{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,21]],"date-time":"2026-04-21T20:37:18Z","timestamp":1776803838051,"version":"3.51.2"},"reference-count":14,"publisher":"American Mathematical Society (AMS)","issue":"312","license":[{"start":{"date-parts":[[2018,9,28]],"date-time":"2018-09-28T00:00:00Z","timestamp":1538092800000},"content-version":"am","delay-in-days":365,"URL":"https:\/\/www.ams.org\/publications\/copyright-and-permissions"}],"funder":[{"DOI":"10.13039\/501100003725","name":"National Research Foundation of Korea","doi-asserted-by":"publisher","award":["2009-0093827"],"award-info":[{"award-number":["2009-0093827"]}],"id":[{"id":"10.13039\/501100003725","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100003725","name":"National Research Foundation of Korea","doi-asserted-by":"publisher","award":["2014-002731"],"award-info":[{"award-number":["2014-002731"]}],"id":[{"id":"10.13039\/501100003725","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100003725","name":"National Research Foundation of Korea","doi-asserted-by":"publisher","award":["RP-Grant 2016"],"award-info":[{"award-number":["RP-Grant 2016"]}],"id":[{"id":"10.13039\/501100003725","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100002630","name":"Ewha Womans University","doi-asserted-by":"publisher","award":["2009-0093827"],"award-info":[{"award-number":["2009-0093827"]}],"id":[{"id":"10.13039\/501100002630","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100002630","name":"Ewha Womans University","doi-asserted-by":"publisher","award":["2014-002731"],"award-info":[{"award-number":["2014-002731"]}],"id":[{"id":"10.13039\/501100002630","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100002630","name":"Ewha Womans University","doi-asserted-by":"publisher","award":["RP-Grant 2016"],"award-info":[{"award-number":["RP-Grant 2016"]}],"id":[{"id":"10.13039\/501100002630","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Math. Comp."],"abstract":"

\n We study a continued fraction\n \n \n \n \n U<\/mml:mi>\n (<\/mml:mo>\n \n \u03c4\n \n <\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n U(\\tau )<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n of order twelve using the modular function theory. We obtain the modular equations of\n \n \n \n \n U<\/mml:mi>\n (<\/mml:mo>\n \n \u03c4\n \n <\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n U(\\tau )<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n by computing the affine models of modular curves\n \n \n \n \n X<\/mml:mi>\n (<\/mml:mo>\n \n \u0393\n \n <\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n X(\\Gamma )<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n with\n \n \n \n \n \n \u0393\n \n <\/mml:mi>\n =<\/mml:mo>\n \n \n \u0393\n \n <\/mml:mi>\n 1<\/mml:mn>\n <\/mml:msub>\n (<\/mml:mo>\n 12<\/mml:mn>\n )<\/mml:mo>\n \n \u2229\n \n <\/mml:mo>\n \n \n \u0393\n \n <\/mml:mi>\n 0<\/mml:mn>\n <\/mml:msub>\n (<\/mml:mo>\n 12<\/mml:mn>\n n<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n \\Gamma = \\Gamma _1 (12) \\cap \\Gamma _0(12n)<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n for any positive integer\n \n \n \n n<\/mml:mi>\n n<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n ; this is a complete extension of the previous result of Mahadeva Naika et al. and Dharmendra et al. to every positive integer\n \n \n \n n<\/mml:mi>\n n<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n . We point out that we provide an explicit construction method for finding the modular equations of\n \n \n \n \n U<\/mml:mi>\n (<\/mml:mo>\n \n \u03c4\n \n <\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n U(\\tau )<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n . We also prove that these modular equations satisfy the Kronecker congruence relations. Furthermore, we show that we can construct the ray class field modulo\n \n \n \n 12<\/mml:mn>\n 12<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n over imaginary quadratic fields by using\n \n \n \n \n U<\/mml:mi>\n (<\/mml:mo>\n \n \u03c4\n \n <\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n U(\\tau )<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n and the value\n \n \n \n \n U<\/mml:mi>\n (<\/mml:mo>\n \n \u03c4\n \n <\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n U(\\tau )<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n at an imaginary quadratic argument is a unit. In addition, if\n \n \n \n \n U<\/mml:mi>\n (<\/mml:mo>\n \n \u03c4\n \n <\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n U(\\tau )<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n is expressed in terms of radicals, then we can express\n \n \n \n \n U<\/mml:mi>\n (<\/mml:mo>\n r<\/mml:mi>\n \n \u03c4\n \n <\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n U(r \\tau )<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n in terms of radicals for a positive rational number\n \n \n \n r<\/mml:mi>\n r<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n .\n <\/p>","DOI":"10.1090\/mcom\/3259","type":"journal-article","created":{"date-parts":[[2017,2,1]],"date-time":"2017-02-01T11:26:57Z","timestamp":1485948417000},"page":"2011-2036","source":"Crossref","is-referenced-by-count":5,"title":["A continued fraction of order twelve as a modular function"],"prefix":"10.1090","volume":"87","author":[{"given":"Yoonjin","family":"Lee","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Yoon Kyung","family":"Park","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"14","published-online":{"date-parts":[[2017,9,28]]},"reference":[{"key":"1","doi-asserted-by":"publisher","first-page":"27","DOI":"10.1515\/CRELLE.2006.063","article-title":"Modular curves and Ramanujan\u2019s continued fraction","volume":"597","author":"Cais, Bryden","year":"2006","journal-title":"J. Reine Angew. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0075-4102","issn-type":"print"},{"issue":"2","key":"2","doi-asserted-by":"publisher","first-page":"199","DOI":"10.1093\/qmath\/han035","article-title":"Construction of class fields over imaginary quadratic fields and applications","volume":"61","author":"Cho, Bumkyu","year":"2010","journal-title":"Q. J. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0033-5606","issn-type":"print"},{"key":"3","unstructured":"B. N. Dharmendra, M. R. Rajesh Kanna and R. Jagadeesh, On continued fraction of order twelve, Pure Math. Sci. 1 (2012), 197\u2013205."},{"issue":"4","key":"4","doi-asserted-by":"publisher","first-page":"922","DOI":"10.1016\/j.jnt.2008.09.018","article-title":"Arithmetic of the Ramanujan-G\u00f6llnitz-Gordon continued fraction","volume":"129","author":"Cho, Bumkyu","year":"2009","journal-title":"J. Number Theory","ISSN":"https:\/\/id.crossref.org\/issn\/0022-314X","issn-type":"print"},{"issue":"3","key":"5","doi-asserted-by":"publisher","first-page":"267","DOI":"10.1007\/s11139-006-8477-7","article-title":"Singular values of the Rogers-Ramanujan continued fraction","volume":"11","author":"Gee, Alice","year":"2006","journal-title":"Ramanujan J.","ISSN":"https:\/\/id.crossref.org\/issn\/1382-4090","issn-type":"print"},{"issue":"3","key":"6","doi-asserted-by":"publisher","first-page":"271","DOI":"10.1007\/BF02568306","article-title":"The equations for modular function fields of principal congruence subgroups of prime level","volume":"90","author":"Ishida, Nobuhiko","year":"1996","journal-title":"Manuscripta Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-2611","issn-type":"print"},{"issue":"6","key":"7","doi-asserted-by":"publisher","first-page":"1389","DOI":"10.2478\/s11533-011-0080-5","article-title":"Generation of Hauptmoduln of \u0393\u2081(\ud835\udc41) by Weierstrass units and application to class fields","volume":"9","author":"Kim, Chang Heon","year":"2011","journal-title":"Cent. Eur. J. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/1895-1074","issn-type":"print"},{"key":"8","series-title":"Grundlehren der Mathematischen Wissenschaften","isbn-type":"print","doi-asserted-by":"crossref","DOI":"10.1007\/978-1-4757-1741-9","volume-title":"Modular units","volume":"244","author":"Kubert, Daniel S.","year":"1981","ISBN":"https:\/\/id.crossref.org\/isbn\/0387905170"},{"issue":"1","key":"9","doi-asserted-by":"publisher","first-page":"373","DOI":"10.1016\/j.jmaa.2016.01.065","article-title":"Modularity of a Ramanujan-Selberg continued fraction","volume":"438","author":"Lee, Yoonjin","year":"2016","journal-title":"J. Math. Anal. Appl.","ISSN":"https:\/\/id.crossref.org\/issn\/0022-247X","issn-type":"print"},{"issue":"3","key":"10","doi-asserted-by":"publisher","first-page":"393","DOI":"10.2478\/s11533-008-0031-y","article-title":"A continued fraction of order twelve","volume":"6","author":"Mahadeva Naika, M. S.","year":"2008","journal-title":"Cent. Eur. J. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/1895-1074","issn-type":"print"},{"issue":"2","key":"11","first-page":"129","article-title":"Some new identities for a continued fraction of order 12","volume":"10","author":"Naika, M. S. Mahadeva","year":"2012","journal-title":"South East Asian J. Math. Math. Sci.","ISSN":"https:\/\/id.crossref.org\/issn\/0972-7752","issn-type":"print"},{"key":"12","unstructured":"M. S. Mahadeva Naika, S. Chandankumar and K. Sushan Bairy, New identites for ratios of Ramanujan\u2019s theta function (communicated)."},{"key":"13","series-title":"CBMS Regional Conference Series in Mathematics","isbn-type":"print","volume-title":"The web of modularity: arithmetic of the coefficients of modular forms and $q$-series","volume":"102","author":"Ono, Ken","year":"2004","ISBN":"https:\/\/id.crossref.org\/isbn\/0821833685"},{"key":"14","series-title":"Kan\\^{o} Memorial Lectures, No. 1","volume-title":"Introduction to the arithmetic theory of automorphic functions","author":"Shimura, Goro","year":"1971"}],"container-title":["Mathematics of Computation"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.ams.org\/mcom\/2018-87-312\/S0025-5718-2017-03259-2\/mcom3259_AM.pdf","content-type":"application\/pdf","content-version":"am","intended-application":"syndication"},{"URL":"http:\/\/www.ams.org\/mcom\/2018-87-312\/S0025-5718-2017-03259-2\/S0025-5718-2017-03259-2.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"},{"URL":"https:\/\/www.ams.org\/mcom\/2018-87-312\/S0025-5718-2017-03259-2\/S0025-5718-2017-03259-2.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,21]],"date-time":"2026-04-21T19:41:33Z","timestamp":1776800493000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.ams.org\/mcom\/2018-87-312\/S0025-5718-2017-03259-2\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2017,9,28]]},"references-count":14,"journal-issue":{"issue":"312","published-print":{"date-parts":[[2018,7]]}},"alternative-id":["S0025-5718-2017-03259-2"],"URL":"https:\/\/doi.org\/10.1090\/mcom\/3259","archive":["CLOCKSS","Portico"],"relation":{},"ISSN":["1088-6842","0025-5718"],"issn-type":[{"value":"1088-6842","type":"electronic"},{"value":"0025-5718","type":"print"}],"subject":[],"published":{"date-parts":[[2017,9,28]]}}}