Mahler measures and $L$-values of elliptic curves over real quadratic fields
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arXiv
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| Format: | Preprint |
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2022
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| _version_ | 1866913221617647616 |
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| author | Tao, Zhengyu Guo, Xuejun Wei, Tao |
| author_facet | Tao, Zhengyu Guo, Xuejun Wei, Tao |
| contents | A famous formula of Rodriguez Villegas shows that the Mahler measures $m(k)$ of $P_k(x,y)=x+1/x+y+1/y+k$ can be written as a Kronecker-Eisenstein series. We prove that the degree of $k$ in Villegas' formula can be bounded by the class numbers of CM points. This fact allows us to systematically derive $28$ new identities linking $m(k)$ to $L$-values of cusp forms. Guided by Beilinson's conjecture, we also prove $5$ formulas that express $L$-values of CM elliptic curves over real quadratic fields to some $2\times 2$ determinants of $m(k)$. This extends a recent work of Guo (the second author of this paper), Ji, Liu, and Qin, in which they dealt with the cases when $k=4\pm 4\sqrt{2}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2209_14717 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Mahler measures and $L$-values of elliptic curves over real quadratic fields Tao, Zhengyu Guo, Xuejun Wei, Tao Number Theory A famous formula of Rodriguez Villegas shows that the Mahler measures $m(k)$ of $P_k(x,y)=x+1/x+y+1/y+k$ can be written as a Kronecker-Eisenstein series. We prove that the degree of $k$ in Villegas' formula can be bounded by the class numbers of CM points. This fact allows us to systematically derive $28$ new identities linking $m(k)$ to $L$-values of cusp forms. Guided by Beilinson's conjecture, we also prove $5$ formulas that express $L$-values of CM elliptic curves over real quadratic fields to some $2\times 2$ determinants of $m(k)$. This extends a recent work of Guo (the second author of this paper), Ji, Liu, and Qin, in which they dealt with the cases when $k=4\pm 4\sqrt{2}$. |
| title | Mahler measures and $L$-values of elliptic curves over real quadratic fields |
| topic | Number Theory |
| url | https://arxiv.org/abs/2209.14717 |