{"id":"https://openalex.org/W7207558088","doi":"https://doi.org/10.48550/arxiv.2609.02592","title":"On the occurrence of congruence multiplicities between Ramanujan's theta functions","display_name":"On the occurrence of congruence multiplicities between Ramanujan's theta functions","publication_year":2026,"publication_date":"2026-09-02","ids":{"openalex":"https://openalex.org/W7207558088","doi":"https://doi.org/10.48550/arxiv.2609.02592"},"language":"en","primary_location":{"id":"pmh:oai:arXiv.org:2609.02592","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.02592","pdf_url":"https://arxiv.org/pdf/2609.02592","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},"volume":null,"type":"preprint","is_traditional":true,"indexed_in":["arxiv","datacite"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://arxiv.org/pdf/2609.02592","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5124102206","display_name":"Shane Chern","orcid":"https://orcid.org/0000-0002-8321-1895","observed_orcids":["https://orcid.org/0000-0002-8321-1895"]},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Chern, Shane","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"middle","author":{"id":"https://openalex.org/A5023334016","display_name":"Nicolas Allen Smoot","orcid":"https://orcid.org/0000-0002-2889-8831","observed_orcids":["https://orcid.org/0000-0002-2889-8831"]},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Smoot, Nicolas Allen","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"last","author":{"id":"https://openalex.org/A5025724291","display_name":"Dazhao Tang","orcid":"https://orcid.org/0000-0003-3281-6912","observed_orcids":["https://orcid.org/0000-0003-3281-6912"]},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Tang, Dazhao","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]}],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":0,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":null,"has_fulltext":true,"cited_by_count":0,"citation_normalized_percentile":null,"cited_by_percentile_year":null,"biblio":{"volume":null,"issue":null,"first_page":null,"last_page":null},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11428","display_name":"Advanced Mathematical Identities","score":1.0,"subfield":{"id":"https://openalex.org/subfields/2602","display_name":"Algebra and Number Theory"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},"topics":[{"id":"https://openalex.org/T11428","display_name":"Advanced Mathematical Identities","score":1.0,"subfield":{"id":"https://openalex.org/subfields/2602","display_name":"Algebra and Number Theory"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T11166","display_name":"Analytic Number Theory Research","score":0.0,"subfield":{"id":"https://openalex.org/subfields/2602","display_name":"Algebra and Number Theory"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T10061","display_name":"Algebraic Geometry and Number Theory","score":0.0,"subfield":{"id":"https://openalex.org/subfields/2608","display_name":"Geometry and Topology"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/ramanujan-s-theta-functions","display_name":"Ramanujan's theta functions","score":0.9610000252723694},{"id":"https://openalex.org/keywords/modular-forms","display_name":"modular forms","score":0.47600001096725464},{"id":"https://openalex.org/keywords/congruence-subgroups","display_name":"congruence subgroups","score":0.6610000133514404},{"id":"https://openalex.org/keywords/function-field-extensions","display_name":"function field extensions","score":0.7260000109672546}],"concepts":[{"id":"https://openalex.org/C9973445","wikidata":"https://www.wikidata.org/wiki/Q8349849","display_name":"Congruence relation","level":2,"score":0.9221000075340271},{"id":"https://openalex.org/C132074034","wikidata":"https://www.wikidata.org/wiki/Q154210","display_name":"Congruence (geometry)","level":2,"score":0.849399983882904},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.7441999912261963},{"id":"https://openalex.org/C54732982","wikidata":"https://www.wikidata.org/wiki/Q1415345","display_name":"Modulo","level":2,"score":0.6104000210762024},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.5893999934196472},{"id":"https://openalex.org/C112343008","wikidata":"https://www.wikidata.org/wiki/Q140744","display_name":"Arithmetic function","level":2,"score":0.45089998841285706},{"id":"https://openalex.org/C2780586882","wikidata":"https://www.wikidata.org/wiki/Q7520643","display_name":"Simple (philosophy)","level":2,"score":0.38119998574256897},{"id":"https://openalex.org/C105546189","wikidata":"https://www.wikidata.org/wiki/Q1154787","display_name":"Theta function","level":2,"score":0.3312999904155731},{"id":"https://openalex.org/C14036430","wikidata":"https://www.wikidata.org/wiki/Q3736076","display_name":"Function (biology)","level":2,"score":0.3203999996185303},{"id":"https://openalex.org/C9652623","wikidata":"https://www.wikidata.org/wiki/Q190109","display_name":"Field (mathematics)","level":2,"score":0.29600000381469727},{"id":"https://openalex.org/C136119220","wikidata":"https://www.wikidata.org/wiki/Q1000660","display_name":"Algebra over a field","level":2,"score":0.29280000925064087},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.29100000858306885},{"id":"https://openalex.org/C156004811","wikidata":"https://www.wikidata.org/wiki/Q2228257","display_name":"Multiplicity (mathematics)","level":2,"score":0.28040000796318054},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.2718000113964081},{"id":"https://openalex.org/C143724316","wikidata":"https://www.wikidata.org/wiki/Q312468","display_name":"Series (stratigraphy)","level":2,"score":0.26420000195503235},{"id":"https://openalex.org/C94375191","wikidata":"https://www.wikidata.org/wiki/Q11205","display_name":"Arithmetic","level":1,"score":0.2563999891281128}],"mesh":[],"locations_count":2,"locations":[{"id":"pmh:oai:arXiv.org:2609.02592","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.02592","pdf_url":"https://arxiv.org/pdf/2609.02592","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},{"id":"doi:10.48550/arxiv.2609.02592","doi":"https://doi.org/10.48550/arxiv.2609.02592","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2609.02592","pdf_url":null,"source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":null,"is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"Preprint"}],"best_oa_location":{"id":"pmh:oai:arXiv.org:2609.02592","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.02592","pdf_url":"https://arxiv.org/pdf/2609.02592","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},"sustainable_development_goals":[],"study_designs":[],"awards":[{"id":"https://openalex.org/G2235460512","display_name":"Multiplicities Between Modular Congruences","funder_award_id":"PAT2615425","funder_id":"https://openalex.org/F4320321181","funder_display_name":"Austrian Science Fund"},{"id":"https://openalex.org/G4232081210","display_name":null,"funder_award_id":"12201093","funder_id":"https://openalex.org/F4320321001","funder_display_name":"National Natural Science Foundation of China"},{"id":"https://openalex.org/G8590477180","display_name":null,"funder_award_id":"10.55776","funder_id":"https://openalex.org/F4320321181","funder_display_name":"Austrian Science Fund"}],"funders":[{"id":"https://openalex.org/F4320321001","display_name":"National Natural Science Foundation of China","ror":"https://ror.org/01h0zpd94"},{"id":"https://openalex.org/F4320321181","display_name":"Austrian Science Fund","ror":"https://ror.org/013tf3c58"},{"id":"https://openalex.org/F4320324805","display_name":"Chongqing Municipal Education Commission","ror":"https://ror.org/031nm5713"}],"has_content":{"pdf":true,"grobid_xml":true},"content_urls":{"pdf":"https://content.openalex.org/works/W7207558088.pdf","grobid_xml":"https://content.openalex.org/works/W7207558088.grobid-xml"},"referenced_works_count":0,"referenced_works":[],"related_works":["https://openalex.org/W4401933130","https://openalex.org/W2295868819","https://openalex.org/W7045429626","https://openalex.org/W2964953621","https://openalex.org/W2226257360","https://openalex.org/W7127106041","https://openalex.org/W4404169289","https://openalex.org/W2133181028","https://openalex.org/W4404306595","https://openalex.org/W2166739744"],"abstract_inverted_index":{"Recently,":[0],"the":[1,41,53,65,76,89,96],"first":[2],"and":[3,17,34,58,75,93],"third":[4],"authors":[5],"initiated":[6],"a":[7,101],"study":[8,66],"of":[9,26,32,67,104],"arithmetical":[10],"relationships":[11],"between":[12,52],"Ramanujan's":[13],"theta":[14,43],"functions":[15],"$\u03c6(-q)$":[16,57],"$\u03c8(q)$.":[18],"In":[19,95],"this":[20],"work,":[21],"we":[22,98],"prove":[23],"eight":[24],"families":[25,55],"internal":[27],"congruences":[28,117],"modulo":[29],"arbitrary":[30],"powers":[31],"$3$":[33],"$5$":[35],"for":[36,56],"infinite":[37],"series":[38],"related":[39],"to":[40,110,115],"two":[42],"functions.":[44],"We":[45,79],"also":[46],"show":[47,99],"that":[48,81],"there":[49],"exist":[50],"isomorphisms":[51,114],"congruence":[54,68,90],"$\u03c8(q)$,":[59],"which":[60],"can":[61],"be":[62],"realized":[63],"by":[64,72],"multiplicities":[69],"previously":[70],"studied":[71],"Garvan,":[73],"Sellers,":[74],"second":[77],"author.":[78],"believe":[80],"such":[82],"equivalences":[83],"are":[84],"exclusive,":[85],"at":[86],"least":[87],"on":[88],"subgroups":[91],"$\u0393_0(6)$":[92],"$\u0393_0(10)$.":[94],"end,":[97],"how":[100],"simple":[102],"manipulation":[103],"function":[105],"field":[106],"extensions":[107],"allows":[108],"us":[109],"predict":[111],"whether":[112],"additional":[113],"our":[116],"occur.":[118]},"counts_by_year":[],"updated_date":"2026-10-28T10:33:24.094632","created_date":"2026-09-04T00:00:00"}