CM points, class numbers, and the Mahler measures of $x^3+y^3+1-kxy$
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 CM 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 $k=\sqrt[3]{729\pm405\sqrt{3}}$, 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 $\mathbb{Q}(\sqrt{3})$.
- Publication:
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arXiv e-prints
- Pub Date:
- October 2023
- DOI:
- 10.48550/arXiv.2310.12510
- arXiv:
- arXiv:2310.12510
- Bibcode:
- 2023arXiv231012510T
- Keywords:
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- Mathematics - Number Theory
- E-Print:
- 19 pages, 2 tables and 2 figures