CM points, class numbers, and the Mahler measures of x3+y3+1βˆ’kxyx^3+y^3+1-kxy

Abstract

We study the Mahler measures of the polynomial family Qk(x,y)=x3+y3+1βˆ’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 CM points with class numbers β©½3\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 a by-product, three conjectural identities of Samart are confirmed. For k=729Β±40533k=\sqrt[3]{729\pm405\sqrt{3}}, we also prove an equality that expresses a 2Γ—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}).Comment: 17 pages, 2 tables. Comments are welcom

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