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Theta series and number fields: theorems and experiments
Let d and n be positive integers and let K be a totally real number field of discriminant d and degree n . We construct a theta series θ K ∈ M d , n , where M d , n is a space of modular forms defined in terms of n and d . Moreover, if d is square free and n is at most 4 then θ K is a complete invar...
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| Published in: | The Ramanujan journal 2021-11, Vol.56 (2), p.613-630 |
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| Main Authors: | , , |
| Format: | Article |
| Language: | English |
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| container_end_page | 630 |
| container_issue | 2 |
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| container_title | The Ramanujan journal |
| container_volume | 56 |
| creator | Barquero-Sanchez, Adrian Mantilla-Soler, Guillermo Ryan, Nathan C. |
| description | Let
d
and
n
be positive integers and let
K
be a totally real number field of discriminant
d
and degree
n
. We construct a theta series
θ
K
∈
M
d
,
n
, where
M
d
,
n
is a space of modular forms defined in terms of
n
and
d
. Moreover, if
d
is square free and
n
is at most 4 then
θ
K
is a complete invariant for
K
. We also investigate whether or not the collection of
θ
-series, associated to the set of isomorphism classes of quartic number fields of a fixed squarefree discriminant
d
, is a linearly independent subset of
M
d
,
4
. This is known to be true if the degree of the number field is less than or equal to 3. We give computational and heuristic evidence suggesting that in degree 4 these theta series should be independent as well. |
| doi_str_mv | 10.1007/s11139-021-00394-y |
| format | article |
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d
and
n
be positive integers and let
K
be a totally real number field of discriminant
d
and degree
n
. We construct a theta series
θ
K
∈
M
d
,
n
, where
M
d
,
n
is a space of modular forms defined in terms of
n
and
d
. Moreover, if
d
is square free and
n
is at most 4 then
θ
K
is a complete invariant for
K
. We also investigate whether or not the collection of
θ
-series, associated to the set of isomorphism classes of quartic number fields of a fixed squarefree discriminant
d
, is a linearly independent subset of
M
d
,
4
. This is known to be true if the degree of the number field is less than or equal to 3. We give computational and heuristic evidence suggesting that in degree 4 these theta series should be independent as well.</description><identifier>ISSN: 1382-4090</identifier><identifier>EISSN: 1572-9303</identifier><identifier>DOI: 10.1007/s11139-021-00394-y</identifier><language>eng</language><publisher>New York: Springer US</publisher><subject>Combinatorics ; Field Theory and Polynomials ; Fourier Analysis ; Functions of a Complex Variable ; Isomorphism ; Mathematics ; Mathematics and Statistics ; Number Theory ; Numbers</subject><ispartof>The Ramanujan journal, 2021-11, Vol.56 (2), p.613-630</ispartof><rights>The Author(s), under exclusive licence to Springer Science+Business Media, LLC part of Springer Nature 2021</rights><rights>The Author(s), under exclusive licence to Springer Science+Business Media, LLC part of Springer Nature 2021.</rights><lds50>peer_reviewed</lds50><oa>free_for_read</oa><woscitedreferencessubscribed>false</woscitedreferencessubscribed><cites>FETCH-LOGICAL-c314t-bb2a777f441ef45c00a21582697a112f549a718df03780631f0328bce0a22e1c3</cites><orcidid>0000-0001-7847-2938 ; 0000-0003-4947-586X</orcidid></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>Barquero-Sanchez, Adrian</creatorcontrib><creatorcontrib>Mantilla-Soler, Guillermo</creatorcontrib><creatorcontrib>Ryan, Nathan C.</creatorcontrib><title>Theta series and number fields: theorems and experiments</title><title>The Ramanujan journal</title><addtitle>Ramanujan J</addtitle><description>Let
d
and
n
be positive integers and let
K
be a totally real number field of discriminant
d
and degree
n
. We construct a theta series
θ
K
∈
M
d
,
n
, where
M
d
,
n
is a space of modular forms defined in terms of
n
and
d
. Moreover, if
d
is square free and
n
is at most 4 then
θ
K
is a complete invariant for
K
. We also investigate whether or not the collection of
θ
-series, associated to the set of isomorphism classes of quartic number fields of a fixed squarefree discriminant
d
, is a linearly independent subset of
M
d
,
4
. This is known to be true if the degree of the number field is less than or equal to 3. We give computational and heuristic evidence suggesting that in degree 4 these theta series should be independent as well.</description><subject>Combinatorics</subject><subject>Field Theory and Polynomials</subject><subject>Fourier Analysis</subject><subject>Functions of a Complex Variable</subject><subject>Isomorphism</subject><subject>Mathematics</subject><subject>Mathematics and Statistics</subject><subject>Number Theory</subject><subject>Numbers</subject><issn>1382-4090</issn><issn>1572-9303</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2021</creationdate><recordtype>article</recordtype><recordid>eNp9kEFLAzEQhYMoWKt_wNOC5-hMkt0k3qRoFQpe6jlkdye2pd2tyRbsvze6gjdPMzDfe_N4jF0j3CKAvkuIKC0HgRxAWsWPJ2yCpRbcSpCneZdGcAUWztlFShsAUCD1hJnligZfJIprSoXv2qI77GqKRVjTtk33xbCiPtJuvNHnPoM76oZ0yc6C3ya6-p1T9vb0uJw988Xr_GX2sOCNRDXwuhZeax2UQgqqbAC8wNKIymqPKEKprNdo2pDTGKgk5kWYuqHMCcJGTtnN6LuP_ceB0uA2_SF2-aUT2ac0qqpspsRINbFPKVJw-5zTx6NDcN8NubEhlxtyPw25YxbJUZQy3L1T_LP-R_UFeHtnzw</recordid><startdate>20211101</startdate><enddate>20211101</enddate><creator>Barquero-Sanchez, Adrian</creator><creator>Mantilla-Soler, Guillermo</creator><creator>Ryan, Nathan C.</creator><general>Springer US</general><general>Springer Nature B.V</general><scope>AAYXX</scope><scope>CITATION</scope><orcidid>https://orcid.org/0000-0001-7847-2938</orcidid><orcidid>https://orcid.org/0000-0003-4947-586X</orcidid></search><sort><creationdate>20211101</creationdate><title>Theta series and number fields: theorems and experiments</title><author>Barquero-Sanchez, Adrian ; Mantilla-Soler, Guillermo ; Ryan, Nathan C.</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-c314t-bb2a777f441ef45c00a21582697a112f549a718df03780631f0328bce0a22e1c3</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2021</creationdate><topic>Combinatorics</topic><topic>Field Theory and Polynomials</topic><topic>Fourier Analysis</topic><topic>Functions of a Complex Variable</topic><topic>Isomorphism</topic><topic>Mathematics</topic><topic>Mathematics and Statistics</topic><topic>Number Theory</topic><topic>Numbers</topic><toplevel>peer_reviewed</toplevel><toplevel>online_resources</toplevel><creatorcontrib>Barquero-Sanchez, Adrian</creatorcontrib><creatorcontrib>Mantilla-Soler, Guillermo</creatorcontrib><creatorcontrib>Ryan, Nathan C.</creatorcontrib><collection>CrossRef</collection><jtitle>The Ramanujan journal</jtitle></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext</fulltext></delivery><addata><au>Barquero-Sanchez, Adrian</au><au>Mantilla-Soler, Guillermo</au><au>Ryan, Nathan C.</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>Theta series and number fields: theorems and experiments</atitle><jtitle>The Ramanujan journal</jtitle><stitle>Ramanujan J</stitle><date>2021-11-01</date><risdate>2021</risdate><volume>56</volume><issue>2</issue><spage>613</spage><epage>630</epage><pages>613-630</pages><issn>1382-4090</issn><eissn>1572-9303</eissn><abstract>Let
d
and
n
be positive integers and let
K
be a totally real number field of discriminant
d
and degree
n
. We construct a theta series
θ
K
∈
M
d
,
n
, where
M
d
,
n
is a space of modular forms defined in terms of
n
and
d
. Moreover, if
d
is square free and
n
is at most 4 then
θ
K
is a complete invariant for
K
. We also investigate whether or not the collection of
θ
-series, associated to the set of isomorphism classes of quartic number fields of a fixed squarefree discriminant
d
, is a linearly independent subset of
M
d
,
4
. This is known to be true if the degree of the number field is less than or equal to 3. We give computational and heuristic evidence suggesting that in degree 4 these theta series should be independent as well.</abstract><cop>New York</cop><pub>Springer US</pub><doi>10.1007/s11139-021-00394-y</doi><tpages>18</tpages><orcidid>https://orcid.org/0000-0001-7847-2938</orcidid><orcidid>https://orcid.org/0000-0003-4947-586X</orcidid><oa>free_for_read</oa></addata></record> |
| fulltext | fulltext |
| identifier | ISSN: 1382-4090 |
| ispartof | The Ramanujan journal, 2021-11, Vol.56 (2), p.613-630 |
| issn | 1382-4090 1572-9303 |
| language | eng |
| recordid | cdi_proquest_journals_2582584669 |
| source | Springer Nature |
| subjects | Combinatorics Field Theory and Polynomials Fourier Analysis Functions of a Complex Variable Isomorphism Mathematics Mathematics and Statistics Number Theory Numbers |
| title | Theta series and number fields: theorems and experiments |
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