{"id":"https://openalex.org/W2016723714","doi":"https://doi.org/10.1090/s0025-5718-04-01644-8","title":"Visible evidence for the Birch and Swinnerton-Dyer conjecture for modular abelian varieties of analytic rank zero","display_name":"Visible evidence for the Birch and Swinnerton-Dyer conjecture for modular abelian varieties of analytic rank zero","publication_year":2004,"publication_date":"2004-05-18","ids":{"openalex":"https://openalex.org/W2016723714","doi":"https://doi.org/10.1090/s0025-5718-04-01644-8","mag":"2016723714"},"language":"en","primary_location":{"id":"doi:10.1090/s0025-5718-04-01644-8","is_oa":true,"landing_page_url":"https://doi.org/10.1090/s0025-5718-04-01644-8","pdf_url":"https://www.ams.org/mcom/2005-74-249/S0025-5718-04-01644-8/S0025-5718-04-01644-8.pdf","source":{"id":"https://openalex.org/S102439543","display_name":"Mathematics of Computation","issn_l":"0025-5718","issn":["0025-5718","1088-6842"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310315719","host_organization_name":"American Mathematical Society","host_organization_lineage":["https://openalex.org/P4310315719"],"host_organization_lineage_names":["American Mathematical Society"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Mathematics of Computation","raw_type":"journal-article"},"type":"article","indexed_in":["crossref"],"open_access":{"is_oa":true,"oa_status":"bronze","oa_url":"https://www.ams.org/mcom/2005-74-249/S0025-5718-04-01644-8/S0025-5718-04-01644-8.pdf","any_repository_has_fulltext":false},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5091041300","display_name":"Amod Agash\u00e9","orcid":null},"institutions":[{"id":"https://openalex.org/I86519309","display_name":"The University of Texas at Austin","ror":"https://ror.org/00hj54h04","country_code":"US","type":"education","lineage":["https://openalex.org/I86519309"]}],"countries":["US"],"is_corresponding":true,"raw_author_name":"Amod Agashe","raw_affiliation_strings":["Department of Mathematics, University of Texas, Austin, Texas 78712","Department of Mathematics, University of Texas, Austin, Texas, 78712"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics, University of Texas, Austin, Texas 78712","institution_ids":["https://openalex.org/I86519309"]},{"raw_affiliation_string":"Department of Mathematics, University of Texas, Austin, Texas, 78712","institution_ids":["https://openalex.org/I86519309"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5071010301","display_name":"William Stein","orcid":null},"institutions":[{"id":"https://openalex.org/I136199984","display_name":"Harvard University","ror":"https://ror.org/03vek6s52","country_code":"US","type":"education","lineage":["https://openalex.org/I136199984"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"William Stein","raw_affiliation_strings":["Department of Mathematics, Harvard University, Cambridge, Massachusetts 02138","Department of Mathematics, Harvard University, Cambridge Massachusetts 02138"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics, Harvard University, Cambridge, Massachusetts 02138","institution_ids":["https://openalex.org/I136199984"]},{"raw_affiliation_string":"Department of Mathematics, Harvard University, Cambridge Massachusetts 02138","institution_ids":["https://openalex.org/I136199984"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":2,"corresponding_author_ids":["https://openalex.org/A5091041300"],"corresponding_institution_ids":["https://openalex.org/I86519309"],"apc_list":null,"apc_paid":null,"fwci":2.5153,"has_fulltext":true,"cited_by_count":48,"citation_normalized_percentile":{"value":0.88182463,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":90,"max":97},"biblio":{"volume":"74","issue":"249","first_page":"455","last_page":"484"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10061","display_name":"Algebraic Geometry and Number Theory","score":0.9998999834060669,"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":0.9998999834060669,"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/T11166","display_name":"Analytic Number Theory Research","score":0.9782999753952026,"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/T11680","display_name":"Advanced Algebra and Geometry","score":0.9623000025749207,"subfield":{"id":"https://openalex.org/subfields/2610","display_name":"Mathematical Physics"},"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/mathematics","display_name":"Mathematics","score":0.8915456533432007},{"id":"https://openalex.org/keywords/conjecture","display_name":"Conjecture","score":0.8597582578659058},{"id":"https://openalex.org/keywords/abelian-group","display_name":"Abelian group","score":0.7306982278823853},{"id":"https://openalex.org/keywords/rank","display_name":"Rank (graph theory)","score":0.6798081994056702},{"id":"https://openalex.org/keywords/quotient","display_name":"Quotient","score":0.6612511277198792},{"id":"https://openalex.org/keywords/order","display_name":"Order (exchange)","score":0.579813539981842},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.541517972946167},{"id":"https://openalex.org/keywords/prime","display_name":"Prime (order theory)","score":0.5394613742828369},{"id":"https://openalex.org/keywords/elliptic-curve","display_name":"Elliptic curve","score":0.5111916661262512},{"id":"https://openalex.org/keywords/zero","display_name":"Zero (linguistics)","score":0.5040813684463501},{"id":"https://openalex.org/keywords/modular-form","display_name":"Modular form","score":0.4420657753944397},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.3499595522880554},{"id":"https://openalex.org/keywords/pure-mathematics","display_name":"Pure mathematics","score":0.3189179301261902}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.8915456533432007},{"id":"https://openalex.org/C2780990831","wikidata":"https://www.wikidata.org/wiki/Q319141","display_name":"Conjecture","level":2,"score":0.8597582578659058},{"id":"https://openalex.org/C136170076","wikidata":"https://www.wikidata.org/wiki/Q181296","display_name":"Abelian group","level":2,"score":0.7306982278823853},{"id":"https://openalex.org/C164226766","wikidata":"https://www.wikidata.org/wiki/Q7293202","display_name":"Rank (graph theory)","level":2,"score":0.6798081994056702},{"id":"https://openalex.org/C199422724","wikidata":"https://www.wikidata.org/wiki/Q41118","display_name":"Quotient","level":2,"score":0.6612511277198792},{"id":"https://openalex.org/C182306322","wikidata":"https://www.wikidata.org/wiki/Q1779371","display_name":"Order (exchange)","level":2,"score":0.579813539981842},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.541517972946167},{"id":"https://openalex.org/C184992742","wikidata":"https://www.wikidata.org/wiki/Q7243229","display_name":"Prime (order theory)","level":2,"score":0.5394613742828369},{"id":"https://openalex.org/C179603306","wikidata":"https://www.wikidata.org/wiki/Q268493","display_name":"Elliptic curve","level":2,"score":0.5111916661262512},{"id":"https://openalex.org/C2780813799","wikidata":"https://www.wikidata.org/wiki/Q3274237","display_name":"Zero (linguistics)","level":2,"score":0.5040813684463501},{"id":"https://openalex.org/C75764964","wikidata":"https://www.wikidata.org/wiki/Q870797","display_name":"Modular form","level":2,"score":0.4420657753944397},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.3499595522880554},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.3189179301261902},{"id":"https://openalex.org/C138885662","wikidata":"https://www.wikidata.org/wiki/Q5891","display_name":"Philosophy","level":0,"score":0.0},{"id":"https://openalex.org/C162324750","wikidata":"https://www.wikidata.org/wiki/Q8134","display_name":"Economics","level":0,"score":0.0},{"id":"https://openalex.org/C10138342","wikidata":"https://www.wikidata.org/wiki/Q43015","display_name":"Finance","level":1,"score":0.0},{"id":"https://openalex.org/C41895202","wikidata":"https://www.wikidata.org/wiki/Q8162","display_name":"Linguistics","level":1,"score":0.0}],"mesh":[],"locations_count":2,"locations":[{"id":"doi:10.1090/s0025-5718-04-01644-8","is_oa":true,"landing_page_url":"https://doi.org/10.1090/s0025-5718-04-01644-8","pdf_url":"https://www.ams.org/mcom/2005-74-249/S0025-5718-04-01644-8/S0025-5718-04-01644-8.pdf","source":{"id":"https://openalex.org/S102439543","display_name":"Mathematics of Computation","issn_l":"0025-5718","issn":["0025-5718","1088-6842"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310315719","host_organization_name":"American Mathematical Society","host_organization_lineage":["https://openalex.org/P4310315719"],"host_organization_lineage_names":["American Mathematical Society"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Mathematics of Computation","raw_type":"journal-article"},{"id":"pmh:oai:CiteSeerX.psu:10.1.1.153.6542","is_oa":false,"landing_page_url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.153.6542","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"http://www.ams.org/mcom/2005-74-249/S0025-5718-04-01644-8/S0025-5718-04-01644-8.pdf","raw_type":"text"}],"best_oa_location":{"id":"doi:10.1090/s0025-5718-04-01644-8","is_oa":true,"landing_page_url":"https://doi.org/10.1090/s0025-5718-04-01644-8","pdf_url":"https://www.ams.org/mcom/2005-74-249/S0025-5718-04-01644-8/S0025-5718-04-01644-8.pdf","source":{"id":"https://openalex.org/S102439543","display_name":"Mathematics of Computation","issn_l":"0025-5718","issn":["0025-5718","1088-6842"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310315719","host_organization_name":"American Mathematical Society","host_organization_lineage":["https://openalex.org/P4310315719"],"host_organization_lineage_names":["American Mathematical Society"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Mathematics of Computation","raw_type":"journal-article"},"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W2016723714.pdf","grobid_xml":"https://content.openalex.org/works/W2016723714.grobid-xml"},"referenced_works_count":46,"referenced_works":["https://openalex.org/W83607088","https://openalex.org/W122794529","https://openalex.org/W429986574","https://openalex.org/W1501002913","https://openalex.org/W1529471319","https://openalex.org/W1530016667","https://openalex.org/W1559532354","https://openalex.org/W1559870539","https://openalex.org/W1879311547","https://openalex.org/W1884403692","https://openalex.org/W1971193247","https://openalex.org/W1976677460","https://openalex.org/W1978008333","https://openalex.org/W2002926215","https://openalex.org/W2006068756","https://openalex.org/W2011844852","https://openalex.org/W2013129046","https://openalex.org/W2037737946","https://openalex.org/W2038913477","https://openalex.org/W2053658212","https://openalex.org/W2056406868","https://openalex.org/W2059292326","https://openalex.org/W2063486345","https://openalex.org/W2072323967","https://openalex.org/W2073082930","https://openalex.org/W2073214093","https://openalex.org/W2086259998","https://openalex.org/W2117791557","https://openalex.org/W2118635106","https://openalex.org/W2172163251","https://openalex.org/W2249385795","https://openalex.org/W2277535877","https://openalex.org/W2335415932","https://openalex.org/W2486137719","https://openalex.org/W2511795521","https://openalex.org/W2604330053","https://openalex.org/W3135300355","https://openalex.org/W4245784540","https://openalex.org/W4285719527","https://openalex.org/W6601579504","https://openalex.org/W6636018733","https://openalex.org/W6638178777","https://openalex.org/W6712560120","https://openalex.org/W6743458583","https://openalex.org/W6818990792","https://openalex.org/W6833398001"],"related_works":["https://openalex.org/W4313895414","https://openalex.org/W2145687427","https://openalex.org/W2951508299","https://openalex.org/W2046460098","https://openalex.org/W3082561679","https://openalex.org/W1994841333","https://openalex.org/W4298378052","https://openalex.org/W2949512258","https://openalex.org/W4294932571","https://openalex.org/W2807815877"],"abstract_inverted_index":{"This":[0],"paper":[1],"provides":[2],"evidence":[3],"for":[4,10,152,283],"the":[5,85,182,292],"Birch":[6],"and":[7,174,282,288],"Swinnerton-Dyer":[8],"conjecture":[9],"analytic":[11,267],"rank":[12,268],"<inline-formula":[13,27,50,87,155,184,204,224,248,269,296,337,350],"content-type=\"math/mathml\">":[14,28,51,88,156,185,205,225,249,270,297,338,351],"<mml:math":[15,29,52,89,157,186,206,226,250,271,298,339,352],"xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"":[16,30,53,90,158,187,207,227,251,272,299,340,353],"alttext=\"0\">":[17,273],"<mml:semantics>":[18,35,61,112,163,191,212,230,256,274,308,342,358],"<mml:mn>0</mml:mn>":[19,65,275],"<mml:annotation":[20,40,73,143,168,197,217,237,261,276,323,344,363],"encoding=\"application/x-tex\">0</mml:annotation>":[21,277],"</mml:semantics>":[22,42,75,146,170,199,219,240,263,278,326,346,365],"</mml:math>":[23,43,76,147,171,200,220,241,264,279,327,347,366],"</inline-formula>":[24,44,77,148,172,201,221,242,265,280,328,348],"abelian":[25,202],"varieties":[26,203],"alttext=\"upper":[31,54,91,159,208,252,354],"A":[32,95,108,160,209,253,303,355],"Subscript":[33,96,106,110,161,210,254,304,356],"f\">":[34,111,162,211,255,357],"<mml:msub>":[36,63,117,129,136,164,213,257,316,359],"<mml:mi>A</mml:mi>":[37,118,137,165,214,258,317,360],"<mml:mi>f</mml:mi>":[38,119,138,166,215,259,318,361],"</mml:msub>":[39,66,120,139,141,167,216,260,319,362],"encoding=\"application/x-tex\">A_f</mml:annotation>":[41,169,218,262,364],"that":[45,332],"are":[46,334],"optimal":[47],"quotients":[48],"of":[49,181,222,246,295],"J":[55],"0":[56],"left-parenthesis":[57,93,301],"upper":[58,94,104,107,287,302],"N":[59],"right-parenthesis\">":[60,307],"<mml:mrow>":[62,113,192,231,309],"<mml:mi>J</mml:mi>":[64],"<mml:mo":[67,70,115,123,314,320],"stretchy=\"false\">(</mml:mo>":[68,116,315],"<mml:mi>N</mml:mi>":[69],"stretchy=\"false\">)</mml:mo>":[71,124,321],"</mml:mrow>":[72,128,140,142,196,236,313,322],"encoding=\"application/x-tex\">J_0(N)</mml:annotation>":[74],"attached":[78],"to":[79],"newforms.":[80],"We":[81,330],"prove":[82],"theorems":[83],"about":[84,177],"ratio":[86],"L":[92],"f":[97,305],"Baseline":[98,306],"comma":[99,189],"1":[100],"right-parenthesis":[101],"slash":[102],"normal":[103],"Omega":[105],"Sub":[109],"<mml:mi>L</mml:mi>":[114],"<mml:mo>,</mml:mo>":[121,194],"<mml:mn>1</mml:mn>":[122],"<mml:mrow":[125,134,310],"class=\"MJX-TeXAtom-ORD\">":[126,135,311],"<mml:mo>/</mml:mo>":[127],"<mml:mi":[130],"mathvariant=\"normal\">":[131],"\u03a9":[132],"</mml:mi>":[133],"encoding=\"application/x-tex\">L(A_f,1)/\\Omega":[144],"_{A_f}</mml:annotation>":[145],",":[149,173,281],"develop":[150],"tools":[151],"computing":[153],"with":[154],"gather":[175],"data":[176],"certain":[178],"arithmetic":[179],"invariants":[180],"nearly":[183],"alttext=\"20":[188],"000\">":[190],"<mml:mn>20</mml:mn>":[193],"<mml:mn>000</mml:mn>":[195],"encoding=\"application/x-tex\">20,000</mml:annotation>":[198],"level":[223],"alttext=\"less-than-or-equal-to":[228],"2333\">":[229],"<mml:mo>":[232],"\u2264":[233],"</mml:mo>":[234],"<mml:mn>2333</mml:mn>":[235],"encoding=\"application/x-tex\">\\leq":[238],"2333</mml:annotation>":[239],".":[243,329],"Over":[244],"half":[245],"these":[247,284],"have":[266],"we":[285],"compute":[286],"lower":[289],"bounds":[290],"on":[291],"conjectural":[293],"order":[294],"alttext=\"\u0428":[300],"<mml:mo>\u0428</mml:mo>":[312],"encoding=\"application/x-tex\">\\Sha":[324],"(A_f)</mml:annotation>":[325],"find":[331],"there":[333],"at":[335],"least":[336],"alttext=\"168\">":[341],"<mml:mn>168</mml:mn>":[343],"encoding=\"application/x-tex\">168</mml:annotation>":[345],"such":[349]},"counts_by_year":[{"year":2025,"cited_by_count":2},{"year":2023,"cited_by_count":1},{"year":2022,"cited_by_count":1},{"year":2021,"cited_by_count":2},{"year":2019,"cited_by_count":1},{"year":2018,"cited_by_count":1},{"year":2016,"cited_by_count":3},{"year":2015,"cited_by_count":1},{"year":2013,"cited_by_count":3},{"year":2012,"cited_by_count":3}],"updated_date":"2026-05-21T06:26:12.895304","created_date":"2025-10-10T00:00:00"}
