CM points, class numbers, and the Mahler measures of \(x^3+y^3+1-kxy\) (Q6622396)

Some conjectures concerning the Mahler's measures of certain classes of Laurent polynomials, including the family\N\[\NQ_{k}(x,y)=x^{3}+y^{3}+1-kxy\in \mathbb{C}[x,y],\N\]\Nwere formulated by \textit{D. W. Boyd} [Experiment. Math. 7, No. 1, 37--82 (1998; Zbl 0932.11069)].\N\NIn the paper under review, the authors apply a method, developed by \textit{ Z. Tao} et al. [``Mahler measures and \(L\)-values of elliptic curves over real quadratic fields'', Preprint, \url{arXiv:2209.14717}], to study the Mahler measures of the polynomials \( Q_{k}(x,y).\) They implement an algorithm to determine complex multiplication points, with class number at most \(3,\) and use these points to derive some formulas that link the Mahler measures of \(Q_{k}(x,y)\) to \(L\)-values of modular forms. This allows them, in particular, to confirm some related conjectural identities of \textit{D. Samart} [Q. J. Math. 74, No. 3, 1187--1208 (2023; Zbl 1548.11139); Can. J. Math. 67, No. 2, 424--449 (2015; Zbl 1344.11070)].