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| Auteurs principaux: | , , |
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| Format: | Preprint |
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2022
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| Accès en ligne: | https://arxiv.org/abs/2211.00305 |
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| _version_ | 1866917728137248768 |
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| author | Jeong, Keunyoung Kwon, Yeong-Wook Park, Junyeong |
| author_facet | Jeong, Keunyoung Kwon, Yeong-Wook Park, Junyeong |
| contents | In this paper, we show the nonvanishing of some Hecke characters on cyclotomic fields. The main ingredient of this paper is a computation of eigenfunctions and the action of Weil representation at some primes including the primes above $2$. As an application, we show that for each isogeny factor of the Jacobian of the $p$-th Fermat curve where $2$ is a quadratic residue modulo $p$, there are infinitely many twists whose analytic rank is zero. Also, for a certain hyperelliptic curve over the $11$-th cyclotomic field whose Jacobian has complex multiplication, there are infinitely many twists whose analytic rank is zero. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2211_00305 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Nonvanishing of $L$-function of some Hecke characters on cyclotomic fields Jeong, Keunyoung Kwon, Yeong-Wook Park, Junyeong Number Theory 11G40, 11G10, 11F27 In this paper, we show the nonvanishing of some Hecke characters on cyclotomic fields. The main ingredient of this paper is a computation of eigenfunctions and the action of Weil representation at some primes including the primes above $2$. As an application, we show that for each isogeny factor of the Jacobian of the $p$-th Fermat curve where $2$ is a quadratic residue modulo $p$, there are infinitely many twists whose analytic rank is zero. Also, for a certain hyperelliptic curve over the $11$-th cyclotomic field whose Jacobian has complex multiplication, there are infinitely many twists whose analytic rank is zero. |
| title | Nonvanishing of $L$-function of some Hecke characters on cyclotomic fields |
| topic | Number Theory 11G40, 11G10, 11F27 |
| url | https://arxiv.org/abs/2211.00305 |