{"id":"https://openalex.org/W7225332891","doi":"https://doi.org/10.5281/zenodo.23990716","title":"Fourteen initial \u0398-data on one curve over Q(i)","display_name":"Fourteen initial \u0398-data on one curve over Q(i)","publication_year":2026,"publication_date":"2026-11-06","ids":{"openalex":"https://openalex.org/W7225332891","doi":"https://doi.org/10.5281/zenodo.23990716"},"language":"en","primary_location":{"id":"doi:10.5281/zenodo.23990716","doi":"https://doi.org/10.5281/zenodo.23990716","is_oa":true,"landing_page_url":"https://doi.org/10.5281/zenodo.23990716","pdf_url":null,"source":{"id":"https://openalex.org/S4306400562","display_name":"Zenodo (CERN European Organization for Nuclear Research)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I67311998","host_organization_name":"European Organization for Nuclear Research","host_organization_lineage":["https://openalex.org/I67311998"],"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"},"volume":null,"type":"preprint","is_traditional":true,"indexed_in":["datacite"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://doi.org/10.5281/zenodo.23990716","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5150242142","display_name":"Toshihisa Urahigashi","orcid":"https://orcid.org/0009-0004-0460-6242","observed_orcids":["https://orcid.org/0009-0004-0460-6242"]},"institutions":[],"countries":[],"is_corresponding":true,"raw_author_name":"Urahigashi, Toshihisa","raw_affiliation_strings":["Independent researcher"],"raw_orcid":"https://orcid.org/0009-0004-0460-6242","affiliations":[{"raw_affiliation_string":"Independent researcher","institution_ids":[]}]}],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":0,"corresponding_author_ids":["https://openalex.org/A5150242142"],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":null,"has_fulltext":false,"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":0.9434000253677368,"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":0.9434000253677368,"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":0.0478999987244606,"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":0.0010000000474974513,"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/inter-universal-teichmuller-theory","display_name":"inter-universal Teichm\u00fcller theory","score":0.9350000023841858},{"id":"https://openalex.org/keywords/field-of-moduli","display_name":"field of moduli","score":0.7300000190734863},{"id":"https://openalex.org/keywords/good-reduction","display_name":"good reduction","score":0.4790000021457672},{"id":"https://openalex.org/keywords/multiplicative-reduction","display_name":"multiplicative reduction","score":0.6140000224113464},{"id":"https://openalex.org/keywords/mochizuki","display_name":"Mochizuki","score":0.578000009059906},{"id":"https://openalex.org/keywords/abc-conjecture","display_name":"abc conjecture","score":0.4729999899864197},{"id":"https://openalex.org/keywords/elliptic-curve-isogenies","display_name":"elliptic curve isogenies","score":0.367000013589859},{"id":"https://openalex.org/keywords/mordell-weil-group","display_name":"Mordell-Weil group","score":0.460999995470047}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.8852999806404114},{"id":"https://openalex.org/C42747912","wikidata":"https://www.wikidata.org/wiki/Q1048447","display_name":"Multiplicative function","level":2,"score":0.477400004863739},{"id":"https://openalex.org/C34388435","wikidata":"https://www.wikidata.org/wiki/Q2267362","display_name":"Bounded function","level":2,"score":0.475600004196167},{"id":"https://openalex.org/C30860621","wikidata":"https://www.wikidata.org/wiki/Q49008","display_name":"Prime factor","level":3,"score":0.42419999837875366},{"id":"https://openalex.org/C92788228","wikidata":"https://www.wikidata.org/wiki/Q3777923","display_name":"Multiplicative group","level":3,"score":0.4016999900341034},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.39500001072883606},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.3700999915599823},{"id":"https://openalex.org/C54732982","wikidata":"https://www.wikidata.org/wiki/Q1415345","display_name":"Modulo","level":2,"score":0.3698999881744385},{"id":"https://openalex.org/C121089165","wikidata":"https://www.wikidata.org/wiki/Q6889796","display_name":"Moduli","level":2,"score":0.36489999294281006},{"id":"https://openalex.org/C169654258","wikidata":"https://www.wikidata.org/wiki/Q12479","display_name":"Number theory","level":2,"score":0.3488999903202057},{"id":"https://openalex.org/C191795146","wikidata":"https://www.wikidata.org/wiki/Q3878446","display_name":"Norm (philosophy)","level":2,"score":0.34630000591278076},{"id":"https://openalex.org/C186027771","wikidata":"https://www.wikidata.org/wiki/Q4008379","display_name":"Valuation (finance)","level":2,"score":0.33000001311302185},{"id":"https://openalex.org/C191421660","wikidata":"https://www.wikidata.org/wiki/Q829546","display_name":"Primality test","level":3,"score":0.3278999924659729},{"id":"https://openalex.org/C9652623","wikidata":"https://www.wikidata.org/wiki/Q190109","display_name":"Field (mathematics)","level":2,"score":0.32420000433921814},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.32420000433921814},{"id":"https://openalex.org/C136815107","wikidata":"https://www.wikidata.org/wiki/Q154242","display_name":"Nothing","level":2,"score":0.3228999972343445},{"id":"https://openalex.org/C184992742","wikidata":"https://www.wikidata.org/wiki/Q7243229","display_name":"Prime (order theory)","level":2,"score":0.3149999976158142},{"id":"https://openalex.org/C187834632","wikidata":"https://www.wikidata.org/wiki/Q188804","display_name":"Factorization","level":2,"score":0.31310001015663147},{"id":"https://openalex.org/C97137487","wikidata":"https://www.wikidata.org/wiki/Q729138","display_name":"Integer (computer science)","level":2,"score":0.3066999912261963},{"id":"https://openalex.org/C182306322","wikidata":"https://www.wikidata.org/wiki/Q1779371","display_name":"Order (exchange)","level":2,"score":0.3061999976634979},{"id":"https://openalex.org/C146324458","wikidata":"https://www.wikidata.org/wiki/Q1202673","display_name":"Uniform norm","level":2,"score":0.2809999883174896},{"id":"https://openalex.org/C136119220","wikidata":"https://www.wikidata.org/wiki/Q1000660","display_name":"Algebra over a field","level":2,"score":0.27079999446868896},{"id":"https://openalex.org/C77553402","wikidata":"https://www.wikidata.org/wiki/Q13222579","display_name":"Upper and lower bounds","level":2,"score":0.2648000121116638},{"id":"https://openalex.org/C167729594","wikidata":"https://www.wikidata.org/wiki/Q17736","display_name":"Axiom","level":2,"score":0.2599000036716461},{"id":"https://openalex.org/C113429393","wikidata":"https://www.wikidata.org/wiki/Q49008","display_name":"Prime number","level":2,"score":0.25369998812675476},{"id":"https://openalex.org/C94375191","wikidata":"https://www.wikidata.org/wiki/Q11205","display_name":"Arithmetic","level":1,"score":0.2522999942302704}],"mesh":[],"locations_count":2,"locations":[{"id":"doi:10.5281/zenodo.23990716","doi":"https://doi.org/10.5281/zenodo.23990716","is_oa":true,"landing_page_url":"https://doi.org/10.5281/zenodo.23990716","pdf_url":null,"source":{"id":"https://openalex.org/S4306400562","display_name":"Zenodo (CERN European Organization for Nuclear Research)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I67311998","host_organization_name":"European Organization for Nuclear Research","host_organization_lineage":["https://openalex.org/I67311998"],"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"},{"id":"doi:10.5281/zenodo.23990717","doi":"https://doi.org/10.5281/zenodo.23990717","is_oa":true,"landing_page_url":"https://doi.org/10.5281/zenodo.23990717","pdf_url":null,"source":{"id":"https://openalex.org/S4306400562","display_name":"Zenodo (CERN European Organization for Nuclear Research)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I67311998","host_organization_name":"European Organization for Nuclear Research","host_organization_lineage":["https://openalex.org/I67311998"],"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":"doi:10.5281/zenodo.23990716","doi":"https://doi.org/10.5281/zenodo.23990716","is_oa":true,"landing_page_url":"https://doi.org/10.5281/zenodo.23990716","pdf_url":null,"source":{"id":"https://openalex.org/S4306400562","display_name":"Zenodo (CERN European Organization for Nuclear Research)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I67311998","host_organization_name":"European Organization for Nuclear Research","host_organization_lineage":["https://openalex.org/I67311998"],"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"},"sustainable_development_goals":[],"study_designs":[],"awards":[],"funders":[],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":0,"referenced_works":[],"related_works":["https://openalex.org/W7211873771","https://openalex.org/W7225042391","https://openalex.org/W7210275725","https://openalex.org/W4412644196","https://openalex.org/W7220471555","https://openalex.org/W7225101354","https://openalex.org/W3135431298","https://openalex.org/W7098311004","https://openalex.org/W942115114","https://openalex.org/W137588603"],"abstract_inverted_index":{"Let":[0],"E:":[1],"y2":[2],"=":[3,11,28],"x(x":[4],"\u2212":[5,7,169,184],"1)(x":[6],"a)":[8],"with":[9,99,340],"a":[10,15,170,185,266],"(41":[12],"+":[13],"26i)472,":[14],"curve":[16,44,106],"over":[17],"Q(i).":[18,82],"We":[19],"prove":[20],"that":[21,63,122,132,211,330],"for":[22,307],"each":[23,187],"prime":[24,86],"\u2113":[25,153],"in":[26,50,237,281,288,295,299],"\u039b":[27],"{89,":[29],"97,":[30],"101,":[31],"103,":[32],"107,":[33],"109,":[34],"113,":[35],"127,":[36],"131,":[37],"137,":[38],"139,":[39],"149,":[40],"151,":[41],"157}":[42],"the":[43,51,67,84,108,115,123,142,146,151,180,193,200,205,208,220,224,235,285,290,296,300,308,316,322,328,332,335,341],"extends":[45],"to":[46,138],"an":[47,319],"initial":[48,140],"\u0398-datum":[49],"sense":[52],"of":[53,56,69,72,79,92,96,114,167,182,202,246,249,258,269,310,321,334,351],"Definition":[54],"3.1":[55],"Mochizuki's":[57],"Inter-universal":[58,73],"Teichm\u00fcller":[59,74],"theory":[60,75],"I,":[61],"and":[62,94,103,107,129,159,174,207,223,241,292,298,318,337,344],"this":[64,212,282,311],"datum":[65],"satisfies":[66],"hypotheses":[68],"Theorem":[70],"1.10":[71],"IV.":[76,251],"The":[77,105,148,165,231,302,313,353],"field":[78],"moduli":[80],"is":[81,89,177,190,199,253,265],"Above":[83],"split":[85,188],"2357":[87,227],"there":[88],"one":[90,95,160],"place":[91,189],"multiplicative":[93],"good":[97],"reduction,":[98],"ramification":[100],"indices":[101],"15\u2113":[102],"1.":[104],"fourteen":[109,232],"levels":[110,124,233],"come":[111],"from":[112,348],"\u00a713":[113],"author's":[116,286],"second":[117],"report":[118],"(doi:10.5281/zenodo.22601333),":[119],"which":[120],"showed":[121],"pass":[125],"five":[126,156],"numerical":[127],"conditions":[128],"said":[130],"explicitly":[131],"they":[133,242],"had":[134],"not":[135,178,215],"been":[136],"verified":[137],"be":[139],"\u0398-data;":[141],"present":[143],"note":[144,317,336],"gives":[145],"verification.":[147],"proof":[149],"uses":[150],"individual":[152],"only":[154],"through":[155,192],"integer":[157],"facts":[158],"Frobenius":[161],"witness":[162],"per":[163],"level.":[164],"norm":[166],"1":[168,183],"has":[171],"1592":[172],"digits,":[173],"its":[175,293,345],"factorization":[176],"used:":[179],"valuation":[181],"at":[186],"bounded":[191],"cyclotomic":[194],"factor":[195],"\u03a6d(\u03c0),":[196],"where":[197],"d":[198],"order":[201],"\u03c0":[203],"modulo":[204],"place,":[206],"138,731":[209],"primes":[210,236],"bound":[213],"does":[214],"exclude":[216],"are":[217,228,234,355],"tested":[218],"directly;":[219],"inert":[221],"places":[222,225],"above":[226],"treated":[229],"directly.":[230],"[\u221a(log":[238],"q\u2200),":[239],"472/3),":[240],"satisfy":[243],"condition":[244],"(C1)":[245],"Corollary":[247,256],"2.2":[248],"IUT":[250,259],"Nothing":[252],"claimed":[254],"about":[255,262],"3.12":[257],"III":[260],"or":[261],"abc.":[263],"This":[264],"preprint.":[267],"Use":[268],"generative":[270],"AI.":[271],"Generative":[272],"AI":[273],"tools":[274],"(large":[275],"language":[276],"models)":[277],"were":[278],"used":[279],"extensively":[280],"work,":[283],"under":[284,357],"direction,":[287],"finding":[289],"example":[291],"proofs,":[294],"computations":[297],"writing.":[301],"author":[303],"takes":[304],"full":[305],"responsibility":[306],"contents":[309],"report.":[312],"deposit":[314],"contains":[315],"archive":[320],"folder":[323],"fourteen-levels/":[324],"(the":[325],"LaTeX":[326],"source,":[327],"program":[329],"recomputes":[331],"arithmetic":[333],"compares":[338],"it":[339],"printed":[342],"numbers,":[343],"output),":[346],"made":[347],"tag":[349],"v0.20.0":[350],"github.com/turahigashi/iut-explicit-initial-theta.":[352],"files":[354],"licensed":[356],"CC":[358],"BY":[359],"4.0.":[360]},"counts_by_year":[],"updated_date":"2026-11-07T11:25:07.637180","created_date":"2026-11-06T00:00:00"}