{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2024,10,11]],"date-time":"2024-10-11T04:24:23Z","timestamp":1728620663604},"reference-count":24,"publisher":"American Mathematical Society (AMS)","issue":"351","license":[{"start":{"date-parts":[[2025,3,12]],"date-time":"2025-03-12T00:00:00Z","timestamp":1741737600000},"content-version":"am","delay-in-days":365,"URL":"https:\/\/www.ams.org\/publications\/copyright-and-permissions"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Math. Comp."],"abstract":"

We study the Mahler measures of the polynomial family \n\n \n \n \n Q<\/mml:mi>\n k<\/mml:mi>\n <\/mml:msub>\n (<\/mml:mo>\n x<\/mml:mi>\n ,<\/mml:mo>\n y<\/mml:mi>\n )<\/mml:mo>\n =<\/mml:mo>\n \n x<\/mml:mi>\n 3<\/mml:mn>\n <\/mml:msup>\n +<\/mml:mo>\n \n y<\/mml:mi>\n 3<\/mml:mn>\n <\/mml:msup>\n +<\/mml:mo>\n 1<\/mml:mn>\n \u2212<\/mml:mo>\n k<\/mml:mi>\n x<\/mml:mi>\n y<\/mml:mi>\n <\/mml:mrow>\n Q_k(x,y) = x^3+y^3+1-kxy<\/mml:annotation>\n <\/mml:semantics>\n<\/mml:math>\n<\/inline-formula> using the method previously developed by the authors. An algorithm is implemented to search for complex multiplication points with class numbers \n\n \n \n \u2a7d<\/mml:mo>\n 3<\/mml:mn>\n <\/mml:mrow>\n \\leqslant 3<\/mml:annotation>\n <\/mml:semantics>\n<\/mml:math>\n<\/inline-formula>, we employ these points to derive interesting formulas that link the Mahler measures of \n\n \n \n \n Q<\/mml:mi>\n k<\/mml:mi>\n <\/mml:msub>\n (<\/mml:mo>\n x<\/mml:mi>\n ,<\/mml:mo>\n y<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n Q_k(x,y)<\/mml:annotation>\n <\/mml:semantics>\n<\/mml:math>\n<\/inline-formula> to \n\n \n L<\/mml:mi>\n L<\/mml:annotation>\n <\/mml:semantics>\n<\/mml:math>\n<\/inline-formula>-values of modular forms. As by-products, some conjectural identities of Samart are confirmed, one of them involves the modified Mahler measure \n\n \n \n \n \n n<\/mml:mi>\n ~<\/mml:mo>\n <\/mml:mover>\n <\/mml:mrow>\n (<\/mml:mo>\n k<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n \\tilde {n}(k)<\/mml:annotation>\n <\/mml:semantics>\n<\/mml:math>\n<\/inline-formula> introduced by Samart recently. For \n\n \n \n k<\/mml:mi>\n =<\/mml:mo>\n \n \n 729<\/mml:mn>\n \u00b1<\/mml:mo>\n 405<\/mml:mn>\n \n 3<\/mml:mn>\n <\/mml:msqrt>\n <\/mml:mrow>\n 3<\/mml:mn>\n <\/mml:mroot>\n <\/mml:mrow>\n k=\\sqrt [3]{729\\pm 405\\sqrt {3}}<\/mml:annotation>\n <\/mml:semantics>\n<\/mml:math>\n<\/inline-formula>, we also prove an equality that expresses a \n\n \n \n 2<\/mml:mn>\n \u00d7<\/mml:mo>\n 2<\/mml:mn>\n <\/mml:mrow>\n 2\\times 2<\/mml:annotation>\n <\/mml:semantics>\n<\/mml:math>\n<\/inline-formula> determinant with entries the Mahler measures of \n\n \n \n \n Q<\/mml:mi>\n k<\/mml:mi>\n <\/mml:msub>\n (<\/mml:mo>\n x<\/mml:mi>\n ,<\/mml:mo>\n y<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n Q_k(x,y)<\/mml:annotation>\n <\/mml:semantics>\n<\/mml:math>\n<\/inline-formula> as some multiple of the \n\n \n L<\/mml:mi>\n L<\/mml:annotation>\n <\/mml:semantics>\n<\/mml:math>\n<\/inline-formula>-value of two isogenous elliptic curves over \n\n \n \n \n Q<\/mml:mi>\n <\/mml:mrow>\n (<\/mml:mo>\n \n 3<\/mml:mn>\n <\/mml:msqrt>\n )<\/mml:mo>\n <\/mml:mrow>\n \\mathbb {Q}(\\sqrt {3})<\/mml:annotation>\n <\/mml:semantics>\n<\/mml:math>\n<\/inline-formula>.<\/p>","DOI":"10.1090\/mcom\/3961","type":"journal-article","created":{"date-parts":[[2024,3,9]],"date-time":"2024-03-09T00:18:23Z","timestamp":1709943503000},"page":"425-446","source":"Crossref","is-referenced-by-count":0,"title":["CM points, class numbers, and the Mahler measures of \ud835\udc65\u00b3+\ud835\udc66\u00b3+1-\ud835\udc58\ud835\udc65\ud835\udc66"],"prefix":"10.1090","volume":"94","author":[{"given":"Zhengyu","family":"Tao","sequence":"first","affiliation":[]},{"given":"Xuejun","family":"Guo","sequence":"additional","affiliation":[]}],"member":"14","published-online":{"date-parts":[[2024,3,12]]},"reference":[{"key":"1","series-title":"Die Grundlehren der mathematischen Wissenschaften, Band 203","doi-asserted-by":"crossref","DOI":"10.1007\/978-3-642-65663-7","volume-title":"Elliptic modular functions: an introduction","author":"Schoeneberg, Bruno","year":"1974"},{"issue":"2","key":"2","doi-asserted-by":"publisher","first-page":"259","DOI":"10.1090\/S0894-0347-97-00228-2","article-title":"Deligne periods of mixed motives, \ud835\udc3e-theory and the entropy of certain \ud835\udc4d\u207f-actions","volume":"10","author":"Deninger, Christopher","year":"1997","journal-title":"J. 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Wei, Mahler measures and \ud835\udc3f-values of elliptic curves over real quadratic fields, Preprint, arXiv:2209.14717, 2022."}],"container-title":["Mathematics of Computation"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.ams.org\/mcom\/2025-94-351\/S0025-5718-2024-03961-3\/S0025-5718-2024-03961-3.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2024,10,10]],"date-time":"2024-10-10T17:50:49Z","timestamp":1728582649000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.ams.org\/mcom\/2025-94-351\/S0025-5718-2024-03961-3\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,3,12]]},"references-count":24,"journal-issue":{"issue":"351","published-print":{"date-parts":[[2025,1]]}},"alternative-id":["S0025-5718-2024-03961-3"],"URL":"https:\/\/doi.org\/10.1090\/mcom\/3961","archive":["CLOCKSS","Portico"],"relation":{},"ISSN":["0025-5718","1088-6842"],"issn-type":[{"type":"print","value":"0025-5718"},{"type":"electronic","value":"1088-6842"}],"subject":[],"published":{"date-parts":[[2024,3,12]]}}}