Modular forms and differential operators View Full Text


Ontology type: schema:ScholarlyArticle     


Article Info

DATE

1994-02

AUTHORS

Don Zagier

ABSTRACT

In 1956, Rankin described which polynomials in the derivatives of modular forms are again modular forms, and in 1977, H Cohen defined for eachn ≥ 0 a bilinear operation which assigns to two modular formsf andg of weightk andl a modular form [f, g]n of weightk +l + 2n. In the present paper we study these “Rankin-Cohen brackets” from two points of view. On the one hand we give various explanations of their modularity and various algebraic relations among them by relating the modular form theory to the theories of theta series, of Jacobi forms, and of pseudodifferential operators. In a different direction, we study the abstract algebraic structure (“RC algebra”) consisting of a graded vector space together with a collection of bilinear operations [,]n of degree + 2n satisfying all of the axioms of the Rankin-Cohen brackets. Under certain hypotheses, these turn out to be equivalent to commutative graded algebras together with a derivationS of degree 2 and an element Φ of degree 4, up to the equivalence relation (∂,Φ) ~ (∂ - ϕE, Φ - ϕ2 + ∂(ϕ)) where ϕ is an element of degree 2 andE is the Fuler operator (= multiplication by the degree). More... »

PAGES

57-75

References to SciGraph publications

Identifiers

URI

http://scigraph.springernature.com/pub.10.1007/bf02830874

DOI

http://dx.doi.org/10.1007/bf02830874

DIMENSIONS

https://app.dimensions.ai/details/publication/pub.1002910991


Indexing Status Check whether this publication has been indexed by Scopus and Web Of Science using the SN Indexing Status Tool
Incoming Citations Browse incoming citations for this publication using opencitations.net

JSON-LD is the canonical representation for SciGraph data.

TIP: You can open this SciGraph record using an external JSON-LD service: JSON-LD Playground Google SDTT

[
  {
    "@context": "https://springernature.github.io/scigraph/jsonld/sgcontext.json", 
    "about": [
      {
        "id": "http://purl.org/au-research/vocabulary/anzsrc-for/2008/0101", 
        "inDefinedTermSet": "http://purl.org/au-research/vocabulary/anzsrc-for/2008/", 
        "name": "Pure Mathematics", 
        "type": "DefinedTerm"
      }, 
      {
        "id": "http://purl.org/au-research/vocabulary/anzsrc-for/2008/01", 
        "inDefinedTermSet": "http://purl.org/au-research/vocabulary/anzsrc-for/2008/", 
        "name": "Mathematical Sciences", 
        "type": "DefinedTerm"
      }
    ], 
    "author": [
      {
        "affiliation": {
          "alternateName": "Max Planck Institute for Mathematics", 
          "id": "https://www.grid.ac/institutes/grid.461798.5", 
          "name": [
            "Max-Planck-Institut f\u00fcr Mathematik, Gottfried-Claren Str. 26, 53225, Bonn, Germany"
          ], 
          "type": "Organization"
        }, 
        "familyName": "Zagier", 
        "givenName": "Don", 
        "id": "sg:person.014615320431.19", 
        "sameAs": [
          "https://app.dimensions.ai/discover/publication?and_facet_researcher=ur.014615320431.19"
        ], 
        "type": "Person"
      }
    ], 
    "citation": [
      {
        "id": "https://doi.org/10.1073/pnas.83.10.3068", 
        "sameAs": [
          "https://app.dimensions.ai/details/publication/pub.1014697058"
        ], 
        "type": "CreativeWork"
      }, 
      {
        "id": "sg:pub.10.1007/bf01436180", 
        "sameAs": [
          "https://app.dimensions.ai/details/publication/pub.1043838697", 
          "https://doi.org/10.1007/bf01436180"
        ], 
        "type": "CreativeWork"
      }, 
      {
        "id": "sg:pub.10.1007/bf01436180", 
        "sameAs": [
          "https://app.dimensions.ai/details/publication/pub.1043838697", 
          "https://doi.org/10.1007/bf01436180"
        ], 
        "type": "CreativeWork"
      }, 
      {
        "id": "sg:pub.10.1007/bfb0095644", 
        "sameAs": [
          "https://app.dimensions.ai/details/publication/pub.1051025175", 
          "https://doi.org/10.1007/bfb0095644"
        ], 
        "type": "CreativeWork"
      }, 
      {
        "id": "sg:pub.10.1007/bfb0095644", 
        "sameAs": [
          "https://app.dimensions.ai/details/publication/pub.1051025175", 
          "https://doi.org/10.1007/bfb0095644"
        ], 
        "type": "CreativeWork"
      }, 
      {
        "id": "https://doi.org/10.4064/aa-27-1-505-519", 
        "sameAs": [
          "https://app.dimensions.ai/details/publication/pub.1091814320"
        ], 
        "type": "CreativeWork"
      }
    ], 
    "datePublished": "1994-02", 
    "datePublishedReg": "1994-02-01", 
    "description": "In 1956, Rankin described which polynomials in the derivatives of modular forms are again modular forms, and in 1977, H Cohen defined for eachn \u2265 0 a bilinear operation which assigns to two modular formsf andg of weightk andl a modular form [f, g]n of weightk +l + 2n. In the present paper we study these \u201cRankin-Cohen brackets\u201d from two points of view. On the one hand we give various explanations of their modularity and various algebraic relations among them by relating the modular form theory to the theories of theta series, of Jacobi forms, and of pseudodifferential operators. In a different direction, we study the abstract algebraic structure (\u201cRC algebra\u201d) consisting of a graded vector space together with a collection of bilinear operations [,]n of degree + 2n satisfying all of the axioms of the Rankin-Cohen brackets. Under certain hypotheses, these turn out to be equivalent to commutative graded algebras together with a derivationS of degree 2 and an element \u03a6 of degree 4, up to the equivalence relation (\u2202,\u03a6) ~ (\u2202 - \u03d5E, \u03a6 - \u03d52 + \u2202(\u03d5)) where \u03d5 is an element of degree 2 andE is the Fuler operator (= multiplication by the degree).", 
    "genre": "research_article", 
    "id": "sg:pub.10.1007/bf02830874", 
    "inLanguage": [
      "en"
    ], 
    "isAccessibleForFree": false, 
    "isPartOf": [
      {
        "id": "sg:journal.1320093", 
        "issn": [
          "2008-1359", 
          "2251-7456"
        ], 
        "name": "Proceedings - Mathematical Sciences", 
        "type": "Periodical"
      }, 
      {
        "issueNumber": "1", 
        "type": "PublicationIssue"
      }, 
      {
        "type": "PublicationVolume", 
        "volumeNumber": "104"
      }
    ], 
    "name": "Modular forms and differential operators", 
    "pagination": "57-75", 
    "productId": [
      {
        "name": "readcube_id", 
        "type": "PropertyValue", 
        "value": [
          "0ddd5305253cc6c98a61727dfe6537b020a44642d296b5f3eea3c68f8be4e6a8"
        ]
      }, 
      {
        "name": "doi", 
        "type": "PropertyValue", 
        "value": [
          "10.1007/bf02830874"
        ]
      }, 
      {
        "name": "dimensions_id", 
        "type": "PropertyValue", 
        "value": [
          "pub.1002910991"
        ]
      }
    ], 
    "sameAs": [
      "https://doi.org/10.1007/bf02830874", 
      "https://app.dimensions.ai/details/publication/pub.1002910991"
    ], 
    "sdDataset": "articles", 
    "sdDatePublished": "2019-04-11T13:35", 
    "sdLicense": "https://scigraph.springernature.com/explorer/license/", 
    "sdPublisher": {
      "name": "Springer Nature - SN SciGraph project", 
      "type": "Organization"
    }, 
    "sdSource": "s3://com-uberresearch-data-dimensions-target-20181106-alternative/cleanup/v134/2549eaecd7973599484d7c17b260dba0a4ecb94b/merge/v9/a6c9fde33151104705d4d7ff012ea9563521a3ce/jats-lookup/v90/0000000370_0000000370/records_46777_00000000.jsonl", 
    "type": "ScholarlyArticle", 
    "url": "http://link.springer.com/10.1007/BF02830874"
  }
]
 

Download the RDF metadata as:  json-ld nt turtle xml License info

HOW TO GET THIS DATA PROGRAMMATICALLY:

JSON-LD is a popular format for linked data which is fully compatible with JSON.

curl -H 'Accept: application/ld+json' 'https://scigraph.springernature.com/pub.10.1007/bf02830874'

N-Triples is a line-based linked data format ideal for batch operations.

curl -H 'Accept: application/n-triples' 'https://scigraph.springernature.com/pub.10.1007/bf02830874'

Turtle is a human-readable linked data format.

curl -H 'Accept: text/turtle' 'https://scigraph.springernature.com/pub.10.1007/bf02830874'

RDF/XML is a standard XML format for linked data.

curl -H 'Accept: application/rdf+xml' 'https://scigraph.springernature.com/pub.10.1007/bf02830874'


 

This table displays all metadata directly associated to this object as RDF triples.

75 TRIPLES      21 PREDICATES      31 URIs      19 LITERALS      7 BLANK NODES

Subject Predicate Object
1 sg:pub.10.1007/bf02830874 schema:about anzsrc-for:01
2 anzsrc-for:0101
3 schema:author N77e8fda2d1ef4397809bdc5ce5963128
4 schema:citation sg:pub.10.1007/bf01436180
5 sg:pub.10.1007/bfb0095644
6 https://doi.org/10.1073/pnas.83.10.3068
7 https://doi.org/10.4064/aa-27-1-505-519
8 schema:datePublished 1994-02
9 schema:datePublishedReg 1994-02-01
10 schema:description In 1956, Rankin described which polynomials in the derivatives of modular forms are again modular forms, and in 1977, H Cohen defined for eachn ≥ 0 a bilinear operation which assigns to two modular formsf andg of weightk andl a modular form [f, g]n of weightk +l + 2n. In the present paper we study these “Rankin-Cohen brackets” from two points of view. On the one hand we give various explanations of their modularity and various algebraic relations among them by relating the modular form theory to the theories of theta series, of Jacobi forms, and of pseudodifferential operators. In a different direction, we study the abstract algebraic structure (“RC algebra”) consisting of a graded vector space together with a collection of bilinear operations [,]n of degree + 2n satisfying all of the axioms of the Rankin-Cohen brackets. Under certain hypotheses, these turn out to be equivalent to commutative graded algebras together with a derivationS of degree 2 and an element Φ of degree 4, up to the equivalence relation (∂,Φ) ~ (∂ - ϕE, Φ - ϕ2 + ∂(ϕ)) where ϕ is an element of degree 2 andE is the Fuler operator (= multiplication by the degree).
11 schema:genre research_article
12 schema:inLanguage en
13 schema:isAccessibleForFree false
14 schema:isPartOf N5e5c624c817542f9b936a19ddbf773d8
15 Nb98bce9d650245b6893a336fac5ac036
16 sg:journal.1320093
17 schema:name Modular forms and differential operators
18 schema:pagination 57-75
19 schema:productId N0b3e12ab16bc4168ad512a7a92f3f7dd
20 N90f609711a2f403a8ccca24a100a7cad
21 Ndd96089576254887a95c225b5082dd77
22 schema:sameAs https://app.dimensions.ai/details/publication/pub.1002910991
23 https://doi.org/10.1007/bf02830874
24 schema:sdDatePublished 2019-04-11T13:35
25 schema:sdLicense https://scigraph.springernature.com/explorer/license/
26 schema:sdPublisher N56428a0dc58e4aeeafcedd076a601b7b
27 schema:url http://link.springer.com/10.1007/BF02830874
28 sgo:license sg:explorer/license/
29 sgo:sdDataset articles
30 rdf:type schema:ScholarlyArticle
31 N0b3e12ab16bc4168ad512a7a92f3f7dd schema:name dimensions_id
32 schema:value pub.1002910991
33 rdf:type schema:PropertyValue
34 N56428a0dc58e4aeeafcedd076a601b7b schema:name Springer Nature - SN SciGraph project
35 rdf:type schema:Organization
36 N5e5c624c817542f9b936a19ddbf773d8 schema:volumeNumber 104
37 rdf:type schema:PublicationVolume
38 N77e8fda2d1ef4397809bdc5ce5963128 rdf:first sg:person.014615320431.19
39 rdf:rest rdf:nil
40 N90f609711a2f403a8ccca24a100a7cad schema:name doi
41 schema:value 10.1007/bf02830874
42 rdf:type schema:PropertyValue
43 Nb98bce9d650245b6893a336fac5ac036 schema:issueNumber 1
44 rdf:type schema:PublicationIssue
45 Ndd96089576254887a95c225b5082dd77 schema:name readcube_id
46 schema:value 0ddd5305253cc6c98a61727dfe6537b020a44642d296b5f3eea3c68f8be4e6a8
47 rdf:type schema:PropertyValue
48 anzsrc-for:01 schema:inDefinedTermSet anzsrc-for:
49 schema:name Mathematical Sciences
50 rdf:type schema:DefinedTerm
51 anzsrc-for:0101 schema:inDefinedTermSet anzsrc-for:
52 schema:name Pure Mathematics
53 rdf:type schema:DefinedTerm
54 sg:journal.1320093 schema:issn 2008-1359
55 2251-7456
56 schema:name Proceedings - Mathematical Sciences
57 rdf:type schema:Periodical
58 sg:person.014615320431.19 schema:affiliation https://www.grid.ac/institutes/grid.461798.5
59 schema:familyName Zagier
60 schema:givenName Don
61 schema:sameAs https://app.dimensions.ai/discover/publication?and_facet_researcher=ur.014615320431.19
62 rdf:type schema:Person
63 sg:pub.10.1007/bf01436180 schema:sameAs https://app.dimensions.ai/details/publication/pub.1043838697
64 https://doi.org/10.1007/bf01436180
65 rdf:type schema:CreativeWork
66 sg:pub.10.1007/bfb0095644 schema:sameAs https://app.dimensions.ai/details/publication/pub.1051025175
67 https://doi.org/10.1007/bfb0095644
68 rdf:type schema:CreativeWork
69 https://doi.org/10.1073/pnas.83.10.3068 schema:sameAs https://app.dimensions.ai/details/publication/pub.1014697058
70 rdf:type schema:CreativeWork
71 https://doi.org/10.4064/aa-27-1-505-519 schema:sameAs https://app.dimensions.ai/details/publication/pub.1091814320
72 rdf:type schema:CreativeWork
73 https://www.grid.ac/institutes/grid.461798.5 schema:alternateName Max Planck Institute for Mathematics
74 schema:name Max-Planck-Institut für Mathematik, Gottfried-Claren Str. 26, 53225, Bonn, Germany
75 rdf:type schema:Organization
 




Preview window. Press ESC to close (or click here)


...