risdate>2024</risdate><volume>10</volume><issue>3</issue><artnum>56</artnum><issn>2522-0160</issn><eissn>2363-9555</eissn><abstract>Let
F
q
be the finite field with
q
≥
5
elements,
A
:
=
F
q
[
T
]
and
F
:
=
F
q
(
T
)
. Assume that
q
is odd and take
|
·
|
to be the absolute value at
∞
that is normalized by
|
T
|
=
q
. Given a pair
w
=
(
g
1
,
g
2
)
∈
A
2
with
g
2
≠
0
, consider the associated Drinfeld module
ϕ
w
:
A
→
A
{
τ
}
of rank 2 defined by
ϕ
T
w
=
T
+
g
1
τ
+
g
2
τ
2
. Fix integers
c
1
,
c
2
≥
1
and define
|
w
|
:
=
max
{
|
g
1
|
1
c
1
,
|
g
2
|
1
c
2
}
. I show that when ordered by height, there is a positive density of pairs
w
=
(
g
1
,
g
2
)
, such that the
T
-adic Galois representation attached to
ϕ
w
is surjective.</abstract><cop>Cham</cop><pub>Springer International Publishing</pub><doi>10.1007/s40993-024-00541-6</doi><orcidid>https://orcid.org/0000-0001-6946-1559</orcidid></addata></record> |
| fulltext | fulltext |
| identifier | ISSN: 2522-0160 |
| ispartof | Research in number theory, 2024-09, Vol.10 (3), Article 56 |
| issn | 2522-0160 2363-9555 |
| language | eng |
| recordid | cdi_proquest_journals_3064985685 |
| source | SpringerLink Journals - AutoHoldings |
| subjects | Density Fields (mathematics) Mathematics Mathematics and Statistics Number Theory Representations |
| title | The T-adic Galois representation is surjective for a positive density of Drinfeld modules |
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