On the coefficients of the Taylor expansion of $L$-functions of elliptic curves
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866917477722619904 |
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| author | Wei, Tong Zhai, Shuai |
| author_facet | Wei, Tong Zhai, Shuai |
| contents | In this paper, we investigate the coefficients of the Taylor expansion of the complex $L$-series of any elliptic curve over $\mathbb{Q}$. We prove that, in the family of quadratic twists by all the discriminants $d$, these coefficients are nonvanishing under GRH when $d$ is sufficiently large. Unconditionally, we obtain a general lower bound for the number of nonvanishing coefficients in the family of quadratic twists, through a series of results from the moments of the central values of the derivatives of quadratic twists of modular $L$-function. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2605_09251 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On the coefficients of the Taylor expansion of $L$-functions of elliptic curves Wei, Tong Zhai, Shuai Number Theory In this paper, we investigate the coefficients of the Taylor expansion of the complex $L$-series of any elliptic curve over $\mathbb{Q}$. We prove that, in the family of quadratic twists by all the discriminants $d$, these coefficients are nonvanishing under GRH when $d$ is sufficiently large. Unconditionally, we obtain a general lower bound for the number of nonvanishing coefficients in the family of quadratic twists, through a series of results from the moments of the central values of the derivatives of quadratic twists of modular $L$-function. |
| title | On the coefficients of the Taylor expansion of $L$-functions of elliptic curves |
| topic | Number Theory |
| url | https://arxiv.org/abs/2605.09251 |