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

Tao, Zhengyu; Guo, Xuejun

doi:10.1090/mcom/3961

CM points, class numbers, and the Mahler measures of $x^3+y^3+1-kxy$
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by Zhengyu Tao and Xuejun Guo

Math. Comp.

DOI: https://doi.org/10.1090/mcom/3961

Published electronically: March 12, 2024

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Abstract:

We study the Mahler measures of the polynomial family $Q_k(x,y) = x^3+y^3+1-kxy$ using the method previously developed by the authors. An algorithm is implemented to search for complex multiplication points with class numbers $\leqslant 3$, we employ these points to derive interesting formulas that link the Mahler measures of $Q_k(x,y)$ to $L$-values of modular forms. As by-products, some conjectural identities of Samart are confirmed, one of them involves the modified Mahler measure $\tilde {n}(k)$ introduced by Samart recently. For $k=\sqrt [3]{729\pm 405\sqrt {3}}$, we also prove an equality that expresses a $2\times 2$ determinant with entries the Mahler measures of $Q_k(x,y)$ as some multiple of the $L$-value of two isogenous elliptic curves over $\mathbb {Q}(\sqrt {3})$.

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