CM points, class numbers, and the Mahler measures of $x^3+y^3+1-kxy$
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| Format: | Preprint |
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2023
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| _version_ | 1866913259557224448 |
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| author | Tao, Zhengyu Guo, Xuejun |
| author_facet | Tao, Zhengyu Guo, Xuejun |
| contents | 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 $\mathbb{Q}(\sqrt{3})$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2310_12510 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | CM points, class numbers, and the Mahler measures of $x^3+y^3+1-kxy$ Tao, Zhengyu Guo, Xuejun Number Theory 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 $\mathbb{Q}(\sqrt{3})$. |
| title | CM points, class numbers, and the Mahler measures of $x^3+y^3+1-kxy$ |
| topic | Number Theory |
| url | https://arxiv.org/abs/2310.12510 |