{"title":"CM points, class numbers, and the Mahler measures of 𝑥³+𝑦³+1-𝑘𝑥𝑦","authors":"Zhengyu Tao, Xuejun Guo","doi":"10.1090/mcom/3961","DOIUrl":null,"url":null,"abstract":"<p>We study the Mahler measures of the polynomial family <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper Q Subscript k Baseline left-parenthesis x comma y right-parenthesis equals x cubed plus y cubed plus 1 minus k x y\">\n <mml:semantics>\n <mml:mrow>\n <mml:msub>\n <mml:mi>Q</mml:mi>\n <mml:mi>k</mml:mi>\n </mml:msub>\n <mml:mo stretchy=\"false\">(</mml:mo>\n <mml:mi>x</mml:mi>\n <mml:mo>,</mml:mo>\n <mml:mi>y</mml:mi>\n <mml:mo stretchy=\"false\">)</mml:mo>\n <mml:mo>=</mml:mo>\n <mml:msup>\n <mml:mi>x</mml:mi>\n <mml:mn>3</mml:mn>\n </mml:msup>\n <mml:mo>+</mml:mo>\n <mml:msup>\n <mml:mi>y</mml:mi>\n <mml:mn>3</mml:mn>\n </mml:msup>\n <mml:mo>+</mml:mo>\n <mml:mn>1</mml:mn>\n <mml:mo>−<!-- − --></mml:mo>\n <mml:mi>k</mml:mi>\n <mml:mi>x</mml:mi>\n <mml:mi>y</mml:mi>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">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 <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"less-than-or-slanted-equals 3\">\n <mml:semantics>\n <mml:mrow>\n <mml:mo>⩽<!-- ⩽ --></mml:mo>\n <mml:mn>3</mml:mn>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">\\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 <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper Q Subscript k Baseline left-parenthesis x comma y right-parenthesis\">\n <mml:semantics>\n <mml:mrow>\n <mml:msub>\n <mml:mi>Q</mml:mi>\n <mml:mi>k</mml:mi>\n </mml:msub>\n <mml:mo stretchy=\"false\">(</mml:mo>\n <mml:mi>x</mml:mi>\n <mml:mo>,</mml:mo>\n <mml:mi>y</mml:mi>\n <mml:mo stretchy=\"false\">)</mml:mo>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">Q_k(x,y)</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> to <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper L\">\n <mml:semantics>\n <mml:mi>L</mml:mi>\n <mml:annotation encoding=\"application/x-tex\">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 <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"ModifyingAbove n With tilde left-parenthesis k right-parenthesis\">\n <mml:semantics>\n <mml:mrow>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mover>\n <mml:mi>n</mml:mi>\n <mml:mo stretchy=\"false\">~<!-- ~ --></mml:mo>\n </mml:mover>\n </mml:mrow>\n <mml:mo stretchy=\"false\">(</mml:mo>\n <mml:mi>k</mml:mi>\n <mml:mo stretchy=\"false\">)</mml:mo>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">\\tilde {n}(k)</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> introduced by Samart recently. For <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"k equals NestedRootIndex 3 NestedStartRoot 729 plus-or-minus 405 StartRoot 3 EndRoot NestedEndRoot\">\n <mml:semantics>\n <mml:mrow>\n <mml:mi>k</mml:mi>\n <mml:mo>=</mml:mo>\n <mml:mroot>\n <mml:mrow>\n <mml:mn>729</mml:mn>\n <mml:mo>±<!-- ± --></mml:mo>\n <mml:mn>405</mml:mn>\n <mml:msqrt>\n <mml:mn>3</mml:mn>\n </mml:msqrt>\n </mml:mrow>\n <mml:mn>3</mml:mn>\n </mml:mroot>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">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 <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"2 times 2\">\n <mml:semantics>\n <mml:mrow>\n <mml:mn>2</mml:mn>\n <mml:mo>×<!-- × --></mml:mo>\n <mml:mn>2</mml:mn>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">2\\times 2</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> determinant with entries the Mahler measures of <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper Q Subscript k Baseline left-parenthesis x comma y right-parenthesis\">\n <mml:semantics>\n <mml:mrow>\n <mml:msub>\n <mml:mi>Q</mml:mi>\n <mml:mi>k</mml:mi>\n </mml:msub>\n <mml:mo stretchy=\"false\">(</mml:mo>\n <mml:mi>x</mml:mi>\n <mml:mo>,</mml:mo>\n <mml:mi>y</mml:mi>\n <mml:mo stretchy=\"false\">)</mml:mo>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">Q_k(x,y)</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> as some multiple of the <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper L\">\n <mml:semantics>\n <mml:mi>L</mml:mi>\n <mml:annotation encoding=\"application/x-tex\">L</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula>-value of two isogenous elliptic curves over <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"double-struck upper Q left-parenthesis StartRoot 3 EndRoot right-parenthesis\">\n <mml:semantics>\n <mml:mrow>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mi mathvariant=\"double-struck\">Q</mml:mi>\n </mml:mrow>\n <mml:mo stretchy=\"false\">(</mml:mo>\n <mml:msqrt>\n <mml:mn>3</mml:mn>\n </mml:msqrt>\n <mml:mo stretchy=\"false\">)</mml:mo>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">\\mathbb {Q}(\\sqrt {3})</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula>.</p>","PeriodicalId":2,"journal":{"name":"ACS Applied Bio Materials","volume":null,"pages":null},"PeriodicalIF":4.6000,"publicationDate":"2024-03-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"ACS Applied Bio Materials","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1090/mcom/3961","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATERIALS SCIENCE, BIOMATERIALS","Score":null,"Total":0}
引用次数: 0
Abstract
We study the Mahler measures of the polynomial family Qk(x,y)=x3+y3+1−kxyQ_k(x,y) = x^3+y^3+1-kxy using the method previously developed by the authors. An algorithm is implemented to search for complex multiplication points with class numbers ⩽3\leqslant 3, we employ these points to derive interesting formulas that link the Mahler measures of Qk(x,y)Q_k(x,y) to LL-values of modular forms. As by-products, some conjectural identities of Samart are confirmed, one of them involves the modified Mahler measure n~(k)\tilde {n}(k) introduced by Samart recently. For k=729±40533k=\sqrt [3]{729\pm 405\sqrt {3}}, we also prove an equality that expresses a 2×22\times 2 determinant with entries the Mahler measures of Qk(x,y)Q_k(x,y) as some multiple of the LL-value of two isogenous elliptic curves over Q(3)\mathbb {Q}(\sqrt {3}).
我们利用作者之前开发的方法研究了多项式族 Q k ( x , y ) = x 3 + y 3 + 1 - k x y Q_k(x,y) = x^3+y^3+1-kxy 的马勒度量。我们利用这些点推导出有趣的公式，将 Q k ( x , y ) Q_k(x,y) 的马勒度量与模块形式的 L L 值联系起来。作为副产品，萨马特的一些猜想得到了证实，其中之一涉及萨马特最近引入的修正马勒度量 n ~ ( k ) （tilde {n}(k)）。对于 k = 729 ± 405 3 3 k=\sqrt [3]{729\pm 405\sqrt {3}} ，我们也证明了表示 n ~ ( k ) 的等式。 对于 k = 729 ± 405 3 3 k= （sqrt [3]{729\pm 405\sqrt {3}}，我们还证明了一个等式，该等式将 Q k ( x , y ) 的马勒度量 Q_k(x,y)的 2 × 2 2 （times 2 ）行列式表达为 Q ( 3 ) 上两个同源椭圆曲线的 L L - 值的某个倍数（\mathbb {Q}(\sqrt {3})）。
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