On $L$-functions of Hecke characters and anticyclotomic towers
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866909419482120192 |
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| author | Jia, Haijun |
| author_facet | Jia, Haijun |
| contents | In this paper, we generalize a work of Rohrlich. Let $K/\mathbb{Q}$ be an imaginary quadratic field and $ϕ$ be a Hecke character of $K$ of infinite type (1,0) whose restriction to $\mathbb{Q}$ is the quadratic character corresponding to $K/\mathbb{Q}$. We consider a class of Hecke characters $χ$, which are anticyclotomic twists of $ϕ$ with ramification in a prescribed finite set of primes. We shall prove the central vanishing order of the Hecke $L$-function $L(s,χ$) attached to each $χ$ is 0 or 1 depending on the root number $W(χ)$ for all but finitely many such $χ$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_05867 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On $L$-functions of Hecke characters and anticyclotomic towers Jia, Haijun Number Theory In this paper, we generalize a work of Rohrlich. Let $K/\mathbb{Q}$ be an imaginary quadratic field and $ϕ$ be a Hecke character of $K$ of infinite type (1,0) whose restriction to $\mathbb{Q}$ is the quadratic character corresponding to $K/\mathbb{Q}$. We consider a class of Hecke characters $χ$, which are anticyclotomic twists of $ϕ$ with ramification in a prescribed finite set of primes. We shall prove the central vanishing order of the Hecke $L$-function $L(s,χ$) attached to each $χ$ is 0 or 1 depending on the root number $W(χ)$ for all but finitely many such $χ$. |
| title | On $L$-functions of Hecke characters and anticyclotomic towers |
| topic | Number Theory |
| url | https://arxiv.org/abs/2412.05867 |