CM points, class numbers, and the Mahler measures of 𝑥³+𝑦³+1-𝑘𝑥𝑦

Abstract

We study the Mahler measures of the polynomial family Q k ( x , y ) = x 3 + y 3 + 1 − k x y Q_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 Q k ( x , y ) Q_k(x,y) to L L -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 ± 405 3 3 k=\sqrt [3]{729\pm 405\sqrt {3}} , we also prove an equality that expresses a 2 × 2 2\times 2 determinant with entries the Mahler measures of Q k ( x , y ) Q_k(x,y) as some multiple of the L L -value of two isogenous elliptic curves over Q ( 3 ) \mathbb {Q}(\sqrt {3}) .