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| Main Author: | |
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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2406.08939 |
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| _version_ | 1866910485327118336 |
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| author | Kwon, Jaesung |
| author_facet | Kwon, Jaesung |
| contents | Let $F$ be a number field, $f$ an algebraic automorphic newform on $\mathrm{GL}(2)$ over $F$, $p$ an odd prime does not divide the class number of $F$ and the level of $f$. We prove that $f$ is determined by its $L$-values twisted by Galois characters $ϕ$ of certain $\mathbb{Z}_p$-extension of $F$. Furthermore, if $F$ is totally real or CM, then under some mild assumption on $f$, the compositum of the Hecke field of $f$ and the cyclotomic field $\mathbb{Q}(ϕ)$ is generated by the algebraic $L$-values of $f$ twisted by Galois characters $ϕ$ of certain $\mathbb{Z}_p$-extension of $F$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_08939 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Generation of cyclotomic Hecke fields by $L$-values of cusp forms on $\mathrm{GL}(2)$ with certain $\mathbb{Z}_p$ twist Kwon, Jaesung Number Theory Let $F$ be a number field, $f$ an algebraic automorphic newform on $\mathrm{GL}(2)$ over $F$, $p$ an odd prime does not divide the class number of $F$ and the level of $f$. We prove that $f$ is determined by its $L$-values twisted by Galois characters $ϕ$ of certain $\mathbb{Z}_p$-extension of $F$. Furthermore, if $F$ is totally real or CM, then under some mild assumption on $f$, the compositum of the Hecke field of $f$ and the cyclotomic field $\mathbb{Q}(ϕ)$ is generated by the algebraic $L$-values of $f$ twisted by Galois characters $ϕ$ of certain $\mathbb{Z}_p$-extension of $F$. |
| title | Generation of cyclotomic Hecke fields by $L$-values of cusp forms on $\mathrm{GL}(2)$ with certain $\mathbb{Z}_p$ twist |
| topic | Number Theory |
| url | https://arxiv.org/abs/2406.08939 |