{"id":"https://openalex.org/W7220745832","doi":"https://doi.org/10.48550/arxiv.2610.06015","title":"When the BSD Real Period Remembers Elliptic Curves over $\\mathbb{Q}$","display_name":"When the BSD Real Period Remembers Elliptic Curves over $\\mathbb{Q}$","publication_year":2026,"publication_date":"2026-10-05","ids":{"openalex":"https://openalex.org/W7220745832","doi":"https://doi.org/10.48550/arxiv.2610.06015"},"language":"en","primary_location":{"id":"pmh:oai:arXiv.org:2610.06015","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2610.06015","pdf_url":"https://arxiv.org/pdf/2610.06015","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":null,"license_id":null,"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/2610.06015","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5053882187","display_name":"Asuka Shiga","orcid":null,"observed_orcids":[]},"institutions":[{"id":"https://openalex.org/I201537933","display_name":"Tohoku University","ror":"https://ror.org/01dq60k83","country_code":"JP","type":"education","lineage":["https://openalex.org/I201537933"]}],"countries":["JP"],"is_corresponding":true,"raw_author_name":"Shiga, Asuka","raw_affiliation_strings":["Mathematical Institute , Graduate School of Science , Tohoku University , 6-3 Aramaki Aza Aoba , Aoba-ku , Sendai , Miyagi 980-8578 , Japan"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Mathematical Institute , Graduate School of Science , Tohoku University , 6-3 Aramaki Aza Aoba , Aoba-ku , Sendai , Miyagi 980-8578 , Japan","institution_ids":["https://openalex.org/I201537933"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":1,"corresponding_author_ids":["https://openalex.org/A5053882187"],"corresponding_institution_ids":["https://openalex.org/I201537933"],"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/T10061","display_name":"Algebraic Geometry and Number Theory","score":0.7217000126838684,"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"}},"topics":[{"id":"https://openalex.org/T10061","display_name":"Algebraic Geometry and Number Theory","score":0.7217000126838684,"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"}},{"id":"https://openalex.org/T11680","display_name":"Advanced Algebra and Geometry","score":0.2207999974489212,"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":0.036400001496076584,"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"}}],"keywords":[{"id":"https://openalex.org/keywords/birch-swinnerton-dyer-conjecture","display_name":"Birch and Swinnerton-Dyer conjecture","score":0.9620000123977661},{"id":"https://openalex.org/keywords/elliptic-curves-over-number-fields","display_name":"elliptic curves over number fields","score":0.8270000219345093},{"id":"https://openalex.org/keywords/tate-shafarevich-group","display_name":"Tate-Shafarevich group","score":0.9769999980926514},{"id":"https://openalex.org/keywords/isogeny-classes","display_name":"isogeny classes","score":0.8199999928474426},{"id":"https://openalex.org/keywords/quadratic-twists","display_name":"quadratic twists","score":0.824999988079071}],"concepts":[{"id":"https://openalex.org/C179603306","wikidata":"https://www.wikidata.org/wiki/Q268493","display_name":"Elliptic curve","level":2,"score":0.7342000007629395},{"id":"https://openalex.org/C131182338","wikidata":"https://www.wikidata.org/wiki/Q7858596","display_name":"Twists of curves","level":5,"score":0.6815999746322632},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.6762999892234802},{"id":"https://openalex.org/C73683783","wikidata":"https://www.wikidata.org/wiki/Q2728886","display_name":"Modular elliptic curve","level":4,"score":0.6638000011444092},{"id":"https://openalex.org/C104993295","wikidata":"https://www.wikidata.org/wiki/Q17023345","display_name":"Hessian form of an elliptic curve","level":5,"score":0.638700008392334},{"id":"https://openalex.org/C2780990831","wikidata":"https://www.wikidata.org/wiki/Q319141","display_name":"Conjecture","level":2,"score":0.5559999942779541},{"id":"https://openalex.org/C2781291010","wikidata":"https://www.wikidata.org/wiki/Q178580","display_name":"Period (music)","level":2,"score":0.5382999777793884},{"id":"https://openalex.org/C121444067","wikidata":"https://www.wikidata.org/wiki/Q2835817","display_name":"Schoof's algorithm","level":4,"score":0.5306000113487244},{"id":"https://openalex.org/C78397625","wikidata":"https://www.wikidata.org/wiki/Q192487","display_name":"Discriminant","level":2,"score":0.5217000246047974},{"id":"https://openalex.org/C162355417","wikidata":"https://www.wikidata.org/wiki/Q2993331","display_name":"Sato\u2013Tate conjecture","level":5,"score":0.5034999847412109},{"id":"https://openalex.org/C157567686","wikidata":"https://www.wikidata.org/wiki/Q7644139","display_name":"Supersingular elliptic curve","level":3,"score":0.49399998784065247},{"id":"https://openalex.org/C23230895","wikidata":"https://www.wikidata.org/wiki/Q104752","display_name":"Coprime integers","level":2,"score":0.40630000829696655},{"id":"https://openalex.org/C97137487","wikidata":"https://www.wikidata.org/wiki/Q729138","display_name":"Integer (computer science)","level":2,"score":0.4020000100135803},{"id":"https://openalex.org/C37055786","wikidata":"https://www.wikidata.org/wiki/Q7843607","display_name":"Tripling-oriented Doche\u2013Icart\u2013Kohel curve","level":5,"score":0.39239999651908875},{"id":"https://openalex.org/C41608201","wikidata":"https://www.wikidata.org/wiki/Q980509","display_name":"Embedding","level":2,"score":0.3889000117778778},{"id":"https://openalex.org/C164810661","wikidata":"https://www.wikidata.org/wiki/Q1358313","display_name":"Ring of integers","level":3,"score":0.35190001130104065},{"id":"https://openalex.org/C129844170","wikidata":"https://www.wikidata.org/wiki/Q41299","display_name":"Quadratic equation","level":2,"score":0.3483000099658966},{"id":"https://openalex.org/C94020503","wikidata":"https://www.wikidata.org/wiki/Q12916","display_name":"Real number","level":2,"score":0.3467000126838684},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.3458999991416931},{"id":"https://openalex.org/C12657307","wikidata":"https://www.wikidata.org/wiki/Q616608","display_name":"Algebraic number field","level":2,"score":0.33889999985694885},{"id":"https://openalex.org/C117121985","wikidata":"https://www.wikidata.org/wiki/Q78380999","display_name":"Elliptic curve point multiplication","level":3,"score":0.3172000050544739},{"id":"https://openalex.org/C12516240","wikidata":"https://www.wikidata.org/wiki/Q5641621","display_name":"Half-period ratio","level":5,"score":0.3075000047683716},{"id":"https://openalex.org/C203436722","wikidata":"https://www.wikidata.org/wiki/Q902950","display_name":"Isomorphism (crystallography)","level":3,"score":0.29829999804496765},{"id":"https://openalex.org/C171036898","wikidata":"https://www.wikidata.org/wiki/Q256355","display_name":"Equivariant map","level":2,"score":0.29679998755455017},{"id":"https://openalex.org/C2777212361","wikidata":"https://www.wikidata.org/wiki/Q5127848","display_name":"Class (philosophy)","level":2,"score":0.29670000076293945},{"id":"https://openalex.org/C2781311116","wikidata":"https://www.wikidata.org/wiki/Q83306","display_name":"Group (periodic table)","level":2,"score":0.27000001072883606},{"id":"https://openalex.org/C9652623","wikidata":"https://www.wikidata.org/wiki/Q190109","display_name":"Field (mathematics)","level":2,"score":0.26589998602867126},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.26489999890327454},{"id":"https://openalex.org/C75432250","wikidata":"https://www.wikidata.org/wiki/Q29654355","display_name":"Real line","level":2,"score":0.26420000195503235},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.2621999979019165},{"id":"https://openalex.org/C94375191","wikidata":"https://www.wikidata.org/wiki/Q11205","display_name":"Arithmetic","level":1,"score":0.2554999887943268},{"id":"https://openalex.org/C2776196297","wikidata":"https://www.wikidata.org/wiki/Q17138781","display_name":"Twist","level":2,"score":0.2540000081062317}],"mesh":[],"locations_count":2,"locations":[{"id":"pmh:oai:arXiv.org:2610.06015","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2610.06015","pdf_url":"https://arxiv.org/pdf/2610.06015","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":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},{"id":"doi:10.48550/arxiv.2610.06015","doi":"https://doi.org/10.48550/arxiv.2610.06015","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2610.06015","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":null,"license_id":null,"version":null,"is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"Preprint"}],"best_oa_location":{"id":"pmh:oai:arXiv.org:2610.06015","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2610.06015","pdf_url":"https://arxiv.org/pdf/2610.06015","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":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},"sustainable_development_goals":[],"study_designs":[],"awards":[],"funders":[],"has_content":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W7220745832.pdf","grobid_xml":"https://content.openalex.org/works/W7220745832.grobid-xml"},"referenced_works_count":1,"referenced_works":["https://openalex.org/W4416050003"],"related_works":["https://openalex.org/W620905233","https://openalex.org/W2076619945","https://openalex.org/W2768161846","https://openalex.org/W2059324275","https://openalex.org/W2964297629","https://openalex.org/W2147662267","https://openalex.org/W1970955288","https://openalex.org/W7204624899","https://openalex.org/W2326710204","https://openalex.org/W42860055"],"abstract_inverted_index":{"We":[0,66,91],"study":[1],"the":[2,6,10,16,30,36,44,77,86,112,120,124,139,152,164],"extent":[3],"to":[4,25,83,104,138,158],"which":[5],"invariants":[7],"appearing":[8],"in":[9],"Birch":[11],"and":[12,19,35,59,119],"Swinnerton-Dyer":[13],"conjecture":[14],"remember":[15,43],"elliptic":[17,45,72,96],"curve,":[18],"we":[20],"give":[21],"several":[22],"affirmative":[23],"answers":[24],"this":[26],"question.":[27],"For":[28],"example,":[29],"real":[31,57,78,110,121,153],"period":[32,79,122],"$\u03a9_E":[33],"=\\frac{\u0393(1/4)^2}{2\\sqrt\u03c0\\,17^{1/4}}$":[34],"Tate--Shafarevich":[37],"group":[38],"$":[39],"\\Sha(E/\\mathbb{Q})\\cong":[40],"\\mathbb{Z}/2\\mathbb{Z}\\times":[41],"\\mathbb{Z}/2\\mathbb{Z}$":[42],"curve":[46,73,97],"$E:":[47],"y^2=x^3+17x$.":[48],"Let":[49],"$K$":[50],"be":[51],"a":[52,56,129],"number":[53,113],"field":[54],"admitting":[55],"embedding,":[58],"fix":[60],"one":[61],"such":[62],"embedding":[63],"$\u03c3\\colon":[64],"K\\hookrightarrow\\mathbb{R}$.":[65],"first":[67],"prove":[68,93],"that,":[69],"for":[70],"an":[71,95],"$E$":[74,81,98,169],"over":[75,99,106,145],"$K$,":[76],"of":[80,89,114,117,123,142],"attached":[82],"$\u03c3$":[84],"determines":[85],"$K$-isogeny":[87],"class":[88],"$E$.":[90,143],"then":[92],"that":[94],"$\\mathbb{Q}$":[100,107],"is":[101],"determined":[102],"up":[103],"isomorphism":[105],"by":[108,128],"its":[109],"period,":[111],"connected":[115],"components":[116],"$E(\\mathbb{R})$,":[118],"quadratic":[125],"twist":[126],"$E^{D}$":[127],"negative":[130],"square-free":[131],"integer":[132],"$D\\equiv":[133],"1":[134],"\\bmod":[135],"4$":[136],"coprime":[137],"minimal":[140],"discriminant":[141],"Moreover,":[144],"$\\mathbb{Q}$,":[146],"both":[147],"statements":[148],"already":[149],"hold":[150],"when":[151],"periods":[154],"are":[155],"known":[156],"only":[157],"sufficiently":[159],"many":[160],"decimal":[161],"places,":[162],"where":[163],"required":[165],"precision":[166],"depends":[167],"on":[168],"(and":[170],"$D$).":[171]},"counts_by_year":[],"updated_date":"2026-11-07T11:25:07.637180","created_date":"2026-10-07T00:00:00"}